Results of Hypergeometric Function I

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
15.1.E1	${\displaystyle{\displaystyle{{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)}}$ \genhyperF{2}{1}@{a,b}{c}{z} = \hyperF@{a}{b}{c}{z}		hypergeom([a , b], [c], z) = hypergeom([a, b], [c], z)	HypergeometricPFQ[{a , b}, {c}, z] == Hypergeometric2F1[a, b, c, z]	Successful	Successful	-	Failed [42 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.1.E1	${\displaystyle{\displaystyle F\left(a,b;c;z\right)=F\left({a,b\atop c};z\right)}}$ \hyperF@{a}{b}{c}{z} = \hyperF@@{a}{b}{c}{z}		hypergeom([a, b], [c], z) = hypergeom([a, b], [c], z)	Hypergeometric2F1[a, b, c, z] == Hypergeometric2F1[a, b, c, z]	Successful	Successful	-	Successful [Tested: 300]
15.1.E2	${\displaystyle{\displaystyle\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;c;z\right)}}$ \frac{\hyperF@{a}{b}{c}{z}}{\EulerGamma@{c}} = \hyperOlverF@{a}{b}{c}{z}	${\displaystyle{\displaystyle\Re c>0,|z|<1}}$	(hypergeom([a, b], [c], z))/(GAMMA(c)) = hypergeom([a, b], [c], z)/GAMMA(c)	Divide[Hypergeometric2F1[a, b, c, z],Gamma[c]] == Hypergeometric2F1Regularized[a, b, c, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 108]
15.1.E2	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\mathbf{F}\left({a,b\atop c};z\right)}}$ \hyperOlverF@{a}{b}{c}{z} = \hyperOlverF@@{a}{b}{c}{z}	${\displaystyle{\displaystyle\Re c>0,|z|<1}}$	hypergeom([a, b], [c], z)/GAMMA(c) = hypergeom([a, b], [c], z)/GAMMA(c)	Hypergeometric2F1Regularized[a, b, c, z] == Hypergeometric2F1Regularized[a, b, c, z]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 108]
15.1.E2	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};z\right)={{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;z\right)}}$ \hyperOlverF@@{a}{b}{c}{z} = \genhyperOlverF{2}{1}@{a,b}{c}{z}	${\displaystyle{\displaystyle\Re c>0,|z|<1}}$	hypergeom([a, b], [c], z)/GAMMA(c) = hypergeom([a , b], [c], z)	Hypergeometric2F1Regularized[a, b, c, z] == HypergeometricPFQRegularized[{a , b}, {c}, z]	Failure	Successful	Failed [175 / 216] Result: -.2039500354 Test Values: {a = -3/2, b = -3/2, c = -3/2, z = 1/2} Result: .227101342 Test Values: {a = -3/2, b = -3/2, c = 3/2, z = 1/2} ... skip entries to safe data	Successful [Tested: 108]
15.2.E1	${\displaystyle{\displaystyle F\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}}{{\left(c\right)_{s}}s!}z^{s}}}$ \hyperF@{a}{b}{c}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}\Pochhammersym{b}{s}}{\Pochhammersym{c}{s}s!}z^{s}		hypergeom([a, b], [c], z) = sum((pochhammer(a, s)*pochhammer(b, s))/(pochhammer(c, s)*factorial(s))*(z)^(s), s = 0..infinity)	Hypergeometric2F1[a, b, c, z] == Sum[Divide[Pochhammer[a, s]*Pochhammer[b, s],Pochhammer[c, s]*(s)!]*(z)^(s), {s, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Skipped - Because timed out	Successful [Tested: 300]
15.2.E2	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}}{\Gamma\left(c+s\right)s!}z^{s}}}$ \hyperOlverF@{a}{b}{c}{z} = \sum_{s=0}^{\infty}\frac{\Pochhammersym{a}{s}\Pochhammersym{b}{s}}{\EulerGamma@{c+s}s!}z^{s}	${\displaystyle{\displaystyle|z|<1,\Re(c+s)>0}}$	hypergeom([a, b], [c], z)/GAMMA(c) = sum((pochhammer(a, s)*pochhammer(b, s))/(GAMMA(c + s)*factorial(s))*(z)^(s), s = 0..infinity)	Hypergeometric2F1Regularized[a, b, c, z] == Sum[Divide[Pochhammer[a, s]*Pochhammer[b, s],Gamma[c + s]*(s)!]*(z)^(s), {s, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Failed [25 / 216] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, 0.5]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[c, -2], Rule[z, 0.5]} ... skip entries to safe data
15.2.E3	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};x+\mathrm{i}0\right)-\mathbf{F}\left({a,b\atop c};x-\mathrm{i}0\right)=\frac{2\pi\mathrm{i}}{\Gamma\left(a\right)\Gamma\left(b\right)}(x-1)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c-a-b+1};1-x\right)}}$ \hyperOlverF@@{a}{b}{c}{x+\iunit 0}-\hyperOlverF@@{a}{b}{c}{x-\iunit 0} = \frac{2\pi\iunit}{\EulerGamma@{a}\EulerGamma@{b}}(x-1)^{c-a-b}\hyperOlverF@@{c-a}{c-b}{c-a-b+1}{1-x}	${\displaystyle{\displaystyle x>1,\Re a>0,\Re b>0,|(x+\mathrm{i}0)|<1,|(x-\mathrm{i}0)|<1,|(1-x)|<1,\Re(c+s)>0,\Re((c-a-b+1)+s)>0}}$	hypergeom([a, b], [c], x + I*0)/GAMMA(c)- hypergeom([a, b], [c], x - I*0)/GAMMA(c) = (2*Pi*I)/(GAMMA(a)*GAMMA(b))*(x - 1)^(c - a - b)* hypergeom([c - a, c - b], [c - a - b + 1], 1 - x)/GAMMA(c - a - b + 1)	Hypergeometric2F1Regularized[a, b, c, x + I*0]- Hypergeometric2F1Regularized[a, b, c, x - I*0] == Divide[2*Pi*I,Gamma[a]*Gamma[b]]*(x - 1)^(c - a - b)* Hypergeometric2F1Regularized[c - a, c - b, c - a - b + 1, 1 - x]	Failure	Failure	Error	Skip - No test values generated
15.2.E3_5	${\displaystyle{\displaystyle\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}\left(a,b;-n;z\right)}}$ \lim_{c\to-n}\frac{\hyperF@{a}{b}{c}{z}}{\EulerGamma@{c}} = \hyperOlverF@{a}{b}{-n}{z}	${\displaystyle{\displaystyle\Re c>0,|z|<1,\Re((-n)+s)>0}}$	limit((hypergeom([a, b], [c], z))/(GAMMA(c)), c = - n) = hypergeom([a, b], [- n], z)/GAMMA(- n)	Limit[Divide[Hypergeometric2F1[a, b, c, z],Gamma[c]], c -> - n, GenerateConditions->None] == Hypergeometric2F1Regularized[a, b, - n, z]	Failure	Successful	Successful [Tested: 0]	Failed [25 / 36] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 3], Rule[z, 0.5]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[n, 3], Rule[z, 0.5]} ... skip entries to safe data
15.2.E3_5	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1)!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right)}}$ \hyperOlverF@{a}{b}{-n}{z} = \frac{\Pochhammersym{a}{n+1}\Pochhammersym{b}{n+1}}{(n+1)!}z^{n+1}\hyperF@{a+n+1}{b+n+1}{n+2}{z}	${\displaystyle{\displaystyle\Re c>0,|z|<1,\Re((-n)+s)>0}}$	hypergeom([a, b], [- n], z)/GAMMA(- n) = (pochhammer(a, n + 1)*pochhammer(b, n + 1))/(factorial(n + 1))*(z)^(n + 1)* hypergeom([a + n + 1, b + n + 1], [n + 2], z)	Hypergeometric2F1Regularized[a, b, - n, z] == Divide[Pochhammer[a, n + 1]*Pochhammer[b, n + 1],(n + 1)!]*(z)^(n + 1)* Hypergeometric2F1[a + n + 1, b + n + 1, n + 2, z]	Failure	Failure	Failed [25 / 36] Result: Float(undefined)+Float(undefined)*I Test Values: {a = -3/2, b = -3/2, z = 1/2, n = 3} Result: Float(undefined)+Float(undefined)*I Test Values: {a = -3/2, b = 3/2, z = 1/2, n = 3} ... skip entries to safe data	Successful [Tested: 180]
15.2.E4	${\displaystyle{\displaystyle F\left(-m,b;c;z\right)=\sum_{n=0}^{m}\frac{{\left(-m\right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}{n!}}z^{n}}}$ \hyperF@{-m}{b}{c}{z} = \sum_{n=0}^{m}\frac{\Pochhammersym{-m}{n}\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}{n!}}z^{n}		hypergeom([- m, b], [c], z) = sum((pochhammer(- m, n)*pochhammer(b, n))/(pochhammer(c, n)*factorial(n))*(z)^(n), n = 0..m)	Hypergeometric2F1[- m, b, c, z] == Sum[Divide[Pochhammer[- m, n]*Pochhammer[b, n],Pochhammer[c, n]*(n)!]*(z)^(n), {n, 0, m}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 300]
15.2.E4	${\displaystyle{\displaystyle\sum_{n=0}^{m}\frac{{\left(-m\right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}{n!}}z^{n}=\sum_{n=0}^{m}(-1)^{n}\genfrac{(}{)}{0.0pt}{}{m}{n}\frac{{\left(b\right)_{n}}}{{\left(c\right)_{n}}}z^{n}}}$ \sum_{n=0}^{m}\frac{\Pochhammersym{-m}{n}\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}{n!}}z^{n} = \sum_{n=0}^{m}(-1)^{n}\binom{m}{n}\frac{\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}}z^{n}		sum((pochhammer(- m, n)*pochhammer(b, n))/(pochhammer(c, n)*factorial(n))*(z)^(n), n = 0..m) = sum((- 1)^(n)*binomial(m,n)*(pochhammer(b, n))/(pochhammer(c, n))*(z)^(n), n = 0..m)	Sum[Divide[Pochhammer[- m, n]*Pochhammer[b, n],Pochhammer[c, n]*(n)!]*(z)^(n), {n, 0, m}, GenerateConditions->None] == Sum[(- 1)^(n)*Binomial[m,n]*Divide[Pochhammer[b, n],Pochhammer[c, n]]*(z)^(n), {n, 0, m}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 300]
15.2.E5	${\displaystyle{\displaystyle F\left({-m,b\atop-m-\ell};z\right)=\lim_{c\to-m-\ell}\left(\lim_{a\to-m}F\left({a,b\atop c};z\right)\right)}}$ \hyperF@@{-m}{b}{-m-\ell}{z} = \lim_{c\to-m-\ell}\left(\lim_{a\to-m}\hyperF@@{a}{b}{c}{z}\right)		hypergeom([- m, b], [- m - ell], z) = limit(limit(hypergeom([a, b], [c], z), a = - m), c = - m - ell)	Hypergeometric2F1[- m, b, - m - \[ScriptL], z] == Limit[Limit[Hypergeometric2F1[a, b, c, z], a -> - m, GenerateConditions->None], c -> - m - \[ScriptL], GenerateConditions->None]	Failure	Successful	Successful [Tested: 126]	Successful [Tested: 126]
15.2.E6	${\displaystyle{\displaystyle F\left({-m,b\atop-m-\ell};z\right)=\lim_{a\to-m}F\left({a,b\atop a-\ell};z\right)}}$ \hyperF@@{-m}{b}{-m-\ell}{z} = \lim_{a\to-m}\hyperF@@{a}{b}{a-\ell}{z}		hypergeom([- m, b], [- m - ell], z) = limit(hypergeom([a, b], [a - ell], z), a = - m)	Hypergeometric2F1[- m, b, - m - \[ScriptL], z] == Limit[Hypergeometric2F1[a, b, a - \[ScriptL], z], a -> - m, GenerateConditions->None]	Failure	Successful	Successful [Tested: 0]	Successful [Tested: 126]
15.4.E1	${\displaystyle{\displaystyle F\left(1,1;2;z\right)=-z^{-1}\ln\left(1-z\right)}}$ \hyperF@{1}{1}{2}{z} = -z^{-1}\ln@{1-z}		hypergeom([1, 1], [2], z) = - (z)^(- 1)* ln(1 - z)	Hypergeometric2F1[1, 1, 2, z] == - (z)^(- 1)* Log[1 - z]	Successful	Successful	-	Successful [Tested: 7]
15.4.E2	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},1;\tfrac{3}{2};z^{2}\right)=\frac{1}{2z}\ln\left(\frac{1+z}{1-z}\right)}}$ \hyperF@{\tfrac{1}{2}}{1}{\tfrac{3}{2}}{z^{2}} = \frac{1}{2z}\ln@{\frac{1+z}{1-z}}		hypergeom([(1)/(2), 1], [(3)/(2)], (z)^(2)) = (1)/(2*z)*ln((1 + z)/(1 - z))	Hypergeometric2F1[Divide[1,2], 1, Divide[3,2], (z)^(2)] == Divide[1,2*z]*Log[Divide[1 + z,1 - z]]	Failure	Failure	Failed [2 / 7] Result: .1e-9-2.094395103*I Test Values: {z = 3/2} Result: 0.-1.570796327*I Test Values: {z = 2}	Failed [2 / 7] Result: Complex[1.1102230246251565*^-16, -2.0943951023931953] Test Values: {Rule[z, 1.5]} Result: Complex[0.0, -1.5707963267948966] Test Values: {Rule[z, 2]}
15.4.E3	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},1;\tfrac{3}{2};-z^{2}\right)=z^{-1}\operatorname{arctan}z}}$ \hyperF@{\tfrac{1}{2}}{1}{\tfrac{3}{2}}{-z^{2}} = z^{-1}\atan@@{z}		hypergeom([(1)/(2), 1], [(3)/(2)], - (z)^(2)) = (z)^(- 1)* arctan(z)	Hypergeometric2F1[Divide[1,2], 1, Divide[3,2], - (z)^(2)] == (z)^(- 1)* ArcTan[z]	Successful	Successful	-	Successful [Tested: 7]
15.4.E4	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};z^{2}\right)=z^{-1}\operatorname{arcsin}z}}$ \hyperF@{\tfrac{1}{2}}{\tfrac{1}{2}}{\tfrac{3}{2}}{z^{2}} = z^{-1}\asin@@{z}		hypergeom([(1)/(2), (1)/(2)], [(3)/(2)], (z)^(2)) = (z)^(- 1)* arcsin(z)	Hypergeometric2F1[Divide[1,2], Divide[1,2], Divide[3,2], (z)^(2)] == (z)^(- 1)* ArcSin[z]	Successful	Successful	-	Successful [Tested: 7]
15.4.E5	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};-z^{2}\right)=z^{-1}\ln\left(z+\sqrt{1+z^{2}}\right)}}$ \hyperF@{\tfrac{1}{2}}{\tfrac{1}{2}}{\tfrac{3}{2}}{-z^{2}} = z^{-1}\ln@{z+\sqrt{1+z^{2}}}		hypergeom([(1)/(2), (1)/(2)], [(3)/(2)], - (z)^(2)) = (z)^(- 1)* ln(z +sqrt(1 + (z)^(2)))	Hypergeometric2F1[Divide[1,2], Divide[1,2], Divide[3,2], - (z)^(2)] == (z)^(- 1)* Log[z +Sqrt[1 + (z)^(2)]]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
15.4#Ex1	${\displaystyle{\displaystyle F\left(a,b;a;z\right)=(1-z)^{-b}}}$ \hyperF@{a}{b}{a}{z} = (1-z)^{-b}		hypergeom([a, b], [a], z) = (1 - z)^(- b)	Hypergeometric2F1[a, b, a, z] == (1 - z)^(- b)	Successful	Successful	-	Successful [Tested: 252]
15.4#Ex2	${\displaystyle{\displaystyle F\left(a,b;b;z\right)=(1-z)^{-a}}}$ \hyperF@{a}{b}{b}{z} = (1-z)^{-a}		hypergeom([a, b], [b], z) = (1 - z)^(- a)	Hypergeometric2F1[a, b, b, z] == (1 - z)^(- a)	Successful	Successful	-	Successful [Tested: 252]
15.4.E7	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};z^{2}\right)=\tfrac{1}{2}\left((1+z)^{-2a}+(1-z)^{-2a}\right)}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{1}{2}}{z^{2}} = \tfrac{1}{2}\left((1+z)^{-2a}+(1-z)^{-2a}\right)		hypergeom([a, (1)/(2)+ a], [(1)/(2)], (z)^(2)) = (1)/(2)*((1 + z)^(- 2*a)+(1 - z)^(- 2*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[1,2], (z)^(2)] == Divide[1,2]*((1 + z)^(- 2*a)+(1 - z)^(- 2*a))	Successful	Successful	-	Successful [Tested: 42]
15.4.E8	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};-{\tan^{2}}z\right)=(\cos z)^{2a}\cos\left(2az\right)}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{1}{2}}{-\tan^{2}@@{z}} = (\cos@@{z})^{2a}\cos@{2az}		hypergeom([a, (1)/(2)+ a], [(1)/(2)], - (tan(z))^(2)) = (cos(z))^(2*a)* cos(2*a*z)	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[1,2], - (Tan[z])^(2)] == (Cos[z])^(2*a)* Cos[2*a*z]	Failure	Failure	Successful [Tested: 42]	Successful [Tested: 42]
15.4.E9	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};z^{2}\right)=\frac{1}{(2-4a)z}\left((1+z)^{1-2a}-(1-z)^{1-2a}\right)}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{3}{2}}{z^{2}} = \frac{1}{(2-4a)z}\left((1+z)^{1-2a}-(1-z)^{1-2a}\right)		hypergeom([a, (1)/(2)+ a], [(3)/(2)], (z)^(2)) = (1)/((2 - 4*a)*z)*((1 + z)^(1 - 2*a)-(1 - z)^(1 - 2*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[3,2], (z)^(2)] == Divide[1,(2 - 4*a)*z]*((1 + z)^(1 - 2*a)-(1 - z)^(1 - 2*a))	Successful	Successful	-	Failed [7 / 42] Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.4.E10	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};-{\tan^{2}}z\right)=(\cos z)^{2a}\frac{\sin\left((1-2a)z\right)}{(1-2a)\sin z}}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{3}{2}}{-\tan^{2}@@{z}} = (\cos@@{z})^{2a}\frac{\sin@{(1-2a)z}}{(1-2a)\sin@@{z}}		hypergeom([a, (1)/(2)+ a], [(3)/(2)], - (tan(z))^(2)) = (cos(z))^(2*a)*(sin((1 - 2*a)*z))/((1 - 2*a)*sin(z))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[3,2], - (Tan[z])^(2)] == (Cos[z])^(2*a)*Divide[Sin[(1 - 2*a)*z],(1 - 2*a)*Sin[z]]	Failure	Failure	Failed [7 / 42] Result: Float(undefined)+Float(undefined)*I Test Values: {a = 1/2, z = 1/2*3^(1/2)+1/2*I} Result: Float(undefined)+Float(undefined)*I Test Values: {a = 1/2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [7 / 42] Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.4.E11	${\displaystyle{\displaystyle F\left(-a,a;\tfrac{1}{2};-z^{2}\right)=\tfrac{1}{2}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a}+\left(\sqrt{1+z^{2}}-z\right)^{2a}\right)}}$ \hyperF@{-a}{a}{\tfrac{1}{2}}{-z^{2}} = \tfrac{1}{2}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a}+\left(\sqrt{1+z^{2}}-z\right)^{2a}\right)		hypergeom([- a, a], [(1)/(2)], - (z)^(2)) = (1)/(2)*((sqrt(1 + (z)^(2))+ z)^(2*a)+(sqrt(1 + (z)^(2))- z)^(2*a))	Hypergeometric2F1[- a, a, Divide[1,2], - (z)^(2)] == Divide[1,2]*((Sqrt[1 + (z)^(2)]+ z)^(2*a)+(Sqrt[1 + (z)^(2)]- z)^(2*a))	Failure	Failure	Successful [Tested: 42]	Successful [Tested: 42]
15.4.E12	${\displaystyle{\displaystyle F\left(-a,a;\tfrac{1}{2};{\sin^{2}}z\right)=\cos\left(2az\right)}}$ \hyperF@{-a}{a}{\tfrac{1}{2}}{\sin^{2}@@{z}} = \cos@{2az}		hypergeom([- a, a], [(1)/(2)], (sin(z))^(2)) = cos(2*a*z)	Hypergeometric2F1[- a, a, Divide[1,2], (Sin[z])^(2)] == Cos[2*a*z]	Failure	Failure	Failed [4 / 42] Result: -1.920340573 Test Values: {a = -3/2, z = 2} Result: -1.920340573 Test Values: {a = 3/2, z = 2} ... skip entries to safe data	Failed [4 / 42] Result: -1.9203405733007322 Test Values: {Rule[a, -1.5], Rule[z, 2]} Result: -1.9203405733007322 Test Values: {Rule[a, 1.5], Rule[z, 2]} ... skip entries to safe data
15.4.E13	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{1}{2};-z^{2}\right)=\frac{1}{2\sqrt{1+z^{2}}}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a-1}+\left(\sqrt{1+z^{2}}-z\right)^{2a-1}\right)}}$ \hyperF@{a}{1-a}{\tfrac{1}{2}}{-z^{2}} = \frac{1}{2\sqrt{1+z^{2}}}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a-1}+\left(\sqrt{1+z^{2}}-z\right)^{2a-1}\right)		hypergeom([a, 1 - a], [(1)/(2)], - (z)^(2)) = (1)/(2*sqrt(1 + (z)^(2)))*((sqrt(1 + (z)^(2))+ z)^(2*a - 1)+(sqrt(1 + (z)^(2))- z)^(2*a - 1))	Hypergeometric2F1[a, 1 - a, Divide[1,2], - (z)^(2)] == Divide[1,2*Sqrt[1 + (z)^(2)]]*((Sqrt[1 + (z)^(2)]+ z)^(2*a - 1)+(Sqrt[1 + (z)^(2)]- z)^(2*a - 1))	Successful	Failure	-	Successful [Tested: 42]
15.4.E14	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{1}{2};{\sin^{2}}z\right)=\frac{\cos\left((2a-1)z\right)}{\cos z}}}$ \hyperF@{a}{1-a}{\tfrac{1}{2}}{\sin^{2}@@{z}} = \frac{\cos@{(2a-1)z}}{\cos@@{z}}		hypergeom([a, 1 - a], [(1)/(2)], (sin(z))^(2)) = (cos((2*a - 1)*z))/(cos(z))	Hypergeometric2F1[a, 1 - a, Divide[1,2], (Sin[z])^(2)] == Divide[Cos[(2*a - 1)*z],Cos[z]]	Failure	Failure	Failed [4 / 42] Result: -.6992725697 Test Values: {a = -3/2, z = 2} Result: -3.141408577 Test Values: {a = 3/2, z = 2} ... skip entries to safe data	Failed [4 / 42] Result: -0.6992725693452728 Test Values: {Rule[a, -1.5], Rule[z, 2]} Result: -3.1414085772561924 Test Values: {Rule[a, 1.5], Rule[z, 2]} ... skip entries to safe data
15.4.E15	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{3}{2};-z^{2}\right)=\frac{1}{(2-4a)z}\left(\left(\sqrt{1+z^{2}}+z\right)^{1-2a}-\left(\sqrt{1+z^{2}}-z\right)^{1-2a}\right)}}$ \hyperF@{a}{1-a}{\tfrac{3}{2}}{-z^{2}} = \frac{1}{(2-4a)z}\left(\left(\sqrt{1+z^{2}}+z\right)^{1-2a}-\left(\sqrt{1+z^{2}}-z\right)^{1-2a}\right)		hypergeom([a, 1 - a], [(3)/(2)], - (z)^(2)) = (1)/((2 - 4*a)*z)*((sqrt(1 + (z)^(2))+ z)^(1 - 2*a)-(sqrt(1 + (z)^(2))- z)^(1 - 2*a))	Hypergeometric2F1[a, 1 - a, Divide[3,2], - (z)^(2)] == Divide[1,(2 - 4*a)*z]*((Sqrt[1 + (z)^(2)]+ z)^(1 - 2*a)-(Sqrt[1 + (z)^(2)]- z)^(1 - 2*a))	Failure	Failure	Failed [7 / 42] Result: Float(undefined)+Float(undefined)*I Test Values: {a = 1/2, z = 1/2*3^(1/2)+1/2*I} Result: Float(undefined)+Float(undefined)*I Test Values: {a = 1/2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [7 / 42] Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.4.E16	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{3}{2};{\sin^{2}}z\right)=\frac{\sin\left((2a-1)z\right)}{(2a-1)\sin z}}}$ \hyperF@{a}{1-a}{\tfrac{3}{2}}{\sin^{2}@@{z}} = \frac{\sin@{(2a-1)z}}{(2a-1)\sin@@{z}}		hypergeom([a, 1 - a], [(3)/(2)], (sin(z))^(2)) = (sin((2*a - 1)*z))/((2*a - 1)*sin(z))	Hypergeometric2F1[a, 1 - a, Divide[3,2], (Sin[z])^(2)] == Divide[Sin[(2*a - 1)*z],(2*a - 1)*Sin[z]]	Successful	Failure	-	Failed [10 / 42] Result: -0.5440234501032235 Test Values: {Rule[a, -1.5], Rule[z, 2]} Result: 0.8322936730942848 Test Values: {Rule[a, 1.5], Rule[z, 2]} ... skip entries to safe data
15.4.E17	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;1+2a;z\right)=\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{-2a}}}$ \hyperF@{a}{\tfrac{1}{2}+a}{1+2a}{z} = \left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{-2a}		hypergeom([a, (1)/(2)+ a], [1 + 2*a], z) = ((1)/(2)+(1)/(2)*sqrt(1 - z))^(- 2*a)	Hypergeometric2F1[a, Divide[1,2]+ a, 1 + 2*a, z] == (Divide[1,2]+Divide[1,2]*Sqrt[1 - z])^(- 2*a)	Successful	Successful	-	Failed [20 / 42] Result: Complex[-0.02099232729741518, 0.019754284780044207] Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.009306933376070914, 0.00445671804147707] Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.4.E18	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;2a;z\right)=\frac{1}{\sqrt{1-z}}\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{1-2a}}}$ \hyperF@{a}{\tfrac{1}{2}+a}{2a}{z} = \frac{1}{\sqrt{1-z}}\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{1-2a}		hypergeom([a, (1)/(2)+ a], [2*a], z) = (1)/(sqrt(1 - z))*((1)/(2)+(1)/(2)*sqrt(1 - z))^(1 - 2*a)	Hypergeometric2F1[a, Divide[1,2]+ a, 2*a, z] == Divide[1,Sqrt[1 - z]]*(Divide[1,2]+Divide[1,2]*Sqrt[1 - z])^(1 - 2*a)	Successful	Successful	-	Failed [17 / 42] Result: Complex[0.009435141098616318, 0.007866769593881467] Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0016763074528445276, -3.2228425612285116*^-4] Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.4.E19	${\displaystyle{\displaystyle F\left(a+1,b;a;z\right)=\left(1-(1-(\ifrac{b}{a}))z\right)(1-z)^{-1-b}}}$ \hyperF@{a+1}{b}{a}{z} = \left(1-(1-(\ifrac{b}{a}))z\right)(1-z)^{-1-b}		hypergeom([a + 1, b], [a], z) = (1 -(1 -((b)/(a)))*z)*(1 - z)^(- 1 - b)	Hypergeometric2F1[a + 1, b, a, z] == (1 -(1 -(Divide[b,a]))*z)*(1 - z)^(- 1 - b)	Successful	Successful	-	Failed [34 / 252] Result: Complex[-0.041984654594830306, 0.03950856956008836] Test Values: {Rule[a, -2], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.018613866752141828, 0.008913436082954251] Test Values: {Rule[a, -2], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.4.E20	${\displaystyle{\displaystyle F\left(a,b;c;1\right)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}}}$ \hyperF@{a}{b}{c}{1} = \frac{\EulerGamma@{c}\EulerGamma@{c-a-b}}{\EulerGamma@{c-a}\EulerGamma@{c-b}}	${\displaystyle{\displaystyle\Re c>0,\Re(c-a-b)>0,\Re(c-a)>0,\Re(c-b)>0}}$	hypergeom([a, b], [c], 1) = (GAMMA(c)*GAMMA(c - a - b))/(GAMMA(c - a)*GAMMA(c - b))	Hypergeometric2F1[a, b, c, 1] == Divide[Gamma[c]*Gamma[c - a - b],Gamma[c - a]*Gamma[c - b]]	Failure	Failure	Successful [Tested: 47]	Successful [Tested: 47]
15.4.E21	${\displaystyle{\displaystyle\lim_{z\to 1-}\frac{F\left(a,b;a+b;z\right)}{-\ln\left(1-z\right)}=\frac{\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}}}$ \lim_{z\to 1-}\frac{\hyperF@{a}{b}{a+b}{z}}{-\ln@{1-z}} = \frac{\EulerGamma@{a+b}}{\EulerGamma@{a}\EulerGamma@{b}}	${\displaystyle{\displaystyle\Re(a+b)>0,\Re a>0,\Re b>0}}$	limit((hypergeom([a, b], [a + b], z))/(- ln(1 - z)), z = 1, left) = (GAMMA(a + b))/(GAMMA(a)*GAMMA(b))	Limit[Divide[Hypergeometric2F1[a, b, a + b, z],- Log[1 - z]], z -> 1, Direction -> "FromBelow", GenerateConditions->None] == Divide[Gamma[a + b],Gamma[a]*Gamma[b]]	Successful	Successful	-	Successful [Tested: 9]
15.4.E22	${\displaystyle{\displaystyle\lim_{z\to 1-}(1-z)^{a+b-c}\left(F\left(a,b;c;z\right)-\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\right)=\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}}}$ \lim_{z\to 1-}(1-z)^{a+b-c}\left(\hyperF@{a}{b}{c}{z}-\frac{\EulerGamma@{c}\EulerGamma@{c-a-b}}{\EulerGamma@{c-a}\EulerGamma@{c-b}}\right) = \frac{\EulerGamma@{c}\EulerGamma@{a+b-c}}{\EulerGamma@{a}\EulerGamma@{b}}	${\displaystyle{\displaystyle\Re c>0,\Re(c-a-b)>0,\Re(c-a)>0,\Re(c-b)>0,\Re(a+b-c)>0,\Re a>0,\Re b>0}}$	limit((1 - z)^(a + b - c)*(hypergeom([a, b], [c], z)-(GAMMA(c)*GAMMA(c - a - b))/(GAMMA(c - a)*GAMMA(c - b))), z = 1, left) = (GAMMA(c)*GAMMA(a + b - c))/(GAMMA(a)*GAMMA(b))	Limit[(1 - z)^(a + b - c)*(Hypergeometric2F1[a, b, c, z]-Divide[Gamma[c]*Gamma[c - a - b],Gamma[c - a]*Gamma[c - b]]), z -> 1, Direction -> "FromBelow", GenerateConditions->None] == Divide[Gamma[c]*Gamma[a + b - c],Gamma[a]*Gamma[b]]	Failure	Aborted	Error	Skipped - Because timed out
15.4.E23	${\displaystyle{\displaystyle\lim_{z\to 1-}\frac{F\left(a,b;c;z\right)}{(1-z)^{c-a-b}}=\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}}}$ \lim_{z\to 1-}\frac{\hyperF@{a}{b}{c}{z}}{(1-z)^{c-a-b}} = \frac{\EulerGamma@{c}\EulerGamma@{a+b-c}}{\EulerGamma@{a}\EulerGamma@{b}}	${\displaystyle{\displaystyle\Re c>0,\Re(a+b-c)>0,\Re a>0,\Re b>0}}$	limit((hypergeom([a, b], [c], z))/((1 - z)^(c - a - b)), z = 1, left) = (GAMMA(c)*GAMMA(a + b - c))/(GAMMA(a)*GAMMA(b))	Limit[Divide[Hypergeometric2F1[a, b, c, z],(1 - z)^(c - a - b)], z -> 1, Direction -> "FromBelow", GenerateConditions->None] == Divide[Gamma[c]*Gamma[a + b - c],Gamma[a]*Gamma[b]]	Failure	Failure	Manual Skip!	Successful [Tested: 23]
15.4.E24	${\displaystyle{\displaystyle F\left(-n,b;c;1\right)=\frac{{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}}}$ \hyperF@{-n}{b}{c}{1} = \frac{\Pochhammersym{c-b}{n}}{\Pochhammersym{c}{n}}		hypergeom([- n, b], [c], 1) = (pochhammer(c - b, n))/(pochhammer(c, n))	Hypergeometric2F1[- n, b, c, 1] == Divide[Pochhammer[c - b, n],Pochhammer[c, n]]	Failure	Failure	Error	Failed [6 / 36] Result: Indeterminate Test Values: {Rule[b, -1.5], Rule[c, -2], Rule[n, 3]} Result: Indeterminate Test Values: {Rule[b, 1.5], Rule[c, -2], Rule[n, 3]} ... skip entries to safe data
15.4.E25	${\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\Gamma\left(a+n\right)\Gamma\left(b+n\right)}{\Gamma\left(c+n\right)\Gamma\left(d+n\right)}=\frac{\pi^{2}}{\sin\left(\pi a\right)\sin\left(\pi b\right)}\*\frac{\Gamma\left(c+d-a-b-1\right)}{\Gamma\left(c-a\right)\Gamma\left(d-a\right)\Gamma\left(c-b\right)\Gamma\left(d-b\right)}}}$ \sum_{n=-\infty}^{\infty}\frac{\EulerGamma@{a+n}\EulerGamma@{b+n}}{\EulerGamma@{c+n}\EulerGamma@{d+n}} = \frac{\pi^{2}}{\sin@{\pi a}\sin@{\pi b}}\*\frac{\EulerGamma@{c+d-a-b-1}}{\EulerGamma@{c-a}\EulerGamma@{d-a}\EulerGamma@{c-b}\EulerGamma@{d-b}}	${\displaystyle{\displaystyle\Re(a+n)>0,\Re(b+n)>0,\Re(c+n)>0,\Re(d+n)>0,\Re(c+d-a-b-1)>0,\Re(c-a)>0,\Re(d-a)>0,\Re(c-b)>0,\Re(d-b)>0}}$	sum((GAMMA(a + n)*GAMMA(b + n))/(GAMMA(c + n)*GAMMA(d + n)), n = - infinity..infinity) = ((Pi)^(2))/(sin(Pi*a)*sin(Pi*b))*(GAMMA(c + d - a - b - 1))/(GAMMA(c - a)*GAMMA(d - a)*GAMMA(c - b)*GAMMA(d - b))	Sum[Divide[Gamma[a + n]*Gamma[b + n],Gamma[c + n]*Gamma[d + n]], {n, - Infinity, Infinity}, GenerateConditions->None] == Divide[(Pi)^(2),Sin[Pi*a]*Sin[Pi*b]]*Divide[Gamma[c + d - a - b - 1],Gamma[c - a]*Gamma[d - a]*Gamma[c - b]*Gamma[d - b]]	Failure	Aborted	Manual Skip!	Failed [160 / 281] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[c, 1.5], Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[c, 1.5], Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.4.E26	${\displaystyle{\displaystyle F\left(a,b;a-b+1;-1\right)=\frac{\Gamma\left(a-b+1\right)\Gamma\left(\tfrac{1}{2}a+1\right)}{\Gamma\left(a+1\right)\Gamma\left(\tfrac{1}{2}a-b+1\right)}}}$ \hyperF@{a}{b}{a-b+1}{-1} = \frac{\EulerGamma@{a-b+1}\EulerGamma@{\tfrac{1}{2}a+1}}{\EulerGamma@{a+1}\EulerGamma@{\tfrac{1}{2}a-b+1}}	${\displaystyle{\displaystyle\Re(a-b+1)>0,\Re(\tfrac{1}{2}a+1)>0,\Re(a+1)>0,\Re(\tfrac{1}{2}a-b+1)>0}}$	hypergeom([a, b], [a - b + 1], - 1) = (GAMMA(a - b + 1)*GAMMA((1)/(2)*a + 1))/(GAMMA(a + 1)*GAMMA((1)/(2)*a - b + 1))	Hypergeometric2F1[a, b, a - b + 1, - 1] == Divide[Gamma[a - b + 1]*Gamma[Divide[1,2]*a + 1],Gamma[a + 1]*Gamma[Divide[1,2]*a - b + 1]]	Successful	Successful	-	Successful [Tested: 17]
15.4.E27	${\displaystyle{\displaystyle F\left(1,a;a+1;-1\right)=\tfrac{1}{2}a\left(\psi\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)-\psi\left(\tfrac{1}{2}a\right)\right)}}$ \hyperF@{1}{a}{a+1}{-1} = \tfrac{1}{2}a\left(\digamma@{\tfrac{1}{2}a+\tfrac{1}{2}}-\digamma@{\tfrac{1}{2}a}\right)		hypergeom([1, a], [a + 1], - 1) = (1)/(2)*a*(Psi((1)/(2)*a +(1)/(2))- Psi((1)/(2)*a))	Hypergeometric2F1[1, a, a + 1, - 1] == Divide[1,2]*a*(PolyGamma[Divide[1,2]*a +Divide[1,2]]- PolyGamma[Divide[1,2]*a])	Successful	Successful	-	Successful [Tested: 6]
15.4.E28	${\displaystyle{\displaystyle F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2};\tfrac{1}{2}\right)=\sqrt{\pi}\frac{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}}}$ \hyperF@{a}{b}{\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}}{\tfrac{1}{2}} = \sqrt{\pi}\frac{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}b+\tfrac{1}{2}}}	${\displaystyle{\displaystyle\Re(\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}a+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}b+\tfrac{1}{2})>0}}$	hypergeom([a, b], [(1)/(2)*a +(1)/(2)*b +(1)/(2)], (1)/(2)) = sqrt(Pi)*(GAMMA((1)/(2)*a +(1)/(2)*b +(1)/(2)))/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*b +(1)/(2)))	Hypergeometric2F1[a, b, Divide[1,2]*a +Divide[1,2]*b +Divide[1,2], Divide[1,2]] == Sqrt[Pi]*Divide[Gamma[Divide[1,2]*a +Divide[1,2]*b +Divide[1,2]],Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*b +Divide[1,2]]]	Successful	Successful	-	Successful [Tested: 15]
15.4.E29	${\displaystyle{\displaystyle F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+1;\tfrac{1}{2}\right)=\frac{2\sqrt{\pi}}{a-b}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+1\right)\*\left(\frac{1}{\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}-\frac{1}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b\right)}\right)}}$ \hyperF@{a}{b}{\tfrac{1}{2}a+\tfrac{1}{2}b+1}{\tfrac{1}{2}} = \frac{2\sqrt{\pi}}{a-b}\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}b+1}\*\left(\frac{1}{\EulerGamma@{\tfrac{1}{2}a}\EulerGamma@{\tfrac{1}{2}b+\tfrac{1}{2}}}-\frac{1}{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}b}}\right)	${\displaystyle{\displaystyle\Re(\tfrac{1}{2}a+\tfrac{1}{2}b+1)>0,\Re(\tfrac{1}{2}a)>0,\Re(\tfrac{1}{2}b+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}a+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}b)>0}}$	hypergeom([a, b], [(1)/(2)*a +(1)/(2)*b + 1], (1)/(2)) = (2*sqrt(Pi))/(a - b)*GAMMA((1)/(2)*a +(1)/(2)*b + 1)*((1)/(GAMMA((1)/(2)*a)*GAMMA((1)/(2)*b +(1)/(2)))-(1)/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*b)))	Hypergeometric2F1[a, b, Divide[1,2]*a +Divide[1,2]*b + 1, Divide[1,2]] == Divide[2*Sqrt[Pi],a - b]*Gamma[Divide[1,2]*a +Divide[1,2]*b + 1]*(Divide[1,Gamma[Divide[1,2]*a]*Gamma[Divide[1,2]*b +Divide[1,2]]]-Divide[1,Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*b]])	Failure	Failure	Failed [3 / 9] Result: Float(-infinity) Test Values: {a = 3/2, b = 3/2} Result: Float(undefined) Test Values: {a = 1/2, b = 1/2} ... skip entries to safe data	Failed [3 / 9] Result: Indeterminate Test Values: {Rule[a, 1.5], Rule[b, 1.5]} Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[b, 0.5]} ... skip entries to safe data
15.4.E30	${\displaystyle{\displaystyle F\left(a,1-a;b;\tfrac{1}{2}\right)=\frac{2^{1-b}\sqrt{\pi}\Gamma\left(b\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b\right)\Gamma\left(\tfrac{1}{2}b-\tfrac{1}{2}a+\tfrac{1}{2}\right)}}}$ \hyperF@{a}{1-a}{b}{\tfrac{1}{2}} = \frac{2^{1-b}\sqrt{\pi}\EulerGamma@{b}}{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}b}\EulerGamma@{\tfrac{1}{2}b-\tfrac{1}{2}a+\tfrac{1}{2}}}	${\displaystyle{\displaystyle\Re b>0,\Re(\tfrac{1}{2}a+\tfrac{1}{2}b)>0,\Re(\tfrac{1}{2}b-\tfrac{1}{2}a+\tfrac{1}{2})>0}}$	hypergeom([a, 1 - a], [b], (1)/(2)) = ((2)^(1 - b)*sqrt(Pi)*GAMMA(b))/(GAMMA((1)/(2)*a +(1)/(2)*b)*GAMMA((1)/(2)*b -(1)/(2)*a +(1)/(2)))	Hypergeometric2F1[a, 1 - a, b, Divide[1,2]] == Divide[(2)^(1 - b)*Sqrt[Pi]*Gamma[b],Gamma[Divide[1,2]*a +Divide[1,2]*b]*Gamma[Divide[1,2]*b -Divide[1,2]*a +Divide[1,2]]]	Successful	Failure	-	Successful [Tested: 10]
15.4.E31	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2}-2a;-\tfrac{1}{3}\right)=\left(\frac{8}{9}\right)^{-2a}\frac{\Gamma\left(\tfrac{4}{3}\right)\Gamma\left(\tfrac{3}{2}-2a\right)}{\Gamma\left(\tfrac{3}{2}\right)\Gamma\left(\tfrac{4}{3}-2a\right)}}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{3}{2}-2a}{-\tfrac{1}{3}} = \left(\frac{8}{9}\right)^{-2a}\frac{\EulerGamma@{\tfrac{4}{3}}\EulerGamma@{\tfrac{3}{2}-2a}}{\EulerGamma@{\tfrac{3}{2}}\EulerGamma@{\tfrac{4}{3}-2a}}	${\displaystyle{\displaystyle\Re(\tfrac{3}{2}-2a)>0,\Re(\tfrac{4}{3}-2a)>0}}$	hypergeom([a, (1)/(2)+ a], [(3)/(2)- 2*a], -(1)/(3)) = ((8)/(9))^(- 2*a)*(GAMMA((4)/(3))*GAMMA((3)/(2)- 2*a))/(GAMMA((3)/(2))*GAMMA((4)/(3)- 2*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[3,2]- 2*a, -Divide[1,3]] == (Divide[8,9])^(- 2*a)*Divide[Gamma[Divide[4,3]]*Gamma[Divide[3,2]- 2*a],Gamma[Divide[3,2]]*Gamma[Divide[4,3]- 2*a]]	Failure	Failure	Successful [Tested: 4]	Successful [Tested: 4]
15.4.E32	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{5}{6}+\tfrac{2}{3}a;\tfrac{1}{9}\right)=\sqrt{\pi}\left(\frac{3}{4}\right)^{a}\frac{\Gamma\left(\tfrac{5}{6}+\tfrac{2}{3}a\right)}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{3}a\right)\Gamma\left(\tfrac{5}{6}+\tfrac{1}{3}a\right)}}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{5}{6}+\tfrac{2}{3}a}{\tfrac{1}{9}} = \sqrt{\pi}\left(\frac{3}{4}\right)^{a}\frac{\EulerGamma@{\tfrac{5}{6}+\tfrac{2}{3}a}}{\EulerGamma@{\tfrac{1}{2}+\tfrac{1}{3}a}\EulerGamma@{\tfrac{5}{6}+\tfrac{1}{3}a}}	${\displaystyle{\displaystyle\Re(\tfrac{5}{6}+\tfrac{2}{3}a)>0,\Re(\tfrac{1}{2}+\tfrac{1}{3}a)>0,\Re(\tfrac{5}{6}+\tfrac{1}{3}a)>0}}$	hypergeom([a, (1)/(2)+ a], [(5)/(6)+(2)/(3)*a], (1)/(9)) = sqrt(Pi)*((3)/(4))^(a)*(GAMMA((5)/(6)+(2)/(3)*a))/(GAMMA((1)/(2)+(1)/(3)*a)*GAMMA((5)/(6)+(1)/(3)*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[5,6]+Divide[2,3]*a, Divide[1,9]] == Sqrt[Pi]*(Divide[3,4])^(a)*Divide[Gamma[Divide[5,6]+Divide[2,3]*a],Gamma[Divide[1,2]+Divide[1,3]*a]*Gamma[Divide[5,6]+Divide[1,3]*a]]	Failure	Failure	Successful [Tested: 4]	Successful [Tested: 4]
15.4.E33	${\displaystyle{\displaystyle F\left(3a,\tfrac{1}{3}+a;\tfrac{2}{3}+2a;e^{\ifrac{\mathrm{i}\pi}{3}}\right)=\sqrt{\pi}e^{\ifrac{\mathrm{i}\pi a}{2}}\left(\frac{16}{27}\right)^{(3a+1)/6}\frac{\Gamma\left(\frac{5}{6}+a\right)}{\Gamma\left(\frac{2}{3}+a\right)\Gamma\left(\frac{2}{3}\right)}}}$ \hyperF@{3a}{\tfrac{1}{3}+a}{\tfrac{2}{3}+2a}{e^{\ifrac{\iunit\pi}{3}}} = \sqrt{\pi}e^{\ifrac{\iunit\pi a}{2}}\left(\frac{16}{27}\right)^{(3a+1)/6}\frac{\EulerGamma@{\frac{5}{6}+a}}{\EulerGamma@{\frac{2}{3}+a}\EulerGamma@{\frac{2}{3}}}	${\displaystyle{\displaystyle\Re(\frac{5}{6}+a)>0,\Re(\frac{2}{3}+a)>0}}$	hypergeom([3*a, (1)/(3)+ a], [(2)/(3)+ 2*a], exp((I*Pi)/(3))) = sqrt(Pi)*exp((I*Pi*a)/(2))*((16)/(27))^((3*a + 1)/6)*(GAMMA((5)/(6)+ a))/(GAMMA((2)/(3)+ a)*GAMMA((2)/(3)))	Hypergeometric2F1[3*a, Divide[1,3]+ a, Divide[2,3]+ 2*a, Exp[Divide[I*Pi,3]]] == Sqrt[Pi]*Exp[Divide[I*Pi*a,2]]*(Divide[16,27])^((3*a + 1)/6)*Divide[Gamma[Divide[5,6]+ a],Gamma[Divide[2,3]+ a]*Gamma[Divide[2,3]]]	Failure	Failure	Successful [Tested: 4]	Successful [Tested: 4]
15.5.E1	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}F\left(a,b;c;z\right)=\frac{ab}{c}F\left(a+1,b+1;c+1;z\right)}}$ \deriv{}{z}\hyperF@{a}{b}{c}{z} = \frac{ab}{c}\hyperF@{a+1}{b+1}{c+1}{z}		diff(hypergeom([a, b], [c], z), z) = (a*b)/(c)*hypergeom([a + 1, b + 1], [c + 1], z)	D[Hypergeometric2F1[a, b, c, z], z] == Divide[a*b,c]*Hypergeometric2F1[a + 1, b + 1, c + 1, z]	Successful	Successful	-	Successful [Tested: 300]
15.5.E2	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}F\left(a,b;c;z\right)=\frac{{\left(a\right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}}\*F\left(a+n,b+n;c+n;z\right)}}$ \deriv[n]{}{z}\hyperF@{a}{b}{c}{z} = \frac{\Pochhammersym{a}{n}\Pochhammersym{b}{n}}{\Pochhammersym{c}{n}}\*\hyperF@{a+n}{b+n}{c+n}{z}		diff(hypergeom([a, b], [c], z), [z$(n)]) = (pochhammer(a, n)*pochhammer(b, n))/(pochhammer(c, n))* hypergeom([a + n, b + n], [c + n], z)	D[Hypergeometric2F1[a, b, c, z], {z, n}] == Divide[Pochhammer[a, n]*Pochhammer[b, n],Pochhammer[c, n]]* Hypergeometric2F1[a + n, b + n, c + n, z]	Successful	Successful	-	Successful [Tested: 300]
15.5.E3	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}F\left(a,b;c;z\right)\right)={\left(a\right)_{n}}z^{a+n-1}F\left(a+n,b;c;z\right)}}$ \left(z\deriv{}{z}z\right)^{n}\left(z^{a-1}\hyperF@{a}{b}{c}{z}\right) = \Pochhammersym{a}{n}z^{a+n-1}\hyperF@{a+n}{b}{c}{z}		(z*diff(z, z))^(n)*((z)^(a - 1)* hypergeom([a, b], [c], z)) = pochhammer(a, n)*(z)^(a + n - 1)* hypergeom([a + n, b], [c], z)	(z*D[z, z])^(n)*((z)^(a - 1)* Hypergeometric2F1[a, b, c, z]) == Pochhammer[a, n]*(z)^(a + n - 1)* Hypergeometric2F1[a + n, b, c, z]	Failure	Failure	Manual Skip!	Failed [298 / 300] Result: Complex[2.047155237894918, -4.15915132240068] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.9084280791008837, -0.4608118321937779] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.5.E4	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}F\left(a,b;c-n;z\right)}}$ \deriv[n]{}{z}\left(z^{c-1}\hyperF@{a}{b}{c}{z}\right) = \Pochhammersym{c-n}{n}z^{c-n-1}\hyperF@{a}{b}{c-n}{z}		diff((z)^(c - 1)* hypergeom([a, b], [c], z), [z$(n)]) = pochhammer(c - n, n)*(z)^(c - n - 1)* hypergeom([a, b], [c - n], z)	D[(z)^(c - 1)* Hypergeometric2F1[a, b, c, z], {z, n}] == Pochhammer[c - n, n]*(z)^(c - n - 1)* Hypergeometric2F1[a, b, c - n, z]	Failure	Aborted	Skipped - Because timed out	Failed [300 / 300] Result: Plus[Complex[-10.313412337740687, -15.40985641083086], Times[Complex[-2.9282032302755074, -10.928203230275509], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[, -1.5], Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[Plus[-1, Times[-1, ], 1], Plus[Power[, 2], Power[, 3], Times[2, , -1.5], Times[2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, , 1], Times[-1, Power[, 2], 1], Times[-1, -1.5, 1], Times[-1, , -1.5, 1], Times[-1, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, Power[, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, Power[, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, <syntaxhighlight lang=mathematica>Result: Plus[Complex[123.08315470740952, 79.99762770469566], Times[Complex[-31.999999999999993, -32.0], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[, -1.5], Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], []], Times[Plus[-1, Times[-1, ], 2], Plus[Power[, 2], Power[, 3], Times[2, , -1.5], Times[2, Power[, 2], -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[-1, , 2], Times[-1, Power[, 2], 2], Times[-1, -1.5, 2], Times[-1, , -1.5, 2], Times[-1, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, Power[, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, Power[, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, , 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, Power[, 2], 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[1, ]]], Times[-1, Plus[1, ], Plus[, -1.5, Times[-1, 2]], Plus[-2, Times[-4, ], Times[-2, Power[, 2]], Times[-3, -1.5], Times[-2, , -1.5], Times[2, 2], Times[2, , 2], Times[-1.5, 2], Times[3, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[6, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, Power[, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, , 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Plus[, -1.5, Times[-1, 2]], Plus[1, , -1.5, Times[-1, 2]], Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Binomial[Plus[-1, -1.5], 2], Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[2], Times[Binomial[Plus[-1, -1.5], 2], Plus[Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, -1.5, 2, Power[Plus[Power[-1.5, 2], Times[-1, -1.5, 2]], -1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Hypergeometric2F1[Plus[1, -1.5], Plus[1, -1.5], Plus[1, -1.5], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.5.E5	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{c-a-1}(1-z)^{a+b-c}F\left(a,b;c;z\right)\right)={\left(c-a\right)_{n}}z^{c-a+n-1}(1-z)^{a-n+b-c}\*F\left(a-n,b;c;z\right)}}$ \left(z\deriv{}{z}z\right)^{n}\left(z^{c-a-1}(1-z)^{a+b-c}\hyperF@{a}{b}{c}{z}\right) = \Pochhammersym{c-a}{n}z^{c-a+n-1}(1-z)^{a-n+b-c}\*\hyperF@{a-n}{b}{c}{z}		(z*diff(z, z))^(n)*((z)^(c - a - 1)*(1 - z)^(a + b - c)* hypergeom([a, b], [c], z)) = pochhammer(c - a, n)*(z)^(c - a + n - 1)*(1 - z)^(a - n + b - c)* hypergeom([a - n, b], [c], z)	(z*D[z, z])^(n)*((z)^(c - a - 1)*(1 - z)^(a + b - c)* Hypergeometric2F1[a, b, c, z]) == Pochhammer[c - a, n]*(z)^(c - a + n - 1)*(1 - z)^(a - n + b - c)* Hypergeometric2F1[a - n, b, c, z]	Failure	Failure	Skipped - Because timed out	Failed [298 / 300] Result: Complex[0.9999999999999999, -5.551115123125783*^-17] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.4330127018922193, 0.24999999999999992] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.5.E6	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left((1-z)^{a+b-c}F\left(a,b;c;z\right)\right)=\frac{{\left(c-a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}(1-z)^{a+b-c-n}\*F\left(a,b;c+n;z\right)}}$ \deriv[n]{}{z}\left((1-z)^{a+b-c}\hyperF@{a}{b}{c}{z}\right) = \frac{\Pochhammersym{c-a}{n}\Pochhammersym{c-b}{n}}{\Pochhammersym{c}{n}}(1-z)^{a+b-c-n}\*\hyperF@{a}{b}{c+n}{z}		diff((1 - z)^(a + b - c)* hypergeom([a, b], [c], z), [z$(n)]) = (pochhammer(c - a, n)*pochhammer(c - b, n))/(pochhammer(c, n))*(1 - z)^(a + b - c - n)* hypergeom([a, b], [c + n], z)	D[(1 - z)^(a + b - c)* Hypergeometric2F1[a, b, c, z], {z, n}] == Divide[Pochhammer[c - a, n]*Pochhammer[c - b, n],Pochhammer[c, n]]*(1 - z)^(a + b - c - n)* Hypergeometric2F1[a, b, c + n, z]	Failure	Aborted	Skipped - Because timed out	Failed [300 / 300] Result: Plus[Complex[0.0, 0.0], Times[Complex[-1.6799040046341822, -2.8501979384465357], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[, -1.5], Plus[, -1.5], Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], 1], Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], []], Times[-1, Plus[-1, Times[-1, ], 1], Plus[Times[-1, ], Times[-2, Power[, 2]], Times[-2, Power[, 3]], Times[-1, , -1.5], Times[-2, Power[, 2], -1.5], Times[-1, , -1.5], Times[-2, Power[, 2], -1.5], Times[-1, , -1.5, -1.5], Times[-1, -1.5], Times[-1, , -1.5], Times[-1, -1.5, -1.5], Times[-1, , -1.5, -1.5], Times[-1, -1.5, -1.5], Times[-1, , -1.5, -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[, 1], Times[2, Power[, 2], 1], Times[, -1.5, 1], Times[, -1.5, 1], Times[-1.5, -1.5, 1], Times[-1.5, 1], Times[, -1.5, 1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[4, , Times[Rational[1,<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[1.2497428237239117, 10.604878809262228], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Plus[, -1.5], Plus[, -1.5], Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], 2], Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], []], Times[-1, Plus[-1, Times[-1, ], 2], Plus[Times[-1, ], Times[-2, Power[, 2]], Times[-2, Power[, 3]], Times[-1, , -1.5], Times[-2, Power[, 2], -1.5], Times[-1, , -1.5], Times[-2, Power[, 2], -1.5], Times[-1, , -1.5, -1.5], Times[-1, -1.5], Times[-1, , -1.5], Times[-1, -1.5, -1.5], Times[-1, , -1.5, -1.5], Times[-1, -1.5, -1.5], Times[-1, , -1.5, -1.5], Power[-1.5, 2], Times[, Power[-1.5, 2]], Times[, 2], Times[2, Power[, 2], 2], Times[, -1.5, 2], Times[, -1.5, 2], Times[-1.5, -1.5, 2], Times[-1.5, 2], Times[, -1.5, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Times[4, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[5, Power[, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, Power[, 3], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[5, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[4, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Power[-1.5, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[, Power[-1.5, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[5, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[4, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Power[-1.5, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[, Power[-1.5, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, Power[, 2], -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, , 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, Power[, 2], 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, , -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, , -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[-1, Times[-1, ], Times[-1, -1.5], Times[-1, -1.5], -1.5, 2], Plus[Times[-1, ], Times[-1, Power[, 2]], Times[-1, -1.5], Times[-1, , -1.5], Times[, 2], Times[-1.5, 2], Times[5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[7, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, Power[, 2], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[3, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-2, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, , -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-3, , 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, 2, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[2, ]]], Times[Plus[1, ], Plus[2, ], Plus[1, , -1.5, -1.5, Times[-1, -1.5], Times[-1, 2]], Plus[2, , -1.5, -1.5, Times[-1, -1.5], Times[-1, 2]], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Binomial[Plus[-1.5, -1.5, Times[-1, -1.5]], 2], Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[2], Times[Binomial[Plus[-1.5, -1.5, Times[-1, -1.5]], 2], Plus[Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, -1.5, Power[-1.5, -1], Power[Plus[1, -1.5, -1.5, Times[-1, -1.5], Times[-1, 2]], -1], 2, Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Hypergeometric2F1[Plus[1, -1.5], Plus[1, -1.5], Plus[1, -1.5], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]]}]][3.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.5.E7	${\displaystyle{\displaystyle\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)\right)^{n}\left((1-z)^{a-1}F\left(a,b;c;z\right)\right)=(-1)^{n}\frac{{\left(a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}(1-z)^{a+n-1}\*F\left(a+n,b;c+n;z\right)}}$ \left((1-z)\deriv{}{z}(1-z)\right)^{n}\left((1-z)^{a-1}\hyperF@{a}{b}{c}{z}\right) = (-1)^{n}\frac{\Pochhammersym{a}{n}\Pochhammersym{c-b}{n}}{\Pochhammersym{c}{n}}(1-z)^{a+n-1}\*\hyperF@{a+n}{b}{c+n}{z}		((1 - z)*diff(1 - z, z))^(n)*((1 - z)^(a - 1)* hypergeom([a, b], [c], z)) = (- 1)^(n)*(pochhammer(a, n)*pochhammer(c - b, n))/(pochhammer(c, n))*(1 - z)^(a + n - 1)* hypergeom([a + n, b], [c + n], z)	((1 - z)*D[1 - z, z])^(n)*((1 - z)^(a - 1)* Hypergeometric2F1[a, b, c, z]) == (- 1)^(n)*Divide[Pochhammer[a, n]*Pochhammer[c - b, n],Pochhammer[c, n]]*(1 - z)^(a + n - 1)* Hypergeometric2F1[a + n, b, c + n, z]	Failure	Failure	Skipped - Because timed out	Failed [300 / 300] Result: Complex[-0.9999999999999999, 5.551115123125783*^-17] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.5669872981077805, -0.24999999999999994] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.5.E8	${\displaystyle{\displaystyle\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)\right)^{n}\left(z^{c-1}(1-z)^{b-c}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}(1-z)^{b-c+n}\*F\left(a-n,b;c-n;z\right)}}$ \left((1-z)\deriv{}{z}(1-z)\right)^{n}\left(z^{c-1}(1-z)^{b-c}\hyperF@{a}{b}{c}{z}\right) = \Pochhammersym{c-n}{n}z^{c-n-1}(1-z)^{b-c+n}\*\hyperF@{a-n}{b}{c-n}{z}		((1 - z)*diff(1 - z, z))^(n)*((z)^(c - 1)*(1 - z)^(b - c)* hypergeom([a, b], [c], z)) = pochhammer(c - n, n)*(z)^(c - n - 1)*(1 - z)^(b - c + n)* hypergeom([a - n, b], [c - n], z)	((1 - z)*D[1 - z, z])^(n)*((z)^(c - 1)*(1 - z)^(b - c)* Hypergeometric2F1[a, b, c, z]) == Pochhammer[c - n, n]*(z)^(c - n - 1)*(1 - z)^(b - c + n)* Hypergeometric2F1[a - n, b, c - n, z]	Failure	Failure	Skipped - Because timed out	Failed [299 / 300] Result: Complex[-7.039508221073909, -1.0669744439111815] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[28.125871703124346, -23.36453828137185] Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.5.E9	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}(1-z)^{a+b-c}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}(1-z)^{a+b-c-n}\*F\left(a-n,b-n;c-n;z\right)}}$ \deriv[n]{}{z}\left(z^{c-1}(1-z)^{a+b-c}\hyperF@{a}{b}{c}{z}\right) = \Pochhammersym{c-n}{n}z^{c-n-1}(1-z)^{a+b-c-n}\*\hyperF@{a-n}{b-n}{c-n}{z}		diff((z)^(c - 1)*(1 - z)^(a + b - c)* hypergeom([a, b], [c], z), [z$(n)]) = pochhammer(c - n, n)*(z)^(c - n - 1)*(1 - z)^(a + b - c - n)* hypergeom([a - n, b - n], [c - n], z)	D[(z)^(c - 1)*(1 - z)^(a + b - c)* Hypergeometric2F1[a, b, c, z], {z, n}] == Pochhammer[c - n, n]*(z)^(c - n - 1)*(1 - z)^(a + b - c - n)* Hypergeometric2F1[a - n, b - n, c - n, z]	Failure	Aborted	Skipped - Because timed out	Failed [300 / 300] Result: Plus[Complex[-7.320508075688771, -27.32050807568877], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[-1, Times[-1, ], -1.5], Plus[-1, Times[-1, ], -1.5], []], Times[Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[3, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[Plus[1, Times[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Plus[-1.5, -1.5, Times[-1, -1.5]]], Power[Tim<syntaxhighlight lang=mathematica>Result: Plus[Complex[139.99999999999997, 139.99999999999997], Times[2.0, DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[-1, Times[-1, ], -1.5], Plus[-1, Times[-1, ], -1.5], []], Times[Plus[1, ], Plus[-2, Times[-1, ], -1.5, Times[3, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[2, , Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], [Plus[1, ]]], Times[Plus[1, ], Plus[2, ], Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], [Plus[2, ]]]], 0], Equal[[0], Times[Power[Plus[1, Times[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Plus[-1.5, -1.5, Times[-1, -1.5]]], Power[Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-1, -1.5]], Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[Plus[1, Times[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Plus[-1.5, -1.5, Times[-1, -1.5]]], Power[Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Plus[-2, -1.5]], Plus[Times[Power[Plus[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], -1], Plus[1, Times[-1, -1.5], Times[-1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Hypergeometric2F1[-1.5, -1.5, -1.5, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Times[-1.5, -1.5, Power[-1.5, -1], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Hypergeometric2F1[Plus[1, -1.5], Plus[1, -1.5], Plus[1, -1.5], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[n, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.5.E10	${\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=z^{n}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}z^{n}}}$ \left(z\deriv{}{z}z\right)^{n} = z^{n}\deriv[n]{}{z}z^{n}		(z*diff(z, z))^(n) = (z)^(n)* diff((z)^(n), [z$(n)])	(z*D[z, z])^(n) == (z)^(n)* D[(z)^(n), {z, n}]	Failure	Failure	Failed [7 / 7] Result: -.1616869430e-8-5.000000005*I Test Values: {z = 1/2*3^(1/2)+1/2*I, n = 3} Result: -5.000000005+.1616869430e-8*I Test Values: {z = -1/2+1/2*I*3^(1/2), n = 3} ... skip entries to safe data	Failed [7 / 7] Result: Complex[0.0, -0.625] Test Values: {Rule[n, 3], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: -0.625 Test Values: {Rule[n, 3], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.5.E11	${\displaystyle{\displaystyle(c-a)F\left(a-1,b;c;z\right)+\left(2a-c+(b-a)z\right)F\left(a,b;c;z\right)+a(z-1)F\left(a+1,b;c;z\right)=0}}$ (c-a)\hyperF@{a-1}{b}{c}{z}+\left(2a-c+(b-a)z\right)\hyperF@{a}{b}{c}{z}+a(z-1)\hyperF@{a+1}{b}{c}{z} = 0		(c - a)*hypergeom([a - 1, b], [c], z)+(2*a - c +(b - a)*z)*hypergeom([a, b], [c], z)+ a*(z - 1)*hypergeom([a + 1, b], [c], z) = 0	(c - a)*Hypergeometric2F1[a - 1, b, c, z]+(2*a - c +(b - a)*z)*Hypergeometric2F1[a, b, c, z]+ a*(z - 1)*Hypergeometric2F1[a + 1, b, c, z] == 0	Successful	Successful	-	Failed [42 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E12	${\displaystyle{\displaystyle(b-a)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-bF\left(a,b+1;c;z\right)=0}}$ (b-a)\hyperF@{a}{b}{c}{z}+a\hyperF@{a+1}{b}{c}{z}-b\hyperF@{a}{b+1}{c}{z} = 0		(b - a)*hypergeom([a, b], [c], z)+ a*hypergeom([a + 1, b], [c], z)- b*hypergeom([a, b + 1], [c], z) = 0	(b - a)*Hypergeometric2F1[a, b, c, z]+ a*Hypergeometric2F1[a + 1, b, c, z]- b*Hypergeometric2F1[a, b + 1, c, z] == 0	Successful	Successful	-	Failed [42 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E13	${\displaystyle{\displaystyle(c-a-b)F\left(a,b;c;z\right)+a(1-z)F\left(a+1,b;c;z\right)-(c-b)F\left(a,b-1;c;z\right)=0}}$ (c-a-b)\hyperF@{a}{b}{c}{z}+a(1-z)\hyperF@{a+1}{b}{c}{z}-(c-b)\hyperF@{a}{b-1}{c}{z} = 0		(c - a - b)*hypergeom([a, b], [c], z)+ a*(1 - z)*hypergeom([a + 1, b], [c], z)-(c - b)*hypergeom([a, b - 1], [c], z) = 0	(c - a - b)*Hypergeometric2F1[a, b, c, z]+ a*(1 - z)*Hypergeometric2F1[a + 1, b, c, z]-(c - b)*Hypergeometric2F1[a, b - 1, c, z] == 0	Successful	Successful	-	Failed [49 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E14	${\displaystyle{\displaystyle c\left(a+(b-c)z\right)F\left(a,b;c;z\right)-ac(1-z)F\left(a+1,b;c;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)=0}}$ c\left(a+(b-c)z\right)\hyperF@{a}{b}{c}{z}-ac(1-z)\hyperF@{a+1}{b}{c}{z}+(c-a)(c-b)z\hyperF@{a}{b}{c+1}{z} = 0		c*(a +(b - c)*z)*hypergeom([a, b], [c], z)- a*c*(1 - z)*hypergeom([a + 1, b], [c], z)+(c - a)*(c - b)*z*hypergeom([a, b], [c + 1], z) = 0	c*(a +(b - c)*z)*Hypergeometric2F1[a, b, c, z]- a*c*(1 - z)*Hypergeometric2F1[a + 1, b, c, z]+(c - a)*(c - b)*z*Hypergeometric2F1[a, b, c + 1, z] == 0	Successful	Successful	-	Failed [49 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E15	${\displaystyle{\displaystyle(c-a-1)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-(c-1)F\left(a,b;c-1;z\right)=0}}$ (c-a-1)\hyperF@{a}{b}{c}{z}+a\hyperF@{a+1}{b}{c}{z}-(c-1)\hyperF@{a}{b}{c-1}{z} = 0		(c - a - 1)*hypergeom([a, b], [c], z)+ a*hypergeom([a + 1, b], [c], z)-(c - 1)*hypergeom([a, b], [c - 1], z) = 0	(c - a - 1)*Hypergeometric2F1[a, b, c, z]+ a*Hypergeometric2F1[a + 1, b, c, z]-(c - 1)*Hypergeometric2F1[a, b, c - 1, z] == 0	Successful	Successful	-	Failed [42 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E16	${\displaystyle{\displaystyle c(1-z)F\left(a,b;c;z\right)-cF\left(a-1,b;c;z\right)+(c-b)zF\left(a,b;c+1;z\right)=0}}$ c(1-z)\hyperF@{a}{b}{c}{z}-c\hyperF@{a-1}{b}{c}{z}+(c-b)z\hyperF@{a}{b}{c+1}{z} = 0		c*(1 - z)*hypergeom([a, b], [c], z)- c*hypergeom([a - 1, b], [c], z)+(c - b)*z*hypergeom([a, b], [c + 1], z) = 0	c*(1 - z)*Hypergeometric2F1[a, b, c, z]- c*Hypergeometric2F1[a - 1, b, c, z]+(c - b)*z*Hypergeometric2F1[a, b, c + 1, z] == 0	Successful	Successful	-	Failed [49 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E17	${\displaystyle{\displaystyle\left(a-1+(b+1-c)z\right)F\left(a,b;c;z\right)+(c-a)F\left(a-1,b;c;z\right)-(c-1)(1-z)F\left(a,b;c-1;z\right)=0}}$ \left(a-1+(b+1-c)z\right)\hyperF@{a}{b}{c}{z}+(c-a)\hyperF@{a-1}{b}{c}{z}-(c-1)(1-z)\hyperF@{a}{b}{c-1}{z} = 0		(a - 1 +(b + 1 - c)*z)*hypergeom([a, b], [c], z)+(c - a)*hypergeom([a - 1, b], [c], z)-(c - 1)*(1 - z)*hypergeom([a, b], [c - 1], z) = 0	(a - 1 +(b + 1 - c)*z)*Hypergeometric2F1[a, b, c, z]+(c - a)*Hypergeometric2F1[a - 1, b, c, z]-(c - 1)*(1 - z)*Hypergeometric2F1[a, b, c - 1, z] == 0	Successful	Successful	-	Failed [42 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E18	${\displaystyle{\displaystyle c(c-1)(z-1)F\left(a,b;c-1;z\right)+{c\left(c-1-(2c-a-b-1)z\right)}F\left(a,b;c;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)=0}}$ c(c-1)(z-1)\hyperF@{a}{b}{c-1}{z}+{c\left(c-1-(2c-a-b-1)z\right)}\hyperF@{a}{b}{c}{z}+(c-a)(c-b)z\hyperF@{a}{b}{c+1}{z} = 0		c*(c - 1)*(z - 1)*hypergeom([a, b], [c - 1], z)+c*(c - 1 -(2*c - a - b - 1)*z)*hypergeom([a, b], [c], z)+(c - a)*(c - b)*z*hypergeom([a, b], [c + 1], z) = 0	c*(c - 1)*(z - 1)*Hypergeometric2F1[a, b, c - 1, z]+c*(c - 1 -(2*c - a - b - 1)*z)*Hypergeometric2F1[a, b, c, z]+(c - a)*(c - b)*z*Hypergeometric2F1[a, b, c + 1, z] == 0	Successful	Successful	-	Failed [49 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E19	${\displaystyle{\displaystyle{z(1-z)(a+1)(b+1)}F\left(a+2,b+2;c+2;z\right)+{(c-(a+b+1)z)(c+1)}F\left(a+1,b+1;c+1;z\right)-{c(c+1)}F\left(a,b;c;z\right)=0}}$ {z(1-z)(a+1)(b+1)}\hyperF@{a+2}{b+2}{c+2}{z}+{(c-(a+b+1)z)(c+1)}\hyperF@{a+1}{b+1}{c+1}{z}-{c(c+1)}\hyperF@{a}{b}{c}{z} = 0		z*(1 - z)*(a + 1)*(b + 1)*hypergeom([a + 2, b + 2], [c + 2], z)+(c -(a + b + 1)*z)*(c + 1)*hypergeom([a + 1, b + 1], [c + 1], z)-c*(c + 1)*hypergeom([a, b], [c], z) = 0	z*(1 - z)*(a + 1)*(b + 1)*Hypergeometric2F1[a + 2, b + 2, c + 2, z]+(c -(a + b + 1)*z)*(c + 1)*Hypergeometric2F1[a + 1, b + 1, c + 1, z]-c*(c + 1)*Hypergeometric2F1[a, b, c, z] == 0	Successful	Successful	-	Failed [42 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E20	${\displaystyle{\displaystyle z(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)F\left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;z\right)}}$ z(1-z)\left(\ideriv{\hyperF@{a}{b}{c}{z}}{z}\right) = (c-a)\hyperF@{a-1}{b}{c}{z}+(a-c+bz)\hyperF@{a}{b}{c}{z}		z*(1 - z)*(diff(hypergeom([a, b], [c], z), z)) = (c - a)*hypergeom([a - 1, b], [c], z)+(a - c + b*z)*hypergeom([a, b], [c], z)	z*(1 - z)*(D[Hypergeometric2F1[a, b, c, z], z]) == (c - a)*Hypergeometric2F1[a - 1, b, c, z]+(a - c + b*z)*Hypergeometric2F1[a, b, c, z]	Successful	Successful	-	Failed [42 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E20	${\displaystyle{\displaystyle(c-a)F\left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;z\right)=(c-b)F\left(a,b-1;c;z\right)+(b-c+az)F\left(a,b;c;z\right)}}$ (c-a)\hyperF@{a-1}{b}{c}{z}+(a-c+bz)\hyperF@{a}{b}{c}{z} = (c-b)\hyperF@{a}{b-1}{c}{z}+(b-c+az)\hyperF@{a}{b}{c}{z}		(c - a)*hypergeom([a - 1, b], [c], z)+(a - c + b*z)*hypergeom([a, b], [c], z) = (c - b)*hypergeom([a, b - 1], [c], z)+(b - c + a*z)*hypergeom([a, b], [c], z)	(c - a)*Hypergeometric2F1[a - 1, b, c, z]+(a - c + b*z)*Hypergeometric2F1[a, b, c, z] == (c - b)*Hypergeometric2F1[a, b - 1, c, z]+(b - c + a*z)*Hypergeometric2F1[a, b, c, z]	Successful	Successful	-	Failed [49 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.5.E21	${\displaystyle{\displaystyle c(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)(c-b)F\left(a,b;c+1;z\right)+c(a+b-c)F\left(a,b;c;z\right)}}$ c(1-z)\left(\ideriv{\hyperF@{a}{b}{c}{z}}{z}\right) = (c-a)(c-b)\hyperF@{a}{b}{c+1}{z}+c(a+b-c)\hyperF@{a}{b}{c}{z}		c*(1 - z)*(diff(hypergeom([a, b], [c], z), z)) = (c - a)*(c - b)*hypergeom([a, b], [c + 1], z)+ c*(a + b - c)*hypergeom([a, b], [c], z)	c*(1 - z)*(D[Hypergeometric2F1[a, b, c, z], z]) == (c - a)*(c - b)*Hypergeometric2F1[a, b, c + 1, z]+ c*(a + b - c)*Hypergeometric2F1[a, b, c, z]	Successful	Successful	-	Failed [49 / 300] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.6.E1	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b\right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t}}$ \hyperOlverF@{a}{b}{c}{z} = \frac{1}{\EulerGamma@{b}\EulerGamma@{c-b}}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re c>\Re b,\Re b>0,\Re(c-b)>0,|z|<1,\Re(c+s)>0}}$	hypergeom([a, b], [c], z)/GAMMA(c) = (1)/(GAMMA(b)*GAMMA(c - b))*int(((t)^(b - 1)*(1 - t)^(c - b - 1))/((1 - z*t)^(a)), t = 0..1)	Hypergeometric2F1Regularized[a, b, c, z] == Divide[1,Gamma[b]*Gamma[c - b]]*Integrate[Divide[(t)^(b - 1)*(1 - t)^(c - b - 1),(1 - z*t)^(a)], {t, 0, 1}, GenerateConditions->None]	Failure	Successful	Failed [18 / 18] Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = 3/2, c = 2, z = 1/2} Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = 1/2, c = 3/2, z = 1/2} ... skip entries to safe data	Successful [Tested: 18]
15.6.E2	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}\Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t}}$ \hyperOlverF@{a}{b}{c}{z} = \frac{\EulerGamma@{1+b-c}}{2\pi\iunit\EulerGamma@{b}}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re b>0,\Re(1+b-c)>0,|z|<1,\Re(c+s)>0}}$	hypergeom([a, b], [c], z)/GAMMA(c) = (GAMMA(1 + b - c))/(2*Pi*I*GAMMA(b))*int(((t)^(b - 1)*(t - 1)^(c - b - 1))/((1 - z*t)^(a)), t = 0..(1 +))	Hypergeometric2F1Regularized[a, b, c, z] == Divide[Gamma[1 + b - c],2*Pi*I*Gamma[b]]*Integrate[Divide[(t)^(b - 1)*(t - 1)^(c - b - 1),(1 - z*t)^(a)], {t, 0, (1 +)}, GenerateConditions->None]	Error	Failure	-	Error
15.6.E3	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=e^{-b\pi\mathrm{i}}\frac{\Gamma\left(1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{\infty}^{(0+)}\frac{t^{b-1}(t+1)^{a-c}}{(t-zt+1)^{a}}\mathrm{d}t}}$ \hyperOlverF@{a}{b}{c}{z} = e^{-b\pi\iunit}\frac{\EulerGamma@{1-b}}{2\pi\iunit\EulerGamma@{c-b}}\int_{\infty}^{(0+)}\frac{t^{b-1}(t+1)^{a-c}}{(t-zt+1)^{a}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re\left(c-b\right)>0,\Re(1-b)>0,\Re(c-b)>0,|z|<1,\Re(c+s)>0}}$	hypergeom([a, b], [c], z)/GAMMA(c) = exp(- b*Pi*I)*(GAMMA(1 - b))/(2*Pi*I*GAMMA(c - b))*int(((t)^(b - 1)*(t + 1)^(a - c))/((t - z*t + 1)^(a)), t = infinity..(0 +))	Hypergeometric2F1Regularized[a, b, c, z] == Exp[- b*Pi*I]*Divide[Gamma[1 - b],2*Pi*I*Gamma[c - b]]*Integrate[Divide[(t)^(b - 1)*(t + 1)^(a - c),(t - z*t + 1)^(a)], {t, Infinity, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
15.6.E4	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=e^{-b\pi\mathrm{i}}\frac{\Gamma\left(1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{1}^{(0+)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t}}$ \hyperOlverF@{a}{b}{c}{z} = e^{-b\pi\iunit}\frac{\EulerGamma@{1-b}}{2\pi\iunit\EulerGamma@{c-b}}\int_{1}^{(0+)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re\left(c-b\right)>0,\Re(1-b)>0,\Re(c-b)>0,|z|<1,\Re(c+s)>0}}$	hypergeom([a, b], [c], z)/GAMMA(c) = exp(- b*Pi*I)*(GAMMA(1 - b))/(2*Pi*I*GAMMA(c - b))*int(((t)^(b - 1)*(1 - t)^(c - b - 1))/((1 - z*t)^(a)), t = 1..(0 +))	Hypergeometric2F1Regularized[a, b, c, z] == Exp[- b*Pi*I]*Divide[Gamma[1 - b],2*Pi*I*Gamma[c - b]]*Integrate[Divide[(t)^(b - 1)*(1 - t)^(c - b - 1),(1 - z*t)^(a)], {t, 1, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
15.6.E5	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=e^{-c\pi\mathrm{i}}\Gamma\left(1-b\right)\Gamma\left(1+b-c\right)\*\frac{1}{4\pi^{2}}\int_{A}^{(0+,1+,0-,1-)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t}}$ \hyperOlverF@{a}{b}{c}{z} = e^{-c\pi\iunit}\EulerGamma@{1-b}\EulerGamma@{1+b-c}\*\frac{1}{4\pi^{2}}\int_{A}^{(0+,1+,0-,1-)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re(1-b)>0,\Re(1+b-c)>0,|z|<1,\Re(c+s)>0}}$	hypergeom([a, b], [c], z)/GAMMA(c) = exp(- c*Pi*I)*GAMMA(1 - b)*GAMMA(1 + b - c)*(1)/(4*(Pi)^(2))*int(((t)^(b - 1)*(1 - t)^(c - b - 1))/((1 - z*t)^(a)), t = A..(0 + , 1 + , 0 - , 1 -))	Hypergeometric2F1Regularized[a, b, c, z] == Exp[- c*Pi*I]*Gamma[1 - b]*Gamma[1 + b - c]*Divide[1,4*(Pi)^(2)]*Integrate[Divide[(t)^(b - 1)*(1 - t)^(c - b - 1),(1 - z*t)^(a)], {t, A, (0 + , 1 + , 0 - , 1 -)}, GenerateConditions->None]	Error	Failure	-	Error
15.6.E6	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)\Gamma\left(b\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma\left(b+t\right)\Gamma\left(-t\right)}{\Gamma\left(c+t\right)}(-z)^{t}\mathrm{d}t}}$ \hyperOlverF@{a}{b}{c}{z} = \frac{1}{2\pi\iunit\EulerGamma@{a}\EulerGamma@{b}}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+t}\EulerGamma@{b+t}\EulerGamma@{-t}}{\EulerGamma@{c+t}}(-z)^{t}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\pi,\Re a>0,\Re b>0,\Re(a+t)>0,\Re(b+t)>0,\Re(-t)>0,\Re(c+t)>0,|z|<1,\Re(c+s)>0}}$	hypergeom([a, b], [c], z)/GAMMA(c) = (1)/(2*Pi*I*GAMMA(a)*GAMMA(b))*int((GAMMA(a + t)*GAMMA(b + t)*GAMMA(- t))/(GAMMA(c + t))*(- z)^(t), t = - I*infinity..I*infinity)	Hypergeometric2F1Regularized[a, b, c, z] == Divide[1,2*Pi*I*Gamma[a]*Gamma[b]]*Integrate[Divide[Gamma[a + t]*Gamma[b + t]*Gamma[- t],Gamma[c + t]]*(- z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]	Failure	Aborted	Manual Skip!	Skipped - Because timed out
15.6.E7	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\Gamma\left(a+t\right)\Gamma\left(b+t\right)\Gamma\left(c-a-b-t\right)\Gamma\left(-t\right)(1-z)^{t}\mathrm{d}t}}$ \hyperOlverF@{a}{b}{c}{z} = \frac{1}{2\pi\iunit\EulerGamma@{a}\EulerGamma@{b}\EulerGamma@{c-a}\EulerGamma@{c-b}}\int_{-\iunit\infty}^{\iunit\infty}\EulerGamma@{a+t}\EulerGamma@{b+t}\EulerGamma@{c-a-b-t}\EulerGamma@{-t}(1-z)^{t}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re(a+t)>0,\Re(b+t)>0,\Re(c-a-b-t)>0,\Re(-t)>0,\Re a>0,\Re b>0,\Re(c-a)>0,\Re(c-b)>0,|z|<1,\Re(c+s)>0}}$	hypergeom([a, b], [c], z)/GAMMA(c) = (1)/(2*Pi*I*GAMMA(a)*GAMMA(b)*GAMMA(c - a)*GAMMA(c - b))*int(GAMMA(a + t)*GAMMA(b + t)*GAMMA(c - a - b - t)*GAMMA(- t)*(1 - z)^(t), t = - I*infinity..I*infinity)	Hypergeometric2F1Regularized[a, b, c, z] == Divide[1,2*Pi*I*Gamma[a]*Gamma[b]*Gamma[c - a]*Gamma[c - b]]*Integrate[Gamma[a + t]*Gamma[b + t]*Gamma[c - a - b - t]*Gamma[- t]*(1 - z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
15.6.E8	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}\mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\mathrm{d}t}}$ \hyperOlverF@{a}{b}{c}{z} = \frac{1}{\EulerGamma@{c-d}}\int_{0}^{1}\hyperOlverF@{a}{b}{d}{zt}t^{d-1}(1-t)^{c-d-1}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re c>\Re d,\Re d>0,\Re(c-d)>0,|z|<1,|(zt)|<1,\Re(c+s)>0,\Re(d+s)>0}}$	hypergeom([a, b], [c], z)/GAMMA(c) = (1)/(GAMMA(c - d))*int(hypergeom([a, b], [d], z*t)/GAMMA(d)*(t)^(d - 1)*(1 - t)^(c - d - 1), t = 0..1)	Hypergeometric2F1Regularized[a, b, c, z] == Divide[1,Gamma[c - d]]*Integrate[Hypergeometric2F1Regularized[a, b, d, z*t]*(t)^(d - 1)*(1 - t)^(c - d - 1), {t, 0, 1}, GenerateConditions->None]	Failure	Successful	Failed [252 / 252] Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = -3/2, c = 3/2, d = 1/2*3^(1/2)+1/2*I, z = 1/2} Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = -3/2, c = 3/2, d = 1/2-1/2*I*3^(1/2), z = 1/2} ... skip entries to safe data	Successful [Tested: 252]
15.6.E9	${\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\int_{0}^{1}\frac{t^{d-1}(1-t)^{c-d-1}}{(1-zt)^{a+b-\lambda}}\mathbf{F}\left({\lambda-a,\lambda-b\atop d};zt\right)\mathbf{F}\left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)z}{1-zt}\right)\mathrm{d}t}}$ \hyperOlverF@{a}{b}{c}{z} = \int_{0}^{1}\frac{t^{d-1}(1-t)^{c-d-1}}{(1-zt)^{a+b-\lambda}}\hyperOlverF@@{\lambda-a}{\lambda-b}{d}{zt}\hyperOlverF@@{a+b-\lambda}{\lambda-d}{c-d}{\frac{(1-t)z}{1-zt}}\diff{t}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re c>\Re d,\Re d>0}}$	hypergeom([a, b], [c], z)/GAMMA(c) = int(((t)^(d - 1)*(1 - t)^(c - d - 1))/((1 - z*t)^(a + b - lambda))*hypergeom([lambda - a, lambda - b], [d], z*t)/GAMMA(d)*hypergeom([a + b - lambda, lambda - d], [c - d], ((1 - t)*z)/(1 - z*t))/GAMMA(c - d), t = 0..1)	Hypergeometric2F1Regularized[a, b, c, z] == Integrate[Divide[(t)^(d - 1)*(1 - t)^(c - d - 1),(1 - z*t)^(a + b - \[Lambda])]*Hypergeometric2F1Regularized[\[Lambda]- a, \[Lambda]- b, d, z*t]*Hypergeometric2F1Regularized[a + b - \[Lambda], \[Lambda]- d, c - d, Divide[(1 - t)*z,1 - z*t]], {t, 0, 1}, GenerateConditions->None]	Failure	Aborted	Skipped - Because timed out	Skipped - Because timed out
15.7#Ex1	${\displaystyle{\displaystyle t_{n}=c+n}}$ t_{n} = c+n		t[n] = c + n	Subscript[t, n] == c + n	Skipped - no semantic math	Skipped - no semantic math	-	-
15.7#Ex2	${\displaystyle{\displaystyle u_{2n+1}=(a+n)(c-b+n)}}$ u_{2n+1} = (a+n)(c-b+n)		u[2*n + 1] = (a + n)*(c - b + n)	Subscript[u, 2*n + 1] == (a + n)*(c - b + n)	Skipped - no semantic math	Skipped - no semantic math	-	-
15.7#Ex3	${\displaystyle{\displaystyle u_{2n}=(b+n)(c-a+n)}}$ u_{2n} = (b+n)(c-a+n)		u[2*n] = (b + n)*(c - a + n)	Subscript[u, 2*n] == (b + n)*(c - a + n)	Skipped - no semantic math	Skipped - no semantic math	-	-
15.7#Ex4	${\displaystyle{\displaystyle v_{n}=c+n+(b-a+n+1)z}}$ v_{n} = c+n+(b-a+n+1)z		v[n] = c + n +(b - a + n + 1)*z	Subscript[v, n] == c + n +(b - a + n + 1)*z	Skipped - no semantic math	Skipped - no semantic math	-	-
15.7#Ex5	${\displaystyle{\displaystyle w_{n}=(b+n)(c-a+n)z}}$ w_{n} = (b+n)(c-a+n)z		w[n] = (b + n)*(c - a + n)*z	Subscript[w, n] == (b + n)*(c - a + n)*z	Skipped - no semantic math	Skipped - no semantic math	-	-
15.7#Ex6	${\displaystyle{\displaystyle x_{n}=c+n-(a+b+2n+1)z}}$ x_{n} = c+n-(a+b+2n+1)z		x[n] = c + n -(a + b + 2*n + 1)*(x + y*I)	Subscript[x, n] == c + n -(a + b + 2*n + 1)*(x + y*I)	Skipped - no semantic math	Skipped - no semantic math	-	-
15.7#Ex7	${\displaystyle{\displaystyle y_{n}=(a+n)(b+n)z(1-z)}}$ y_{n} = (a+n)(b+n)z(1-z)		y[n] = (a + n)*(b + n)*(x + y*I)*(1 -(x + y*I))	Subscript[y, n] == (a + n)*(b + n)*(x + y*I)*(1 -(x + y*I))	Skipped - no semantic math	Skipped - no semantic math	-	-
15.8.E1	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};z\right)=(1-z)^{-a}\mathbf{F}\left({a,c-b\atop c};\frac{z}{z-1}\right)}}$ \hyperOlverF@@{a}{b}{c}{z} = (1-z)^{-a}\hyperOlverF@@{a}{c-b}{c}{\frac{z}{z-1}}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}$	hypergeom([a, b], [c], z)/GAMMA(c) = (1 - z)^(- a)* hypergeom([a, c - b], [c], (z)/(z - 1))/GAMMA(c)	Hypergeometric2F1Regularized[a, b, c, z] == (1 - z)^(- a)* Hypergeometric2F1Regularized[a, c - b, c, Divide[z,z - 1]]	Failure	Failure	Error	Failed [1 / 300] Result: Complex[-0.028209479177387697, -0.04886025119029158] Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[c, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]]}
15.8.E1	${\displaystyle{\displaystyle(1-z)^{-a}\mathbf{F}\left({a,c-b\atop c};\frac{z}{z-1}\right)=(1-z)^{-b}\mathbf{F}\left({c-a,b\atop c};\frac{z}{z-1}\right)}}$ (1-z)^{-a}\hyperOlverF@@{a}{c-b}{c}{\frac{z}{z-1}} = (1-z)^{-b}\hyperOlverF@@{c-a}{b}{c}{\frac{z}{z-1}}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}$	(1 - z)^(- a)* hypergeom([a, c - b], [c], (z)/(z - 1))/GAMMA(c) = (1 - z)^(- b)* hypergeom([c - a, b], [c], (z)/(z - 1))/GAMMA(c)	(1 - z)^(- a)* Hypergeometric2F1Regularized[a, c - b, c, Divide[z,z - 1]] == (1 - z)^(- b)* Hypergeometric2F1Regularized[c - a, b, c, Divide[z,z - 1]]	Failure	Failure	Error	Successful [Tested: 300]
15.8.E1	${\displaystyle{\displaystyle(1-z)^{-b}\mathbf{F}\left({c-a,b\atop c};\frac{z}{z-1}\right)=(1-z)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c};z\right)}}$ (1-z)^{-b}\hyperOlverF@@{c-a}{b}{c}{\frac{z}{z-1}} = (1-z)^{c-a-b}\hyperOlverF@@{c-a}{c-b}{c}{z}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}$	(1 - z)^(- b)* hypergeom([c - a, b], [c], (z)/(z - 1))/GAMMA(c) = (1 - z)^(c - a - b)* hypergeom([c - a, c - b], [c], z)/GAMMA(c)	(1 - z)^(- b)* Hypergeometric2F1Regularized[c - a, b, c, Divide[z,z - 1]] == (1 - z)^(c - a - b)* Hypergeometric2F1Regularized[c - a, c - b, c, z]	Failure	Failure	Error	Failed [1 / 300] Result: Complex[0.02820947917738814, 0.04886025119029169] Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[c, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]]}
15.8.E2	${\displaystyle{\displaystyle\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=\frac{(-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}\mathbf{F}\left({a,a-c+1\atop a-b+1};\frac{1}{z}\right)-\frac{(-z)^{-b}}{\Gamma\left(a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({b,b-c+1\atop b-a+1};\frac{1}{z}\right)}}$ \frac{\sin@{\pi(b-a)}}{\pi}\hyperOlverF@@{a}{b}{c}{z} = \frac{(-z)^{-a}}{\EulerGamma@{b}\EulerGamma@{c-a}}\hyperOlverF@@{a}{a-c+1}{a-b+1}{\frac{1}{z}}-\frac{(-z)^{-b}}{\EulerGamma@{a}\EulerGamma@{c-b}}\hyperOlverF@@{b}{b-c+1}{b-a+1}{\frac{1}{z}}	${\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\pi,\Re b>0,\Re(c-a)>0,\Re a>0,\Re(c-b)>0}}$	(sin(Pi*(b - a)))/(Pi)*hypergeom([a, b], [c], z)/GAMMA(c) = ((- z)^(- a))/(GAMMA(b)*GAMMA(c - a))*hypergeom([a, a - c + 1], [a - b + 1], (1)/(z))/GAMMA(a - b + 1)-((- z)^(- b))/(GAMMA(a)*GAMMA(c - b))*hypergeom([b, b - c + 1], [b - a + 1], (1)/(z))/GAMMA(b - a + 1)	Divide[Sin[Pi*(b - a)],Pi]*Hypergeometric2F1Regularized[a, b, c, z] == Divide[(- z)^(- a),Gamma[b]*Gamma[c - a]]*Hypergeometric2F1Regularized[a, a - c + 1, a - b + 1, Divide[1,z]]-Divide[(- z)^(- b),Gamma[a]*Gamma[c - b]]*Hypergeometric2F1Regularized[b, b - c + 1, b - a + 1, Divide[1,z]]	Failure	Failure	Error	Skip - No test values generated
15.8.E3	${\displaystyle{\displaystyle\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=\frac{(1-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}\mathbf{F}\left({a,c-b\atop a-b+1};\frac{1}{1-z}\right)-\frac{(1-z)^{-b}}{\Gamma\left(a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({b,c-a\atop b-a+1};\frac{1}{1-z}\right)}}$ \frac{\sin@{\pi(b-a)}}{\pi}\hyperOlverF@@{a}{b}{c}{z} = \frac{(1-z)^{-a}}{\EulerGamma@{b}\EulerGamma@{c-a}}\hyperOlverF@@{a}{c-b}{a-b+1}{\frac{1}{1-z}}-\frac{(1-z)^{-b}}{\EulerGamma@{a}\EulerGamma@{c-b}}\hyperOlverF@@{b}{c-a}{b-a+1}{\frac{1}{1-z}}	${\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\pi,\Re b>0,\Re(c-a)>0,\Re a>0,\Re(c-b)>0}}$	(sin(Pi*(b - a)))/(Pi)*hypergeom([a, b], [c], z)/GAMMA(c) = ((1 - z)^(- a))/(GAMMA(b)*GAMMA(c - a))*hypergeom([a, c - b], [a - b + 1], (1)/(1 - z))/GAMMA(a - b + 1)-((1 - z)^(- b))/(GAMMA(a)*GAMMA(c - b))*hypergeom([b, c - a], [b - a + 1], (1)/(1 - z))/GAMMA(b - a + 1)	Divide[Sin[Pi*(b - a)],Pi]*Hypergeometric2F1Regularized[a, b, c, z] == Divide[(1 - z)^(- a),Gamma[b]*Gamma[c - a]]*Hypergeometric2F1Regularized[a, c - b, a - b + 1, Divide[1,1 - z]]-Divide[(1 - z)^(- b),Gamma[a]*Gamma[c - b]]*Hypergeometric2F1Regularized[b, c - a, b - a + 1, Divide[1,1 - z]]	Failure	Failure	Error	Successful [Tested: 10]
15.8.E4	${\displaystyle{\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=\frac{1}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({a,b\atop a+b-c+1};1-z\right)-\frac{(1-z)^{c-a-b}}{\Gamma\left(a\right)\Gamma\left(b\right)}\mathbf{F}\left({c-a,c-b\atop c-a-b+1};1-z\right)}}$ \frac{\sin@{\pi(c-a-b)}}{\pi}\hyperOlverF@@{a}{b}{c}{z} = \frac{1}{\EulerGamma@{c-a}\EulerGamma@{c-b}}\hyperOlverF@@{a}{b}{a+b-c+1}{1-z}-\frac{(1-z)^{c-a-b}}{\EulerGamma@{a}\EulerGamma@{b}}\hyperOlverF@@{c-a}{c-b}{c-a-b+1}{1-z}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)|<\pi,\Re(c-a)>0,\Re(c-b)>0,\Re a>0,\Re b>0,|z|<1,|(1-z)|<1,\Re(c+s)>0,\Re((a+b-c+1)+s)>0,\Re((c-a-b+1)+s)>0}}$	(sin(Pi*(c - a - b)))/(Pi)*hypergeom([a, b], [c], z)/GAMMA(c) = (1)/(GAMMA(c - a)*GAMMA(c - b))*hypergeom([a, b], [a + b - c + 1], 1 - z)/GAMMA(a + b - c + 1)-((1 - z)^(c - a - b))/(GAMMA(a)*GAMMA(b))*hypergeom([c - a, c - b], [c - a - b + 1], 1 - z)/GAMMA(c - a - b + 1)	Divide[Sin[Pi*(c - a - b)],Pi]*Hypergeometric2F1Regularized[a, b, c, z] == Divide[1,Gamma[c - a]*Gamma[c - b]]*Hypergeometric2F1Regularized[a, b, a + b - c + 1, 1 - z]-Divide[(1 - z)^(c - a - b),Gamma[a]*Gamma[b]]*Hypergeometric2F1Regularized[c - a, c - b, c - a - b + 1, 1 - z]	Failure	Failure	Failed [2 / 5] Result: Float(undefined)+Float(undefined)*I Test Values: {a = 3/2, b = 3/2, c = 2, z = 1/2} Result: Float(undefined)+Float(undefined)*I Test Values: {a = 1/2, b = 1/2, c = 2, z = 1/2}	Successful [Tested: 15]
15.8.E5	${\displaystyle{\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=\frac{z^{-a}}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({a,a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)-\frac{(1-z)^{c-a-b}z^{a-c}}{\Gamma\left(a\right)\Gamma\left(b\right)}\mathbf{F}\left({c-a,1-a\atop c-a-b+1};1-\frac{1}{z}\right)}}$ \frac{\sin@{\pi(c-a-b)}}{\pi}\hyperOlverF@@{a}{b}{c}{z} = \frac{z^{-a}}{\EulerGamma@{c-a}\EulerGamma@{c-b}}\hyperOlverF@@{a}{a-c+1}{a+b-c+1}{1-\frac{1}{z}}-\frac{(1-z)^{c-a-b}z^{a-c}}{\EulerGamma@{a}\EulerGamma@{b}}\hyperOlverF@@{c-a}{1-a}{c-a-b+1}{1-\frac{1}{z}}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)|<\pi,\Re(c-a)>0,\Re(c-b)>0,\Re a>0,\Re b>0}}$	(sin(Pi*(c - a - b)))/(Pi)*hypergeom([a, b], [c], z)/GAMMA(c) = ((z)^(- a))/(GAMMA(c - a)*GAMMA(c - b))*hypergeom([a, a - c + 1], [a + b - c + 1], 1 -(1)/(z))/GAMMA(a + b - c + 1)-((1 - z)^(c - a - b)* (z)^(a - c))/(GAMMA(a)*GAMMA(b))*hypergeom([c - a, 1 - a], [c - a - b + 1], 1 -(1)/(z))/GAMMA(c - a - b + 1)	Divide[Sin[Pi*(c - a - b)],Pi]*Hypergeometric2F1Regularized[a, b, c, z] == Divide[(z)^(- a),Gamma[c - a]*Gamma[c - b]]*Hypergeometric2F1Regularized[a, a - c + 1, a + b - c + 1, 1 -Divide[1,z]]-Divide[(1 - z)^(c - a - b)* (z)^(a - c),Gamma[a]*Gamma[b]]*Hypergeometric2F1Regularized[c - a, 1 - a, c - a - b + 1, 1 -Divide[1,z]]	Failure	Failure	Error	Skip - No test values generated
15.8.E6	${\displaystyle{\displaystyle F\left({-m,b\atop c};z\right)=\frac{(b)_{m}}{(c)_{m}}(-z)^{m}F\left({-m,1-c-m\atop 1-b-m};\frac{1}{z}\right)}}$ \hyperF@@{-m}{b}{c}{z} = \frac{(b)_{m}}{(c)_{m}}(-z)^{m}\hyperF@@{-m}{1-c-m}{1-b-m}{\frac{1}{z}}		hypergeom([- m, b], [c], z) = (b[m])/(c[m])*(- z)^(m)* hypergeom([- m, 1 - c - m], [1 - b - m], (1)/(z))	Hypergeometric2F1[- m, b, c, z] == Divide[Subscript[b, m],Subscript[c, m]]*(- z)^(m)* Hypergeometric2F1[- m, 1 - c - m, 1 - b - m, Divide[1,z]]	Failure	Failure	Error	Failed [252 / 300] Result: Plus[Complex[1.4330127018922194, 0.24999999999999997], Times[Complex[1.4330127018922196, 0.25], Subscript[-1.5, 1], Power[Subscript[1.5, 1], -1]]] Test Values: {Rule[b, -1.5], Rule[c, 1.5], Rule[m, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Plus[Complex[1.8910254037844387, 0.5433012701892219], Times[Complex[-9.455127018922195, -2.7165063509461094], Subscript[-1.5, 2], Power[Subscript[1.5, 2], -1]]] Test Values: {Rule[b, -1.5], Rule[c, 1.5], Rule[m, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.8.E6	${\displaystyle{\displaystyle\frac{(b)_{m}}{(c)_{m}}(-z)^{m}F\left({-m,1-c-m\atop 1-b-m};\frac{1}{z}\right)=\frac{(b)_{m}}{(c)_{m}}(1-z)^{m}F\left({-m,c-b\atop 1-b-m};\frac{1}{1-z}\right)}}$ \frac{(b)_{m}}{(c)_{m}}(-z)^{m}\hyperF@@{-m}{1-c-m}{1-b-m}{\frac{1}{z}} = \frac{(b)_{m}}{(c)_{m}}(1-z)^{m}\hyperF@@{-m}{c-b}{1-b-m}{\frac{1}{1-z}}		(b[m])/(c[m])*(- z)^(m)* hypergeom([- m, 1 - c - m], [1 - b - m], (1)/(z)) = (b[m])/(c[m])*(1 - z)^(m)* hypergeom([- m, c - b], [1 - b - m], (1)/(1 - z))	Divide[Subscript[b, m],Subscript[c, m]]*(- z)^(m)* Hypergeometric2F1[- m, 1 - c - m, 1 - b - m, Divide[1,z]] == Divide[Subscript[b, m],Subscript[c, m]]*(1 - z)^(m)* Hypergeometric2F1[- m, c - b, 1 - b - m, Divide[1,1 - z]]	Failure	Failure	Error	Failed [164 / 300] Result: Times[Complex[0.0, -5.551115123125783*^-17], Subscript[-1.5, 1], Power[Subscript[1.5, 1], -1]] Test Values: {Rule[b, -1.5], Rule[c, 1.5], Rule[m, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Times[Complex[0.0, 4.440892098500626*^-16], Subscript[-1.5, 2], Power[Subscript[1.5, 2], -1]] Test Values: {Rule[b, -1.5], Rule[c, 1.5], Rule[m, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.8.E7	${\displaystyle{\displaystyle F\left({-m,b\atop c};z\right)=\frac{(c-b)_{m}}{(c)_{m}}F\left({-m,b\atop b-c-m+1};1-z\right)}}$ \hyperF@@{-m}{b}{c}{z} = \frac{(c-b)_{m}}{(c)_{m}}\hyperF@@{-m}{b}{b-c-m+1}{1-z}		hypergeom([- m, b], [c], z) = (c - b[m])/(c[m])*hypergeom([- m, b], [b - c - m + 1], 1 - z)	Hypergeometric2F1[- m, b, c, z] == Divide[Subscript[c - b, m],Subscript[c, m]]*Hypergeometric2F1[- m, b, b - c - m + 1, 1 - z]	Failure	Failure	Error	Failed [300 / 300] Result: DirectedInfinity[] Test Values: {Rule[b, -1.5], Rule[c, -1.5], Rule[m, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: DirectedInfinity[] Test Values: {Rule[b, -1.5], Rule[c, -1.5], Rule[m, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.8.E7	${\displaystyle{\displaystyle\frac{(c-b)_{m}}{(c)_{m}}F\left({-m,b\atop b-c-m+1};1-z\right)=\frac{(c-b)_{m}}{(c)_{m}}z^{m}F\left({-m,1-c-m\atop b-c-m+1};1-\frac{1}{z}\right)}}$ \frac{(c-b)_{m}}{(c)_{m}}\hyperF@@{-m}{b}{b-c-m+1}{1-z} = \frac{(c-b)_{m}}{(c)_{m}}z^{m}\hyperF@@{-m}{1-c-m}{b-c-m+1}{1-\frac{1}{z}}		(c - b[m])/(c[m])*hypergeom([- m, b], [b - c - m + 1], 1 - z) = (c - b[m])/(c[m])*(z)^(m)* hypergeom([- m, 1 - c - m], [b - c - m + 1], 1 -(1)/(z))	Divide[Subscript[c - b, m],Subscript[c, m]]*Hypergeometric2F1[- m, b, b - c - m + 1, 1 - z] == Divide[Subscript[c - b, m],Subscript[c, m]]*(z)^(m)* Hypergeometric2F1[- m, 1 - c - m, b - c - m + 1, 1 -Divide[1,z]]	Failure	Failure	Error	Failed [206 / 300] Result: Indeterminate Test Values: {Rule[b, -1.5], Rule[c, -1.5], Rule[m, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[b, -1.5], Rule[c, -1.5], Rule[m, 2], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
15.8.E8	${\displaystyle{\displaystyle\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(-z)^{-a}}{\Gamma\left(a+m\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(m-k-1)!}{k!\Gamma\left(c-a-k\right)}z^{-k}+\frac{(-z)^{-a}}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}}{k!(k+m)!\Gamma\left(c-a-k-m\right)}(-1)^{k}z^{-k-m}\*\left(\ln\left(-z\right)+\psi\left(k+1\right)+\psi\left(k+m+1\right)-\psi\left(a+k+m\right)-\psi\left(c-a-k-m\right)\right)}}$ \hyperOlverF@@{a}{a+m}{c}{z} = \frac{(-z)^{-a}}{\EulerGamma@{a+m}}\sum_{k=0}^{m-1}\frac{(a)_{k}(m-k-1)!}{k!\EulerGamma@{c-a-k}}z^{-k}+\frac{(-z)^{-a}}{\EulerGamma@{a}}\sum_{k=0}^{\infty}\frac{(a+m)_{k}}{k!(k+m)!\EulerGamma@{c-a-k-m}}(-1)^{k}z^{-k-m}\*\left(\ln@{-z}+\digamma@{k+1}+\digamma@{k+m+1}-\digamma@{a+k+m}-\digamma@{c-a-k-m}\right)	${\displaystyle{\displaystyle|z|>1,|\operatorname{ph}\left(-z\right)|<\pi,\Re(a+m)>0,\Re(c-a-k)>0,\Re a>0,\Re(c-a-k-m)>0,|z|<1,\Re(c+s)>0}}$	hypergeom([a, a + m], [c], z)/GAMMA(c) = ((- z)^(- a))/(GAMMA(a + m))*sum((a[k]*factorial(m - k - 1))/(factorial(k)*GAMMA(c - a - k))*(z)^(- k), k = 0..m - 1)+((- z)^(- a))/(GAMMA(a))*sum((a + m[k])/(factorial(k)*factorial(k + m)*GAMMA(c - a - k - m))*(- 1)^(k)* (z)^(- k - m)*(ln(- z)+ Psi(k + 1)+ Psi(k + m + 1)- Psi(a + k + m)- Psi(c - a - k - m)), k = 0..infinity)	Hypergeometric2F1Regularized[a, a + m, c, z] == Divide[(- z)^(- a),Gamma[a + m]]*Sum[Divide[Subscript[a, k]*(m - k - 1)!,(k)!*Gamma[c - a - k]]*(z)^(- k), {k, 0, m - 1}, GenerateConditions->None]+Divide[(- z)^(- a),Gamma[a]]*Sum[Divide[Subscript[a + m, k],(k)!*(k + m)!*Gamma[c - a - k - m]]*(- 1)^(k)* (z)^(- k - m)*(Log[- z]+ PolyGamma[k + 1]+ PolyGamma[k + m + 1]- PolyGamma[a + k + m]- PolyGamma[c - a - k - m]), {k, 0, Infinity}, GenerateConditions->None]	Error	Failure	-	Skip - No test values generated
15.8.E9	${\displaystyle{\displaystyle\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(1-z)^{-a}}{\Gamma\left(a+m\right)\Gamma\left(c-a\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(c-a-m)_{k}(m-k-1)!}{k!}(z-1)^{-k}+\frac{(-1)^{m}(1-z)^{-a-m}}{\Gamma\left(a\right)\Gamma\left(c-a-m\right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}(c-a)_{k}}{k!(k+m)!}(1-z)^{-k}\*(\ln\left(1-z\right)+\psi\left(k+1\right)+\psi\left(k+m+1\right)-\psi\left(a+k+m\right)-\psi\left(c-a+k\right))}}$ \hyperOlverF@@{a}{a+m}{c}{z} = \frac{(1-z)^{-a}}{\EulerGamma@{a+m}\EulerGamma@{c-a}}\sum_{k=0}^{m-1}\frac{(a)_{k}(c-a-m)_{k}(m-k-1)!}{k!}(z-1)^{-k}+\frac{(-1)^{m}(1-z)^{-a-m}}{\EulerGamma@{a}\EulerGamma@{c-a-m}}\sum_{k=0}^{\infty}\frac{(a+m)_{k}(c-a)_{k}}{k!(k+m)!}(1-z)^{-k}\*(\ln@{1-z}+\digamma@{k+1}+\digamma@{k+m+1}-\digamma@{a+k+m}-\digamma@{c-a+k})	${\displaystyle{\displaystyle|z-1|>1,|\operatorname{ph}\left(1-z\right)|<\pi,\Re(a+m)>0,\Re(c-a)>0,\Re a>0,\Re(c-a-m)>0,|z|<1,\Re(c+s)>0}}$	hypergeom([a, a + m], [c], z)/GAMMA(c) = ((1 - z)^(- a))/(GAMMA(a + m)*GAMMA(c - a))*sum((a[k]*c - a - m[k]*factorial(m - k - 1))/(factorial(k))*(z - 1)^(- k), k = 0..m - 1)+((- 1)^(m)*(1 - z)^(- a - m))/(GAMMA(a)*GAMMA(c - a - m))*sum((a + m[k]*c - a[k])/(factorial(k)*factorial(k + m))*(1 - z)^(- k)*(ln(1 - z)+ Psi(k + 1)+ Psi(k + m + 1)- Psi(a + k + m)- Psi(c - a + k)), k = 0..infinity)	Hypergeometric2F1Regularized[a, a + m, c, z] == Divide[(1 - z)^(- a),Gamma[a + m]*Gamma[c - a]]*Sum[Divide[Subscript[a, k]*Subscript[c - a - m, k]*(m - k - 1)!,(k)!]*(z - 1)^(- k), {k, 0, m - 1}, GenerateConditions->None]+Divide[(- 1)^(m)*(1 - z)^(- a - m),Gamma[a]*Gamma[c - a - m]]*Sum[Divide[Subscript[a + m, k]*Subscript[c - a, k],(k)!*(k + m)!]*(1 - z)^(- k)*(Log[1 - z]+ PolyGamma[k + 1]+ PolyGamma[k + m + 1]- PolyGamma[a + k + m]- PolyGamma[c - a + k]), {k, 0, Infinity}, GenerateConditions->None]	Error	Aborted	-	Failed [2 / 2] Result: Plus[Complex[0.8934823398107985, 0.11625604883874943], Times[Complex[0.18357341911556996, 0.10033661972146816], NSum[Times[Power[Plus[1, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]], Times[-1, k]], Power[Factorial[k], -1], Power[Factorial[Plus[1, k]], -1], Plus[Log[Plus[1, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]], PolyGamma[0, Plus[1, k]], Times[-2, PolyGamma[0, Plus[1.5, k]]], PolyGamma[0, Plus[2, k]]], Power[Subscript[1.5, k], 2]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], Times[Complex[-1.0916552187951503, -0.18372460978003777], Power[Subscript[0.5, 0], 2]]], {Rule[a, 0.5], Rule[c, 2], Rule[m, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} Result: Plus[Complex[0.8646684259719354, -0.05865467444211362], Times[Complex[0.17537516348927204, -0.04648067160197167], NSum[Times[Power[Plus[1, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]], Times[-1, k]], Power[Factorial[k], -1], Power[Factorial[Plus[1, k]], -1], Plus[Log[Plus[1, Times[Rational[-1, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]]], PolyGamma[0, Plus[1, k]], Times[-2, PolyGamma[0, Plus[1.5, k]]], PolyGamma[0, Plus[2, k]]], Power[Subscript[1.5, k], 2]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], Times[Complex[-1.0517400191081774, 0.0910544077031535], Power[Subscript[0.5, 0], 2]]], {Rule[a, 0.5], Rule[c, 2], Rule[m, 1], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]]}
15.8.E10	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{1}{\Gamma\left(a+m\right)\Gamma\left(b+m\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(b)_{k}(m-k-1)!}{k!}(z-1)^{k}-\frac{(z-1)^{m}}{\Gamma\left(a\right)\Gamma\left(b\right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}(b+m)_{k}}{k!(k+m)!}(1-z)^{k}\*\left(\ln\left(1-z\right)-\psi\left(k+1\right)-\psi\left(k+m+1\right)+\psi\left(a+k+m\right)+\psi\left(b+k+m\right)\right)}}$ \hyperOlverF@@{a}{b}{a+b+m}{z} = \frac{1}{\EulerGamma@{a+m}\EulerGamma@{b+m}}\sum_{k=0}^{m-1}\frac{(a)_{k}(b)_{k}(m-k-1)!}{k!}(z-1)^{k}-\frac{(z-1)^{m}}{\EulerGamma@{a}\EulerGamma@{b}}\sum_{k=0}^{\infty}\frac{(a+m)_{k}(b+m)_{k}}{k!(k+m)!}(1-z)^{k}\*\left(\ln@{1-z}-\digamma@{k+1}-\digamma@{k+m+1}+\digamma@{a+k+m}+\digamma@{b+k+m}\right)	${\displaystyle{\displaystyle|z-1|<1,|\operatorname{ph}\left(1-z\right)|<\pi,\Re(a+m)>0,\Re(b+m)>0,\Re a>0,\Re b>0,|z|<1,\Re((a+b+m)+s)>0}}$	hypergeom([a, b], [a + b + m], z)/GAMMA(a + b + m) = (1)/(GAMMA(a + m)*GAMMA(b + m))*sum((a[k]*b[k]*factorial(m - k - 1))/(factorial(k))*(z - 1)^(k), k = 0..m - 1)-((z - 1)^(m))/(GAMMA(a)*GAMMA(b))*sum((a + m[k]*b + m[k])/(factorial(k)*factorial(k + m))*(1 - z)^(k)*(ln(1 - z)- Psi(k + 1)- Psi(k + m + 1)+ Psi(a + k + m)+ Psi(b + k + m)), k = 0..infinity)	Hypergeometric2F1Regularized[a, b, a + b + m, z] == Divide[1,Gamma[a + m]*Gamma[b + m]]*Sum[Divide[Subscript[a, k]*Subscript[b, k]*(m - k - 1)!,(k)!]*(z - 1)^(k), {k, 0, m - 1}, GenerateConditions->None]-Divide[(z - 1)^(m),Gamma[a]*Gamma[b]]*Sum[Divide[Subscript[a + m, k]*Subscript[b + m, k],(k)!*(k + m)!]*(1 - z)^(k)*(Log[1 - z]- PolyGamma[k + 1]- PolyGamma[k + m + 1]+ PolyGamma[a + k + m]+ PolyGamma[b + k + m]), {k, 0, Infinity}, GenerateConditions->None]	Error	Failure	-	Skipped - Because timed out
15.8.E11	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{z^{-a}}{\Gamma\left(a+m\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(m-k-1)!}{k!\Gamma\left(b+m-k\right)}\left(1-\frac{1}{z}\right)^{k}-\frac{z^{-a}}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}}{k!(k+m)!\Gamma\left(b-k\right)}(-1)^{k}\left(1-\frac{1}{z}\right)^{k+m}\*\left(\ln\left(\frac{1-z}{z}\right)-\psi\left(k+1\right)-\psi\left(k+m+1\right)+\psi\left(a+k+m\right)+\psi\left(b-k\right)\right)}}$ \hyperOlverF@@{a}{b}{a+b+m}{z} = \frac{z^{-a}}{\EulerGamma@{a+m}}\sum_{k=0}^{m-1}\frac{(a)_{k}(m-k-1)!}{k!\EulerGamma@{b+m-k}}\left(1-\frac{1}{z}\right)^{k}-\frac{z^{-a}}{\EulerGamma@{a}}\sum_{k=0}^{\infty}\frac{(a+m)_{k}}{k!(k+m)!\EulerGamma@{b-k}}(-1)^{k}\left(1-\frac{1}{z}\right)^{k+m}\*\left(\ln\left(\frac{1-z}{z}\right)-\digamma@{k+1}-\digamma@{k+m+1}+\digamma@{a+k+m}+\digamma@{b-k}\right)	${\displaystyle{\displaystyle\Re z>\tfrac{1}{2},|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)|<\pi,\Re(a+m)>0,\Re(b+m-k)>0,\Re a>0,\Re(b-k)>0,|z|<1,\Re((a+b+m)+s)>0}}$	hypergeom([a, b], [a + b + m], z)/GAMMA(a + b + m) = ((z)^(- a))/(GAMMA(a + m))*sum((a[k]*factorial(m - k - 1))/(factorial(k)*GAMMA(b + m - k))*(1 -(1)/(z))^(k), k = 0..m - 1)-((z)^(- a))/(GAMMA(a))*sum((a + m[k])/(factorial(k)*factorial(k + m)*GAMMA(b - k))*(- 1)^(k)*(1 -(1)/(z))^(k + m)*(ln((1 - z)/(z))- Psi(k + 1)- Psi(k + m + 1)+ Psi(a + k + m)+ Psi(b - k)), k = 0..infinity)	Hypergeometric2F1Regularized[a, b, a + b + m, z] == Divide[(z)^(- a),Gamma[a + m]]*Sum[Divide[Subscript[a, k]*(m - k - 1)!,(k)!*Gamma[b + m - k]]*(1 -Divide[1,z])^(k), {k, 0, m - 1}, GenerateConditions->None]-Divide[(z)^(- a),Gamma[a]]*Sum[Divide[Subscript[a + m, k],(k)!*(k + m)!*Gamma[b - k]]*(- 1)^(k)*(1 -Divide[1,z])^(k + m)*(Log[Divide[1 - z,z]]- PolyGamma[k + 1]- PolyGamma[k + m + 1]+ PolyGamma[a + k + m]+ PolyGamma[b - k]), {k, 0, Infinity}, GenerateConditions->None]	Translation Error	Translation Error	-	-
15.8.E13	${\displaystyle{\displaystyle F\left({a,b\atop 2b};z\right)=\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop b+\tfrac{1}{2}};\left(\frac{z}{2-z}\right)^{2}\right)}}$ \hyperF@@{a}{b}{2b}{z} = \left(1-\tfrac{1}{2}z\right)^{-a}\hyperF@@{\tfrac{1}{2}a}{\tfrac{1}{2}a+\tfrac{1}{2}}{b+\tfrac{1}{2}}{\left(\frac{z}{2-z}\right)^{2}}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}$	hypergeom([a, b], [2*b], z) = (1 -(1)/(2)*z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a +(1)/(2)], [b +(1)/(2)], ((z)/(2 - z))^(2))	Hypergeometric2F1[a, b, 2*b, z] == (1 -Divide[1,2]*z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a +Divide[1,2], b +Divide[1,2], (Divide[z,2 - z])^(2)]	Failure	Failure	Failed [74 / 180] Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I} Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = -3/2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [67 / 180] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.8.E14	${\displaystyle{\displaystyle F\left({a,b\atop 2b};z\right)=\left(1-z\right)^{-\ifrac{a}{2}}F\left({\tfrac{1}{2}a,b-\tfrac{1}{2}a\atop b+\tfrac{1}{2}};\frac{z^{2}}{4z-4}\right)}}$ \hyperF@@{a}{b}{2b}{z} = \left(1-z\right)^{-\ifrac{a}{2}}\hyperF@@{\tfrac{1}{2}a}{b-\tfrac{1}{2}a}{b+\tfrac{1}{2}}{\frac{z^{2}}{4z-4}}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}$	hypergeom([a, b], [2*b], z) = (1 - z)^(-(a)/(2))* hypergeom([(1)/(2)*a, b -(1)/(2)*a], [b +(1)/(2)], ((z)^(2))/(4*z - 4))	Hypergeometric2F1[a, b, 2*b, z] == (1 - z)^(-Divide[a,2])* Hypergeometric2F1[Divide[1,2]*a, b -Divide[1,2]*a, b +Divide[1,2], Divide[(z)^(2),4*z - 4]]	Failure	Failure	Failed [74 / 180] Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = -3/2, z = 1/2*3^(1/2)+1/2*I} Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = -3/2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [67 / 180] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.8.E15	${\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=(1+z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}\atop a-b+1};\frac{4z}{(1+z)^{2}}\right)}}$ \hyperF@@{a}{b}{a-b+1}{z} = (1+z)^{-a}\hyperF@@{\frac{1}{2}a}{\frac{1}{2}a+\frac{1}{2}}{a-b+1}{\frac{4z}{(1+z)^{2}}}	${\displaystyle{\displaystyle|z|<1}}$	hypergeom([a, b], [a - b + 1], z) = (1 + z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a +(1)/(2)], [a - b + 1], (4*z)/((1 + z)^(2)))	Hypergeometric2F1[a, b, a - b + 1, z] == (1 + z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a +Divide[1,2], a - b + 1, Divide[4*z,(1 + z)^(2)]]	Failure	Failure	Failed [6 / 36] Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = 3/2, z = 1/2} Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = -1/2, z = 1/2} ... skip entries to safe data	Failed [30 / 180] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.8.E16	${\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a-b+\frac{1}{2}\atop a-b+1};\frac{-4z}{(1-z)^{2}}\right)}}$ \hyperF@@{a}{b}{a-b+1}{z} = (1-z)^{-a}\hyperF@@{\frac{1}{2}a}{\frac{1}{2}a-b+\frac{1}{2}}{a-b+1}{\frac{-4z}{(1-z)^{2}}}	${\displaystyle{\displaystyle|z|<1}}$	hypergeom([a, b], [a - b + 1], z) = (1 - z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a - b +(1)/(2)], [a - b + 1], (- 4*z)/((1 - z)^(2)))	Hypergeometric2F1[a, b, a - b + 1, z] == (1 - z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a - b +Divide[1,2], a - b + 1, Divide[- 4*z,(1 - z)^(2)]]	Failure	Failure	Failed [6 / 36] Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = 3/2, z = 1/2} Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = -1/2, z = 1/2} ... skip entries to safe data	Failed [30 / 180] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.8.E17	${\displaystyle{\displaystyle F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)=(1-2z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}\atop\frac{1}{2}(a+b+1)};\frac{4z(z-1)}{(1-2z)^{2}}\right)}}$ \hyperF@@{a}{b}{\frac{1}{2}(a+b+1)}{z} = (1-2z)^{-a}\hyperF@@{\frac{1}{2}a}{\frac{1}{2}a+\frac{1}{2}}{\frac{1}{2}(a+b+1)}{\frac{4z(z-1)}{(1-2z)^{2}}}		hypergeom([a, b], [(1)/(2)*(a + b + 1)], z) = (1 - 2*z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a +(1)/(2)], [(1)/(2)*(a + b + 1)], (4*z*(z - 1))/((1 - 2*z)^(2)))	Hypergeometric2F1[a, b, Divide[1,2]*(a + b + 1), z] == (1 - 2*z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a +Divide[1,2], Divide[1,2]*(a + b + 1), Divide[4*z*(z - 1),(1 - 2*z)^(2)]]	Failure	Failure	Successful [Tested: 36]	Failed [3 / 36] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 0.5], Rule[z, 0]} Result: Indeterminate Test Values: {Rule[a, -0.5], Rule[b, -0.5], Rule[z, 0]} ... skip entries to safe data
15.8.E18	${\displaystyle{\displaystyle F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)=F\left({\frac{1}{2}a,\frac{1}{2}b\atop\frac{1}{2}(a+b+1)};4z(1-z)\right)}}$ \hyperF@@{a}{b}{\frac{1}{2}(a+b+1)}{z} = \hyperF@@{\frac{1}{2}a}{\frac{1}{2}b}{\frac{1}{2}(a+b+1)}{4z(1-z)}		hypergeom([a, b], [(1)/(2)*(a + b + 1)], z) = hypergeom([(1)/(2)*a, (1)/(2)*b], [(1)/(2)*(a + b + 1)], 4*z*(1 - z))	Hypergeometric2F1[a, b, Divide[1,2]*(a + b + 1), z] == Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*b, Divide[1,2]*(a + b + 1), 4*z*(1 - z)]	Failure	Failure	Successful [Tested: 36]	Failed [3 / 36] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 0.5], Rule[z, 0]} Result: Indeterminate Test Values: {Rule[a, -0.5], Rule[b, -0.5], Rule[z, 0]} ... skip entries to safe data
15.8.E19	${\displaystyle{\displaystyle F\left({a,1-a\atop c};z\right)=(1-2z)^{1-a-c}(1-z)^{c-1}F\left({\frac{1}{2}(a+c),\frac{1}{2}(a+c-1)\atop c};\frac{4z(z-1)}{(1-2z)^{2}}\right)}}$ \hyperF@@{a}{1-a}{c}{z} = (1-2z)^{1-a-c}(1-z)^{c-1}\hyperF@@{\frac{1}{2}(a+c)}{\frac{1}{2}(a+c-1)}{c}{\frac{4z(z-1)}{(1-2z)^{2}}}		hypergeom([a, 1 - a], [c], z) = (1 - 2*z)^(1 - a - c)*(1 - z)^(c - 1)* hypergeom([(1)/(2)*(a + c), (1)/(2)*(a + c - 1)], [c], (4*z*(z - 1))/((1 - 2*z)^(2)))	Hypergeometric2F1[a, 1 - a, c, z] == (1 - 2*z)^(1 - a - c)*(1 - z)^(c - 1)* Hypergeometric2F1[Divide[1,2]*(a + c), Divide[1,2]*(a + c - 1), c, Divide[4*z*(z - 1),(1 - 2*z)^(2)]]	Failure	Failure	Successful [Tested: 36]	Successful [Tested: 36]
15.8.E20	${\displaystyle{\displaystyle F\left({a,1-a\atop c};z\right)=(1-z)^{c-1}F\left({\frac{1}{2}(c-a),\frac{1}{2}(a+c-1)\atop c};4z(1-z)\right)}}$ \hyperF@@{a}{1-a}{c}{z} = (1-z)^{c-1}\hyperF@@{\frac{1}{2}(c-a)}{\frac{1}{2}(a+c-1)}{c}{4z(1-z)}		hypergeom([a, 1 - a], [c], z) = (1 - z)^(c - 1)* hypergeom([(1)/(2)*(c - a), (1)/(2)*(a + c - 1)], [c], 4*z*(1 - z))	Hypergeometric2F1[a, 1 - a, c, z] == (1 - z)^(c - 1)* Hypergeometric2F1[Divide[1,2]*(c - a), Divide[1,2]*(a + c - 1), c, 4*z*(1 - z)]	Failure	Failure	Successful [Tested: 36]	Successful [Tested: 36]
15.8.E21	${\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=\left(1+\sqrt{z}\right)^{-2a}F\left({a,a-b+\tfrac{1}{2}\atop 2a-2b+1};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right)}}$ \hyperF@@{a}{b}{a-b+1}{z} = \left(1+\sqrt{z}\right)^{-2a}\hyperF@@{a}{a-b+\tfrac{1}{2}}{2a-2b+1}{\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|z|<1}}$	hypergeom([a, b], [a - b + 1], z) = (1 +sqrt(z))^(- 2*a)* hypergeom([a, a - b +(1)/(2)], [2*a - 2*b + 1], (4*sqrt(z))/((1 +sqrt(z))^(2)))	Hypergeometric2F1[a, b, a - b + 1, z] == (1 +Sqrt[z])^(- 2*a)* Hypergeometric2F1[a, a - b +Divide[1,2], 2*a - 2*b + 1, Divide[4*Sqrt[z],(1 +Sqrt[z])^(2)]]	Failure	Failure	Failed [11 / 36] Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = 3/2, z = 1/2} Result: Float(infinity)+Float(infinity)*I Test Values: {a = -3/2, b = -1/2, z = 1/2} ... skip entries to safe data	Failed [55 / 180] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.8.E22	${\displaystyle{\displaystyle F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\left(\frac{\sqrt{1-z^{-1}}-1}{\sqrt{1-z^{-1}}+1}\right)^{a}F\left({a,\tfrac{1}{2}(a+b)\atop a+b};\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}}\right)}}$ \hyperF@@{a}{b}{\tfrac{1}{2}(a+b+1)}{z} = \left(\frac{\sqrt{1-z^{-1}}-1}{\sqrt{1-z^{-1}}+1}\right)^{a}\hyperF@@{a}{\tfrac{1}{2}(a+b)}{a+b}{\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}}}	${\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\pi}}$	hypergeom([a, b], [(1)/(2)*(a + b + 1)], z) = ((sqrt(1 - (z)^(- 1))- 1)/(sqrt(1 - (z)^(- 1))+ 1))^(a)* hypergeom([a, (1)/(2)*(a + b)], [a + b], (4*sqrt(1 - (z)^(- 1)))/((sqrt(1 - (z)^(- 1))+ 1)^(2)))	Hypergeometric2F1[a, b, Divide[1,2]*(a + b + 1), z] == (Divide[Sqrt[1 - (z)^(- 1)]- 1,Sqrt[1 - (z)^(- 1)]+ 1])^(a)* Hypergeometric2F1[a, Divide[1,2]*(a + b), a + b, Divide[4*Sqrt[1 - (z)^(- 1)],(Sqrt[1 - (z)^(- 1)]+ 1)^(2)]]	Failure	Failure	Error	Failed [36 / 36] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[z, 0]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[z, 0]} ... skip entries to safe data
15.8.E23	${\displaystyle{\displaystyle F\left({a,1-a\atop c};z\right)=\left(\sqrt{1-z^{-1}}-1\right)^{1-a}\left(\sqrt{1-z^{-1}}+1\right)^{a-2c+1}\left(1-z^{-1}\right)^{c-1}F\left({c-a,c-\tfrac{1}{2}\atop 2c-1};\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}}\right)}}$ \hyperF@@{a}{1-a}{c}{z} = \left(\sqrt{1-z^{-1}}-1\right)^{1-a}\left(\sqrt{1-z^{-1}}+1\right)^{a-2c+1}\left(1-z^{-1}\right)^{c-1}\hyperF@@{c-a}{c-\tfrac{1}{2}}{2c-1}{\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}}}	${\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\pi}}$	hypergeom([a, 1 - a], [c], z) = (sqrt(1 - (z)^(- 1))- 1)^(1 - a)*(sqrt(1 - (z)^(- 1))+ 1)^(a - 2*c + 1)*(1 - (z)^(- 1))^(c - 1)* hypergeom([c - a, c -(1)/(2)], [2*c - 1], (4*sqrt(1 - (z)^(- 1)))/((sqrt(1 - (z)^(- 1))+ 1)^(2)))	Hypergeometric2F1[a, 1 - a, c, z] == (Sqrt[1 - (z)^(- 1)]- 1)^(1 - a)*(Sqrt[1 - (z)^(- 1)]+ 1)^(a - 2*c + 1)*(1 - (z)^(- 1))^(c - 1)* Hypergeometric2F1[c - a, c -Divide[1,2], 2*c - 1, Divide[4*Sqrt[1 - (z)^(- 1)],(Sqrt[1 - (z)^(- 1)]+ 1)^(2)]]	Failure	Failure	Error	Failed [36 / 36] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[c, -1.5], Rule[z, 0]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[c, 1.5], Rule[z, 0]} ... skip entries to safe data
15.8.E24	${\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}\frac{\Gamma\left(a-b+1\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}a-b+1\right)}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a-b+\tfrac{1}{2}\atop\tfrac{1}{2}};\left(\frac{z+1}{z-1}\right)^{2}\right)+(1+z)(1-z)^{-a-1}\frac{\Gamma\left(a-b+1\right)\Gamma\left(-\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}a-b+\tfrac{1}{2}\right)}F\left({\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}a-b+1\atop\tfrac{3}{2}};\left(\frac{z+1}{z-1}\right)^{2}\right)}}$ \hyperF@@{a}{b}{a-b+1}{z} = (1-z)^{-a}\frac{\EulerGamma@{a-b+1}\EulerGamma@{\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}a-b+1}}\hyperF@@{\tfrac{1}{2}a}{\tfrac{1}{2}a-b+\tfrac{1}{2}}{\tfrac{1}{2}}{\left(\frac{z+1}{z-1}\right)^{2}}+(1+z)(1-z)^{-a-1}\frac{\EulerGamma@{a-b+1}\EulerGamma@{-\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}a}\EulerGamma@{\tfrac{1}{2}a-b+\tfrac{1}{2}}}\hyperF@@{\tfrac{1}{2}a+\tfrac{1}{2}}{\tfrac{1}{2}a-b+1}{\tfrac{3}{2}}{\left(\frac{z+1}{z-1}\right)^{2}}	${\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\pi,\Re(a-b+1)>0,\Re(\tfrac{1}{2}a+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}a-b+1)>0,\Re(\tfrac{1}{2}a)>0,\Re(\tfrac{1}{2}a-b+\tfrac{1}{2})>0}}$	hypergeom([a, b], [a - b + 1], z) = (1 - z)^(- a)*(GAMMA(a - b + 1)*GAMMA((1)/(2)))/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*a - b + 1))*hypergeom([(1)/(2)*a, (1)/(2)*a - b +(1)/(2)], [(1)/(2)], ((z + 1)/(z - 1))^(2))+(1 + z)*(1 - z)^(- a - 1)*(GAMMA(a - b + 1)*GAMMA(-(1)/(2)))/(GAMMA((1)/(2)*a)*GAMMA((1)/(2)*a - b +(1)/(2)))*hypergeom([(1)/(2)*a +(1)/(2), (1)/(2)*a - b + 1], [(3)/(2)], ((z + 1)/(z - 1))^(2))	Hypergeometric2F1[a, b, a - b + 1, z] == (1 - z)^(- a)*Divide[Gamma[a - b + 1]*Gamma[Divide[1,2]],Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*a - b + 1]]*Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a - b +Divide[1,2], Divide[1,2], (Divide[z + 1,z - 1])^(2)]+(1 + z)*(1 - z)^(- a - 1)*Divide[Gamma[a - b + 1]*Gamma[-Divide[1,2]],Gamma[Divide[1,2]*a]*Gamma[Divide[1,2]*a - b +Divide[1,2]]]*Hypergeometric2F1[Divide[1,2]*a +Divide[1,2], Divide[1,2]*a - b + 1, Divide[3,2], (Divide[z + 1,z - 1])^(2)]	Failure	Failure	Error	Skip - No test values generated
15.8.E25	${\displaystyle{\displaystyle F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\frac{\Gamma\left(\tfrac{1}{2}(a+b+1)\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}F\left({\tfrac{1}{2}a,\tfrac{1}{2}b\atop\tfrac{1}{2}};(1-2z)^{2}\right)+(1-2z)\frac{\Gamma\left(\tfrac{1}{2}(a+b+1)\right)\Gamma\left(-\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}b\right)}F\left({\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}b+\tfrac{1}{2}\atop\tfrac{3}{2}};(1-2z)^{2}\right)}}$ \hyperF@@{a}{b}{\tfrac{1}{2}(a+b+1)}{z} = \frac{\EulerGamma@{\tfrac{1}{2}(a+b+1)}\EulerGamma@{\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}b+\tfrac{1}{2}}}\hyperF@@{\tfrac{1}{2}a}{\tfrac{1}{2}b}{\tfrac{1}{2}}{(1-2z)^{2}}+(1-2z)\frac{\EulerGamma@{\tfrac{1}{2}(a+b+1)}\EulerGamma@{-\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}a}\EulerGamma@{\tfrac{1}{2}b}}\hyperF@@{\tfrac{1}{2}a+\tfrac{1}{2}}{\tfrac{1}{2}b+\tfrac{1}{2}}{\tfrac{3}{2}}{(1-2z)^{2}}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)|<\pi,\Re(\tfrac{1}{2}(a+b+1))>0,\Re(\tfrac{1}{2}a+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}b+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}a)>0,\Re(\tfrac{1}{2}b)>0}}$	hypergeom([a, b], [(1)/(2)*(a + b + 1)], z) = (GAMMA((1)/(2)*(a + b + 1))*GAMMA((1)/(2)))/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*b +(1)/(2)))*hypergeom([(1)/(2)*a, (1)/(2)*b], [(1)/(2)], (1 - 2*z)^(2))+(1 - 2*z)*(GAMMA((1)/(2)*(a + b + 1))*GAMMA(-(1)/(2)))/(GAMMA((1)/(2)*a)*GAMMA((1)/(2)*b))*hypergeom([(1)/(2)*a +(1)/(2), (1)/(2)*b +(1)/(2)], [(3)/(2)], (1 - 2*z)^(2))	Hypergeometric2F1[a, b, Divide[1,2]*(a + b + 1), z] == Divide[Gamma[Divide[1,2]*(a + b + 1)]*Gamma[Divide[1,2]],Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*b +Divide[1,2]]]*Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*b, Divide[1,2], (1 - 2*z)^(2)]+(1 - 2*z)*Divide[Gamma[Divide[1,2]*(a + b + 1)]*Gamma[-Divide[1,2]],Gamma[Divide[1,2]*a]*Gamma[Divide[1,2]*b]]*Hypergeometric2F1[Divide[1,2]*a +Divide[1,2], Divide[1,2]*b +Divide[1,2], Divide[3,2], (1 - 2*z)^(2)]	Failure	Failure	Error	Skip - No test values generated
15.8.E26	${\displaystyle{\displaystyle F\left({a,1-a\atop c};z\right)=(1-z)^{c-1}\frac{\Gamma\left(c\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}(c-a+1)\right)\Gamma\left(\tfrac{1}{2}c+\tfrac{1}{2}a\right)}F\left({\tfrac{1}{2}c-\tfrac{1}{2}a,\tfrac{1}{2}c+\tfrac{1}{2}a-\tfrac{1}{2}\atop\tfrac{1}{2}};(1-2z)^{2}\right)+(1-2z)(1-z)^{c-1}\frac{\Gamma\left(c\right)\Gamma\left(-\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}c-\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}(c+a-1)\right)}F\left({\tfrac{1}{2}c-\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}c+\tfrac{1}{2}a\atop\tfrac{3}{2}};(1-2z)^{2}\right)}}$ \hyperF@@{a}{1-a}{c}{z} = (1-z)^{c-1}\frac{\EulerGamma@{c}\EulerGamma@{\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}(c-a+1)}\EulerGamma@{\tfrac{1}{2}c+\tfrac{1}{2}a}}\hyperF@@{\tfrac{1}{2}c-\tfrac{1}{2}a}{\tfrac{1}{2}c+\tfrac{1}{2}a-\tfrac{1}{2}}{\tfrac{1}{2}}{(1-2z)^{2}}+(1-2z)(1-z)^{c-1}\frac{\EulerGamma@{c}\EulerGamma@{-\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}c-\tfrac{1}{2}a}\EulerGamma@{\tfrac{1}{2}(c+a-1)}}\hyperF@@{\tfrac{1}{2}c-\tfrac{1}{2}a+\tfrac{1}{2}}{\tfrac{1}{2}c+\tfrac{1}{2}a}{\tfrac{3}{2}}{(1-2z)^{2}}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)|<\pi,\Re c>0,\Re(\tfrac{1}{2}(c-a+1))>0,\Re(\tfrac{1}{2}c+\tfrac{1}{2}a)>0,\Re(\tfrac{1}{2}c-\tfrac{1}{2}a)>0,\Re(\tfrac{1}{2}(c+a-1))>0}}$	hypergeom([a, 1 - a], [c], z) = (1 - z)^(c - 1)*(GAMMA(c)*GAMMA((1)/(2)))/(GAMMA((1)/(2)*(c - a + 1))*GAMMA((1)/(2)*c +(1)/(2)*a))*hypergeom([(1)/(2)*c -(1)/(2)*a, (1)/(2)*c +(1)/(2)*a -(1)/(2)], [(1)/(2)], (1 - 2*z)^(2))+(1 - 2*z)*(1 - z)^(c - 1)*(GAMMA(c)*GAMMA(-(1)/(2)))/(GAMMA((1)/(2)*c -(1)/(2)*a)*GAMMA((1)/(2)*(c + a - 1)))*hypergeom([(1)/(2)*c -(1)/(2)*a +(1)/(2), (1)/(2)*c +(1)/(2)*a], [(3)/(2)], (1 - 2*z)^(2))	Hypergeometric2F1[a, 1 - a, c, z] == (1 - z)^(c - 1)*Divide[Gamma[c]*Gamma[Divide[1,2]],Gamma[Divide[1,2]*(c - a + 1)]*Gamma[Divide[1,2]*c +Divide[1,2]*a]]*Hypergeometric2F1[Divide[1,2]*c -Divide[1,2]*a, Divide[1,2]*c +Divide[1,2]*a -Divide[1,2], Divide[1,2], (1 - 2*z)^(2)]+(1 - 2*z)*(1 - z)^(c - 1)*Divide[Gamma[c]*Gamma[-Divide[1,2]],Gamma[Divide[1,2]*c -Divide[1,2]*a]*Gamma[Divide[1,2]*(c + a - 1)]]*Hypergeometric2F1[Divide[1,2]*c -Divide[1,2]*a +Divide[1,2], Divide[1,2]*c +Divide[1,2]*a, Divide[3,2], (1 - 2*z)^(2)]	Failure	Failure	Error	Skip - No test values generated
15.8.E27	${\displaystyle{\displaystyle\frac{2\Gamma\left(\tfrac{1}{2}\right)\Gamma\left(a+b+\tfrac{1}{2}\right)}{\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left(b+\tfrac{1}{2}\right)}F\left(a,b;\tfrac{1}{2};z\right)=F\left(2a,2b;a+b+\tfrac{1}{2};\tfrac{1}{2}-\tfrac{1}{2}\sqrt{z}\right)+F\left(2a,2b;a+b+\tfrac{1}{2};\tfrac{1}{2}+\tfrac{1}{2}\sqrt{z}\right)}}$ \frac{2\EulerGamma@{\tfrac{1}{2}}\EulerGamma@{a+b+\tfrac{1}{2}}}{\EulerGamma@{a+\tfrac{1}{2}}\EulerGamma@{b+\tfrac{1}{2}}}\hyperF@{a}{b}{\tfrac{1}{2}}{z} = \hyperF@{2a}{2b}{a+b+\tfrac{1}{2}}{\tfrac{1}{2}-\tfrac{1}{2}\sqrt{z}}+\hyperF@{2a}{2b}{a+b+\tfrac{1}{2}}{\tfrac{1}{2}+\tfrac{1}{2}\sqrt{z}}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)|<\pi,\Re(a+b+\tfrac{1}{2})>0,\Re(a+\tfrac{1}{2})>0,\Re(b+\tfrac{1}{2})>0}}$	(2*GAMMA((1)/(2))*GAMMA(a + b +(1)/(2)))/(GAMMA(a +(1)/(2))*GAMMA(b +(1)/(2)))*hypergeom([a, b], [(1)/(2)], z) = hypergeom([2*a, 2*b], [a + b +(1)/(2)], (1)/(2)-(1)/(2)*sqrt(z))+ hypergeom([2*a, 2*b], [a + b +(1)/(2)], (1)/(2)+(1)/(2)*sqrt(z))	Divide[2*Gamma[Divide[1,2]]*Gamma[a + b +Divide[1,2]],Gamma[a +Divide[1,2]]*Gamma[b +Divide[1,2]]]*Hypergeometric2F1[a, b, Divide[1,2], z] == Hypergeometric2F1[2*a, 2*b, a + b +Divide[1,2], Divide[1,2]-Divide[1,2]*Sqrt[z]]+ Hypergeometric2F1[2*a, 2*b, a + b +Divide[1,2], Divide[1,2]+Divide[1,2]*Sqrt[z]]	Failure	Failure	Successful [Tested: 45]	Successful [Tested: 45]
15.8.E28	${\displaystyle{\displaystyle\frac{2\sqrt{z}\Gamma\left(-\tfrac{1}{2}\right)\Gamma\left(a+b-\tfrac{1}{2}\right)}{\Gamma\left(a-\tfrac{1}{2}\right)\Gamma\left(b-\tfrac{1}{2}\right)}F\left(a,b;\tfrac{3}{2};z\right)=F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}-\tfrac{1}{2}\sqrt{z}\right)-F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}+\tfrac{1}{2}\sqrt{z}\right)}}$ \frac{2\sqrt{z}\EulerGamma@{-\tfrac{1}{2}}\EulerGamma@{a+b-\tfrac{1}{2}}}{\EulerGamma@{a-\tfrac{1}{2}}\EulerGamma@{b-\tfrac{1}{2}}}\hyperF@{a}{b}{\tfrac{3}{2}}{z} = \hyperF@{2a-1}{2b-1}{a+b-\tfrac{1}{2}}{\tfrac{1}{2}-\tfrac{1}{2}\sqrt{z}}-\hyperF@{2a-1}{2b-1}{a+b-\tfrac{1}{2}}{\tfrac{1}{2}+\tfrac{1}{2}\sqrt{z}}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)|<\pi,\Re(a+b-\tfrac{1}{2})>0,\Re(a-\tfrac{1}{2})>0,\Re(b-\tfrac{1}{2})>0}}$	(2*sqrt(z)*GAMMA(-(1)/(2))*GAMMA(a + b -(1)/(2)))/(GAMMA(a -(1)/(2))*GAMMA(b -(1)/(2)))*hypergeom([a, b], [(3)/(2)], z) = hypergeom([2*a - 1, 2*b - 1], [a + b -(1)/(2)], (1)/(2)-(1)/(2)*sqrt(z))- hypergeom([2*a - 1, 2*b - 1], [a + b -(1)/(2)], (1)/(2)+(1)/(2)*sqrt(z))	Divide[2*Sqrt[z]*Gamma[-Divide[1,2]]*Gamma[a + b -Divide[1,2]],Gamma[a -Divide[1,2]]*Gamma[b -Divide[1,2]]]*Hypergeometric2F1[a, b, Divide[3,2], z] == Hypergeometric2F1[2*a - 1, 2*b - 1, a + b -Divide[1,2], Divide[1,2]-Divide[1,2]*Sqrt[z]]- Hypergeometric2F1[2*a - 1, 2*b - 1, a + b -Divide[1,2], Divide[1,2]+Divide[1,2]*Sqrt[z]]	Failure	Failure	Error	Skip - No test values generated
15.8.E29	${\displaystyle{\displaystyle F\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a+\tfrac{2}{3}};z\right)=\left(1+\sqrt{z}\right)^{-2a}\*F\left({a,\tfrac{2}{3}a+\tfrac{1}{6}\atop\tfrac{4}{3}a+\tfrac{1}{3}};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right)}}$ \hyperF@@{a}{\tfrac{1}{3}a+\tfrac{1}{3}}{\tfrac{2}{3}a+\tfrac{2}{3}}{z} = \left(1+\sqrt{z}\right)^{-2a}\*\hyperF@@{a}{\tfrac{2}{3}a+\tfrac{1}{6}}{\tfrac{4}{3}a+\tfrac{1}{3}}{\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}}		hypergeom([a, (1)/(3)*a +(1)/(3)], [(2)/(3)*a +(2)/(3)], z) = (1 +sqrt(z))^(- 2*a)* hypergeom([a, (2)/(3)*a +(1)/(6)], [(4)/(3)*a +(1)/(3)], (4*sqrt(z))/((1 +sqrt(z))^(2)))	Hypergeometric2F1[a, Divide[1,3]*a +Divide[1,3], Divide[2,3]*a +Divide[2,3], z] == (1 +Sqrt[z])^(- 2*a)* Hypergeometric2F1[a, Divide[2,3]*a +Divide[1,6], Divide[4,3]*a +Divide[1,3], Divide[4*Sqrt[z],(1 +Sqrt[z])^(2)]]	Failure	Failure	Failed [25 / 42] Result: .2121145592-.5120898515*I Test Values: {a = -3/2, z = 1/2*3^(1/2)+1/2*I} Result: 2.582409423-.3e-9*I Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [10 / 42] Result: Complex[-0.4773575227812281, -0.2756024942774353] Test Values: {Rule[a, -1.5], Rule[z, 1.5]} Result: Complex[-1.2380680865464244, -0.7147989430426637] Test Values: {Rule[a, -1.5], Rule[z, 2]} ... skip entries to safe data
15.8.E30	${\displaystyle{\displaystyle\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop\tfrac{1}{3}a+\tfrac{5}{6}};\left(\frac{z}{2-z}\right)^{2}\right)=F\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a+\tfrac{2}{3}};z\right)}}$ \left(1-\tfrac{1}{2}z\right)^{-a}\hyperF@@{\tfrac{1}{2}a}{\tfrac{1}{2}a+\tfrac{1}{2}}{\tfrac{1}{3}a+\tfrac{5}{6}}{\left(\frac{z}{2-z}\right)^{2}} = \hyperF@@{a}{\tfrac{1}{3}a+\tfrac{1}{3}}{\tfrac{2}{3}a+\tfrac{2}{3}}{z}		(1 -(1)/(2)*z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a +(1)/(2)], [(1)/(3)*a +(5)/(6)], ((z)/(2 - z))^(2)) = hypergeom([a, (1)/(3)*a +(1)/(3)], [(2)/(3)*a +(2)/(3)], z)	(1 -Divide[1,2]*z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a +Divide[1,2], Divide[1,3]*a +Divide[5,6], (Divide[z,2 - z])^(2)] == Hypergeometric2F1[a, Divide[1,3]*a +Divide[1,3], Divide[2,3]*a +Divide[2,3], z]	Failure	Failure	Failed [6 / 42] Result: Float(undefined)+Float(undefined)*I Test Values: {a = -3/2, z = 2} Result: Float(infinity)+Float(infinity)*I Test Values: {a = 3/2, z = 2} ... skip entries to safe data	Failed [6 / 42] Result: Complex[-0.7147989430426644, 0.7147989430426637] Test Values: {Rule[a, -1.5], Rule[z, 2]} Result: DirectedInfinity[] Test Values: {Rule[a, 1.5], Rule[z, 2]} ... skip entries to safe data
15.8.E30	${\displaystyle{\displaystyle F\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a+\tfrac{2}{3}};z\right)=(1+z)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop\tfrac{2}{3}a+\tfrac{2}{3}};\frac{4z}{(1+z)^{2}}\right)}}$ \hyperF@@{a}{\tfrac{1}{3}a+\tfrac{1}{3}}{\tfrac{2}{3}a+\tfrac{2}{3}}{z} = (1+z)^{-a}\hyperF@@{\tfrac{1}{2}a}{\tfrac{1}{2}a+\tfrac{1}{2}}{\tfrac{2}{3}a+\tfrac{2}{3}}{\frac{4z}{(1+z)^{2}}}		hypergeom([a, (1)/(3)*a +(1)/(3)], [(2)/(3)*a +(2)/(3)], z) = (1 + z)^(- a)* hypergeom([(1)/(2)*a, (1)/(2)*a +(1)/(2)], [(2)/(3)*a +(2)/(3)], (4*z)/((1 + z)^(2)))	Hypergeometric2F1[a, Divide[1,3]*a +Divide[1,3], Divide[2,3]*a +Divide[2,3], z] == (1 + z)^(- a)* Hypergeometric2F1[Divide[1,2]*a, Divide[1,2]*a +Divide[1,2], Divide[2,3]*a +Divide[2,3], Divide[4*z,(1 + z)^(2)]]	Failure	Failure	Failed [30 / 42] Result: .2121145619-.5120898515*I Test Values: {a = -3/2, z = 1/2*3^(1/2)+1/2*I} Result: 2.582409420-.7e-9*I Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [10 / 42] Result: Complex[-0.477357522781229, -0.2756024942774353] Test Values: {Rule[a, -1.5], Rule[z, 1.5]} Result: Complex[-1.238068086546428, -0.7147989430426637] Test Values: {Rule[a, -1.5], Rule[z, 2]} ... skip entries to safe data
15.8.E31	${\displaystyle{\displaystyle F\left({3a,3a+\frac{1}{2}\atop 4a+\frac{2}{3}};z\right)=\left(1-\tfrac{9}{8}z\right)^{-2a}\*F\left({a,a+\frac{1}{2}\atop 2a+\frac{5}{6}};\frac{27z^{2}(z-1)}{(9z-8)^{2}}\right)}}$ \hyperF@@{3a}{3a+\frac{1}{2}}{4a+\frac{2}{3}}{z} = \left(1-\tfrac{9}{8}z\right)^{-2a}\*\hyperF@@{a}{a+\frac{1}{2}}{2a+\frac{5}{6}}{\frac{27z^{2}(z-1)}{(9z-8)^{2}}}		hypergeom([3*a, 3*a +(1)/(2)], [4*a +(2)/(3)], z) = (1 -(9)/(8)*z)^(- 2*a)* hypergeom([a, a +(1)/(2)], [2*a +(5)/(6)], (27*(z)^(2)*(z - 1))/((9*z - 8)^(2)))	Hypergeometric2F1[3*a, 3*a +Divide[1,2], 4*a +Divide[2,3], z] == (1 -Divide[9,8]*z)^(- 2*a)* Hypergeometric2F1[a, a +Divide[1,2], 2*a +Divide[5,6], Divide[27*(z)^(2)*(z - 1),(9*z - 8)^(2)]]	Failure	Failure	Successful [Tested: 6]	Successful [Tested: 6]
15.8.E32	${\displaystyle{\displaystyle\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3a}}\left(\frac{1}{\Gamma\left(a+\frac{2}{3}\right)\Gamma\left(\frac{2}{3}\right)}F\left({a,a+\frac{1}{3}\atop\frac{2}{3}};z^{-3}\right)+\frac{e^{\frac{1}{3}\pi\mathrm{i}}}{z\Gamma\left(a\right)\Gamma\left(\frac{4}{3}\right)}F\left({a+\frac{1}{3},a+\frac{2}{3}\atop\frac{4}{3}};z^{-3}\right)\right)=\frac{3^{\frac{3}{2}a+\frac{1}{2}}e^{\frac{1}{2}a\pi\mathrm{i}}\Gamma\left(a+\frac{1}{3}\right)(1-\zeta)^{a}}{2\pi\Gamma\left(2a+\frac{2}{3}\right)(-\zeta)^{2a}}F\left({a+\frac{1}{3},3a\atop 2a+\frac{2}{3}};\zeta^{-1}\right)}}$ \frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3a}}\left(\frac{1}{\EulerGamma@{a+\frac{2}{3}}\EulerGamma@{\frac{2}{3}}}\hyperF@@{a}{a+\frac{1}{3}}{\frac{2}{3}}{z^{-3}}+\frac{e^{\frac{1}{3}\pi\iunit}}{z\EulerGamma@{a}\EulerGamma@{\frac{4}{3}}}\hyperF@@{a+\frac{1}{3}}{a+\frac{2}{3}}{\frac{4}{3}}{z^{-3}}\right) = \frac{3^{\frac{3}{2}a+\frac{1}{2}}e^{\frac{1}{2}a\pi\iunit}\EulerGamma@{a+\frac{1}{3}}(1-\zeta)^{a}}{2\pi\EulerGamma@{2a+\frac{2}{3}}(-\zeta)^{2a}}\hyperF@@{a+\frac{1}{3}}{3a}{2a+\frac{2}{3}}{\zeta^{-1}}	${\displaystyle{\displaystyle|z|>1,|\operatorname{ph}\left(-z\right)|<\frac{1}{3}\pi,\Re(a+\frac{2}{3})>0,\Re a>0,\Re(a+\frac{1}{3})>0,\Re(2a+\frac{2}{3})>0}}$	((1 - (z)^(3))^(a))/((- z)^(3*a))*((1)/(GAMMA(a +(2)/(3))*GAMMA((2)/(3)))*hypergeom([a, a +(1)/(3)], [(2)/(3)], (z)^(- 3))+(exp((1)/(3)*Pi*I))/(z*GAMMA(a)*GAMMA((4)/(3)))*hypergeom([a +(1)/(3), a +(2)/(3)], [(4)/(3)], (z)^(- 3))) = ((3)^((3)/(2)*a +(1)/(2))* exp((1)/(2)*a*Pi*I)*GAMMA(a +(1)/(3))*(1 - zeta)^(a))/(2*Pi*GAMMA(2*a +(2)/(3))*(- zeta)^(2*a))*hypergeom([a +(1)/(3), 3*a], [2*a +(2)/(3)], (zeta)^(- 1))	Divide[(1 - (z)^(3))^(a),(- z)^(3*a)]*(Divide[1,Gamma[a +Divide[2,3]]*Gamma[Divide[2,3]]]*Hypergeometric2F1[a, a +Divide[1,3], Divide[2,3], (z)^(- 3)]+Divide[Exp[Divide[1,3]*Pi*I],z*Gamma[a]*Gamma[Divide[4,3]]]*Hypergeometric2F1[a +Divide[1,3], a +Divide[2,3], Divide[4,3], (z)^(- 3)]) == Divide[(3)^(Divide[3,2]*a +Divide[1,2])* Exp[Divide[1,2]*a*Pi*I]*Gamma[a +Divide[1,3]]*(1 - \[Zeta])^(a),2*Pi*Gamma[2*a +Divide[2,3]]*(- \[Zeta])^(2*a)]*Hypergeometric2F1[a +Divide[1,3], 3*a, 2*a +Divide[2,3], \[Zeta]^(- 1)]	Failure	Failure	Error	Skip - No test values generated
15.8.E33	${\displaystyle{\displaystyle F\left({\frac{1}{3},\frac{2}{3}\atop 1};1-\left(\frac{1-z}{1+2z}\right)^{3}\right)=(1+2z)F\left({\frac{1}{3},\frac{2}{3}\atop 1};z^{3}\right)}}$ \hyperF@@{\frac{1}{3}}{\frac{2}{3}}{1}{1-\left(\frac{1-z}{1+2z}\right)^{3}} = (1+2z)\hyperF@@{\frac{1}{3}}{\frac{2}{3}}{1}{z^{3}}		hypergeom([(1)/(3), (2)/(3)], [1], 1 -((1 - z)/(1 + 2*z))^(3)) = (1 + 2*z)*hypergeom([(1)/(3), (2)/(3)], [1], (z)^(3))	Hypergeometric2F1[Divide[1,3], Divide[2,3], 1, 1 -(Divide[1 - z,1 + 2*z])^(3)] == (1 + 2*z)*Hypergeometric2F1[Divide[1,3], Divide[2,3], 1, (z)^(3)]	Failure	Failure	Failed [6 / 7] Result: .2094462e-2-1.732617448*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -.350667893-11.44453323*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [4 / 7] Result: Complex[0.23768141357499772, -1.326441364739111] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} Result: Complex[0.2791710117197028, 0.7366165529284218] Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]]} ... skip entries to safe data
15.9.E1	${\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}F\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right)}}$ \JacobipolyP{\alpha}{\beta}{n}@{x} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\hyperF@@{-n}{n+\alpha+\beta+1}{\alpha+1}{\frac{1-x}{2}}		JacobiP(n, alpha, beta, x) = (pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n, n + alpha + beta + 1], [alpha + 1], (1 - x)/(2))	JacobiP[n, \[Alpha], \[Beta], x] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*Hypergeometric2F1[- n, n + \[Alpha]+ \[Beta]+ 1, \[Alpha]+ 1, Divide[1 - x,2]]	Successful	Successful	-	Successful [Tested: 81]
15.9.E2	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}F\left({-n,n+2\lambda\atop\lambda+\frac{1}{2}};\frac{1-x}{2}\right)}}$ \ultrasphpoly{\lambda}{n}@{x} = \frac{\Pochhammersym{2\lambda}{n}}{n!}\hyperF@@{-n}{n+2\lambda}{\lambda+\frac{1}{2}}{\frac{1-x}{2}}		GegenbauerC(n, lambda, x) = (pochhammer(2*lambda, n))/(factorial(n))*hypergeom([- n, n + 2*lambda], [lambda +(1)/(2)], (1 - x)/(2))	GegenbauerC[n, \[Lambda], x] == Divide[Pochhammer[2*\[Lambda], n],(n)!]*Hypergeometric2F1[- n, n + 2*\[Lambda], \[Lambda]+Divide[1,2], Divide[1 - x,2]]	Successful	Successful	-	Failed [15 / 90] Result: 0.375 Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[λ, -1.5]} Result: 0.4375 Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -1.5]} ... skip entries to safe data
15.9.E3	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}F\left({-\frac{1}{2}n,\frac{1}{2}(1-n)\atop 1-\lambda-n};\frac{1}{x^{2}}\right)}}$ \ultrasphpoly{\lambda}{n}@{x} = (2x)^{n}\frac{\Pochhammersym{\lambda}{n}}{n!}\hyperF@@{-\frac{1}{2}n}{\frac{1}{2}(1-n)}{1-\lambda-n}{\frac{1}{x^{2}}}		GegenbauerC(n, lambda, x) = (2*x)^(n)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([-(1)/(2)*n, (1)/(2)*(1 - n)], [1 - lambda - n], (1)/((x)^(2)))	GegenbauerC[n, \[Lambda], x] == (2*x)^(n)*Divide[Pochhammer[\[Lambda], n],(n)!]*Hypergeometric2F1[-Divide[1,2]*n, Divide[1,2]*(1 - n), 1 - \[Lambda]- n, Divide[1,(x)^(2)]]	Failure	Failure	Failed [3 / 90] Result: Float(undefined)+Float(undefined)*I Test Values: {lambda = -2, x = 3/2, n = 3} Result: Float(undefined)+Float(undefined)*I Test Values: {lambda = -2, x = 1/2, n = 3} ... skip entries to safe data	Failed [3 / 90] Result: Indeterminate Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -2]} Result: Indeterminate Test Values: {Rule[n, 3], Rule[x, 0.5], Rule[λ, -2]} ... skip entries to safe data
15.9.E4	${\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta\right)=e^{n\mathrm{i}\theta}\frac{{\left(\lambda\right)_{n}}}{n!}F\left({-n,\lambda\atop 1-\lambda-n};e^{-2\mathrm{i}\theta}\right)}}$ \ultrasphpoly{\lambda}{n}@{\cos@@{\theta}} = e^{n\iunit\theta}\frac{\Pochhammersym{\lambda}{n}}{n!}\hyperF@@{-n}{\lambda}{1-\lambda-n}{e^{-2\iunit\theta}}		GegenbauerC(n, lambda, cos(theta)) = exp(n*I*theta)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([- n, lambda], [1 - lambda - n], exp(- 2*I*theta))	GegenbauerC[n, \[Lambda], Cos[\[Theta]]] == Exp[n*I*\[Theta]]*Divide[Pochhammer[\[Lambda], n],(n)!]*Hypergeometric2F1[- n, \[Lambda], 1 - \[Lambda]- n, Exp[- 2*I*\[Theta]]]	Failure	Failure	Failed [10 / 300] Result: Float(undefined)+Float(undefined)*I Test Values: {lambda = -2, theta = 1/2*3^(1/2)+1/2*I, n = 3} Result: Float(undefined)+Float(undefined)*I Test Values: {lambda = -2, theta = -1/2+1/2*I*3^(1/2), n = 3} ... skip entries to safe data	Failed [10 / 300] Result: Indeterminate Test Values: {Rule[n, 3], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[λ, -2]} Result: Indeterminate Test Values: {Rule[n, 3], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]], Rule[λ, -2]} ... skip entries to safe data
15.9.E5	${\displaystyle{\displaystyle T_{n}\left(x\right)=F\left({-n,n\atop\frac{1}{2}};\frac{1-x}{2}\right)}}$ \ChebyshevpolyT{n}@{x} = \hyperF@@{-n}{n}{\frac{1}{2}}{\frac{1-x}{2}}		ChebyshevT(n, x) = hypergeom([- n, n], [(1)/(2)], (1 - x)/(2))	ChebyshevT[n, x] == Hypergeometric2F1[- n, n, Divide[1,2], Divide[1 - x,2]]	Successful	Successful	-	Successful [Tested: 9]
15.9.E6	${\displaystyle{\displaystyle U_{n}\left(x\right)=(n+1)F\left({-n,n+2\atop\frac{3}{2}};\frac{1-x}{2}\right)}}$ \ChebyshevpolyU{n}@{x} = (n+1)\hyperF@@{-n}{n+2}{\frac{3}{2}}{\frac{1-x}{2}}		ChebyshevU(n, x) = (n + 1)*hypergeom([- n, n + 2], [(3)/(2)], (1 - x)/(2))	ChebyshevU[n, x] == (n + 1)*Hypergeometric2F1[- n, n + 2, Divide[3,2], Divide[1 - x,2]]	Successful	Failure	-	Successful [Tested: 9]
15.9.E7	${\displaystyle{\displaystyle P_{n}\left(x\right)=F\left({-n,n+1\atop 1};\frac{1-x}{2}\right)}}$ \LegendrepolyP{n}@{x} = \hyperF@@{-n}{n+1}{1}{\frac{1-x}{2}}		LegendreP(n, x) = hypergeom([- n, n + 1], [1], (1 - x)/(2))	LegendreP[n, x] == Hypergeometric2F1[- n, n + 1, 1, Divide[1 - x,2]]	Successful	Successful	-	Successful [Tested: 9]
15.9.E11	${\displaystyle{\displaystyle\phi^{(\alpha,\beta)}_{\lambda}\left(t\right)=F\left({\tfrac{1}{2}(\alpha+\beta+1-\mathrm{i}\lambda),\tfrac{1}{2}(\alpha+\beta+1+\mathrm{i}\lambda)\atop\alpha+1};-{\sinh^{2}}t\right)}}$ \Jacobiphi{\alpha}{\beta}{\lambda}@{t} = \hyperF@@{\tfrac{1}{2}(\alpha+\beta+1-\iunit\lambda)}{\tfrac{1}{2}(\alpha+\beta+1+\iunit\lambda)}{\alpha+1}{-\sinh^{2}@@{t}}		hypergeom([((alpha)+(beta)+1-I*(lambda))/2, ((alpha)+(beta)+1+I*(lambda))], [(alpha)+1], -sinh(t)^2) = hypergeom([(1)/(2)*(alpha + beta + 1 - I*lambda), (1)/(2)*(alpha + beta + 1 + I*lambda)], [alpha + 1], - (sinh(t))^(2))	Error	Failure	Missing Macro Error	Failed [288 / 300] Result: -.4877482336e-1+.1329197787e-1*I Test Values: {alpha = 3/2, beta = 3/2, lambda = 1/2*3^(1/2)+1/2*I, t = -3/2} Result: -.4877482336e-1+.1329197787e-1*I Test Values: {alpha = 3/2, beta = 3/2, lambda = 1/2*3^(1/2)+1/2*I, t = 3/2} ... skip entries to safe data	-
15.9.E15	${\displaystyle{\displaystyle C^{(\lambda)}_{\alpha}\left(z\right)=\frac{\Gamma\left(\alpha+2\lambda\right)}{\Gamma\left(2\lambda\right)\Gamma\left(\alpha+1\right)}F\left({-\alpha,\alpha+2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-z}{2}\right)}}$ \ultrasphpoly{\lambda}{\alpha}@{z} = \frac{\EulerGamma@{\alpha+2\lambda}}{\EulerGamma@{2\lambda}\EulerGamma@{\alpha+1}}\hyperF@@{-\alpha}{\alpha+2\lambda}{\lambda+\tfrac{1}{2}}{\frac{1-z}{2}}	${\displaystyle{\displaystyle\Re(\alpha+2\lambda)>0,\Re(2\lambda)>0,\Re(\alpha+1)>0}}$	GegenbauerC(alpha, lambda, z) = (GAMMA(alpha + 2*lambda))/(GAMMA(2*lambda)*GAMMA(alpha + 1))*hypergeom([- alpha, alpha + 2*lambda], [lambda +(1)/(2)], (1 - z)/(2))	GegenbauerC[\[Alpha], \[Lambda], z] == Divide[Gamma[\[Alpha]+ 2*\[Lambda]],Gamma[2*\[Lambda]]*Gamma[\[Alpha]+ 1]]*Hypergeometric2F1[- \[Alpha], \[Alpha]+ 2*\[Lambda], \[Lambda]+Divide[1,2], Divide[1 - z,2]]	Successful	Successful	-	Successful [Tested: 105]
15.9.E16	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop 2b};z\right)=\frac{\sqrt{\pi}}{\Gamma\left(b\right)}z^{-b+(\ifrac{1}{2})}(1-z)^{(b-a-(\ifrac{1}{2}))/2}\*P^{-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\frac{2-z}{2\sqrt{1-z}}\right)}}$ \hyperOlverF@@{a}{b}{2b}{z} = \frac{\sqrt{\pi}}{\EulerGamma@{b}}z^{-b+(\ifrac{1}{2})}(1-z)^{(b-a-(\ifrac{1}{2}))/2}\*\assLegendreP[-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{\frac{2-z}{2\sqrt{1-z}}}	${\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,|1-z|<1,\Re b>0,|z|<1,\Re((2b)+s)>0}}$	hypergeom([a, b], [2*b], z)/GAMMA(2*b) = (sqrt(Pi))/(GAMMA(b))*(z)^(- b +((1)/(2)))*(1 - z)^((b - a -((1)/(2)))/2)* LegendreP(a - b -((1)/(2)), - b +((1)/(2)), (2 - z)/(2*sqrt(1 - z)))	Hypergeometric2F1Regularized[a, b, 2*b, z] == Divide[Sqrt[Pi],Gamma[b]]*(z)^(- b +(Divide[1,2]))*(1 - z)^((b - a -(Divide[1,2]))/2)* LegendreP[a - b -(Divide[1,2]), - b +(Divide[1,2]), 3, Divide[2 - z,2*Sqrt[1 - z]]]	Failure	Failure	Successful [Tested: 6]	Successful [Tested: 18]
15.9.E17	${\displaystyle{\displaystyle\mathbf{F}\left({a,a+\tfrac{1}{2}\atop c};z\right)=2^{c-1}z^{\ifrac{(1-c)}{2}}(1-z)^{-a+(\ifrac{(c-1)}{2})}\*P^{1-c}_{2a-c}\left(\frac{1}{\sqrt{1-z}}\right)}}$ \hyperOlverF@@{a}{a+\tfrac{1}{2}}{c}{z} = 2^{c-1}z^{\ifrac{(1-c)}{2}}(1-z)^{-a+(\ifrac{(c-1)}{2})}\*\assLegendreP[1-c]{2a-c}@{\frac{1}{\sqrt{1-z}}}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)|<\pi,|z|<1,\Re(c+s)>0}}$	hypergeom([a, a +(1)/(2)], [c], z)/GAMMA(c) = (2)^(c - 1)* (z)^((1 - c)/(2))*(1 - z)^(- a +((c - 1)/(2)))* LegendreP(2*a - c, 1 - c, (1)/(sqrt(1 - z)))	Hypergeometric2F1Regularized[a, a +Divide[1,2], c, z] == (2)^(c - 1)* (z)^(Divide[1 - c,2])*(1 - z)^(- a +(Divide[c - 1,2]))* LegendreP[2*a - c, 1 - c, 3, Divide[1,Sqrt[1 - z]]]	Failure	Failure	Error	Successful [Tested: 180]
15.9.E18	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a+b+\tfrac{1}{2}};z\right)=2^{a+b-(\ifrac{1}{2})}(-z)^{(-a-b+(\ifrac{1}{2}))/2}\*P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{1-z}\right)}}$ \hyperOlverF@@{a}{b}{a+b+\tfrac{1}{2}}{z} = 2^{a+b-(\ifrac{1}{2})}(-z)^{(-a-b+(\ifrac{1}{2}))/2}\*\assLegendreP[-a-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{\sqrt{1-z}}	${\displaystyle{\displaystyle\left|\operatorname{ph}\left(-z\right)\right|<\pi,|z|<1,\Re((a+b+\tfrac{1}{2})+s)>0}}$	hypergeom([a, b], [a + b +(1)/(2)], z)/GAMMA(a + b +(1)/(2)) = (2)^(a + b -((1)/(2)))*(- z)^((- a - b +((1)/(2)))/2)* LegendreP(a - b -((1)/(2)), - a - b +((1)/(2)), sqrt(1 - z))	Hypergeometric2F1Regularized[a, b, a + b +Divide[1,2], z] == (2)^(a + b -(Divide[1,2]))*(- z)^((- a - b +(Divide[1,2]))/2)* LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[1,2]), 3, Sqrt[1 - z]]	Failure	Failure	Error	Successful [Tested: 144]
15.9.E19	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a-b+1};z\right)=z^{\ifrac{(b-a)}{2}}(1-z)^{-b}\*P^{b-a}_{-b}\left(\frac{1+z}{1-z}\right)}}$ \hyperOlverF@@{a}{b}{a-b+1}{z} = z^{\ifrac{(b-a)}{2}}(1-z)^{-b}\*\assLegendreP[b-a]{-b}@{\frac{1+z}{1-z}}	${\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)|<\pi,|z|<1,\Re((a-b+1)+s)>0}}$	hypergeom([a, b], [a - b + 1], z)/GAMMA(a - b + 1) = (z)^((b - a)/(2))*(1 - z)^(- b)* LegendreP(- b, b - a, (1 + z)/(1 - z))	Hypergeometric2F1Regularized[a, b, a - b + 1, z] == (z)^(Divide[b - a,2])*(1 - z)^(- b)* LegendreP[- b, b - a, 3, Divide[1 + z,1 - z]]	Successful	Failure	-	Successful [Tested: 180]
15.9.E20	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\left(-z(1-z)\right)^{\ifrac{(1-a-b)}{4}}\*P^{\ifrac{(1-a-b)}{2}}_{\ifrac{(a-b-1)}{2}}\left(1-2z\right)}}$ \hyperOlverF@@{a}{b}{\tfrac{1}{2}(a+b+1)}{z} = \left(-z(1-z)\right)^{\ifrac{(1-a-b)}{4}}\*\assLegendreP[\ifrac{(1-a-b)}{2}]{\ifrac{(a-b-1)}{2}}@{1-2z}	${\displaystyle{\displaystyle\left|\operatorname{ph}\left(-z\right)\right|<\pi,|z|<1,\Re((\tfrac{1}{2}(a+b+1))+s)>0}}$	hypergeom([a, b], [(1)/(2)*(a + b + 1)], z)/GAMMA((1)/(2)*(a + b + 1)) = (- z*(1 - z))^((1 - a - b)/(4))* LegendreP((a - b - 1)/(2), (1 - a - b)/(2), 1 - 2*z)	Hypergeometric2F1Regularized[a, b, Divide[1,2]*(a + b + 1), z] == (- z*(1 - z))^(Divide[1 - a - b,4])* LegendreP[Divide[a - b - 1,2], Divide[1 - a - b,2], 3, 1 - 2*z]	Failure	Failure	Error	Successful [Tested: 144]
15.9.E21	${\displaystyle{\displaystyle\mathbf{F}\left({a,1-a\atop c};z\right)=\left(\frac{-z}{1-z}\right)^{\ifrac{(1-c)}{2}}\*P^{1-c}_{-a}\left(1-2z\right)}}$ \hyperOlverF@@{a}{1-a}{c}{z} = \left(\frac{-z}{1-z}\right)^{\ifrac{(1-c)}{2}}\*\assLegendreP[1-c]{-a}@{1-2z}	${\displaystyle{\displaystyle\left|\operatorname{ph}\left(-z\right)\right|<\pi,|z|<1,\Re(c+s)>0}}$	hypergeom([a, 1 - a], [c], z)/GAMMA(c) = ((- z)/(1 - z))^((1 - c)/(2))* LegendreP(- a, 1 - c, 1 - 2*z)	Hypergeometric2F1Regularized[a, 1 - a, c, z] == (Divide[- z,1 - z])^(Divide[1 - c,2])* LegendreP[- a, 1 - c, 3, 1 - 2*z]	Failure	Successful	Error	-
15.9.E22	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}};z\right)=\frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left(b+\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{+\pi\mathrm{i}(a+b-(\ifrac{1}{2}))}P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(-\sqrt{z}\right)+P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{z}\right)\right)}}$ \hyperOlverF@@{a}{b}{\tfrac{1}{2}}{z} = \frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\EulerGamma@{a+\tfrac{1}{2}}\EulerGamma@{b+\tfrac{1}{2}}\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{+\pi\iunit(a+b-(\ifrac{1}{2}))}\assLegendreP[-a-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{-\sqrt{z}}+\assLegendreP[-a-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{\sqrt{z}}\right)	${\displaystyle{\displaystyle a\neq-\frac{1}{2},b\neq-\frac{1}{2},0<|\operatorname{ph}z|,|\operatorname{ph}z|<\pi,\Re(a+\tfrac{1}{2})>0,\Re(b+\tfrac{1}{2})>0,|z|<1,\Re((\tfrac{1}{2})+s)>0}}$	hypergeom([a, b], [(1)/(2)], z)/GAMMA((1)/(2)) = ((2)^(a + b -((3)/(2))))/(Pi)*GAMMA(a +(1)/(2))*GAMMA(b +(1)/(2))*(z - 1)^((- a - b +((1)/(2)))/2)*(exp(+ Pi*I*(a + b -((1)/(2))))*LegendreP(a - b -((1)/(2)), - a - b +((1)/(2)), -sqrt(z))+ LegendreP(a - b -((1)/(2)), - a - b +((1)/(2)), sqrt(z)))	Hypergeometric2F1Regularized[a, b, Divide[1,2], z] == Divide[(2)^(a + b -(Divide[3,2])),Pi]*Gamma[a +Divide[1,2]]*Gamma[b +Divide[1,2]]*(z - 1)^((- a - b +(Divide[1,2]))/2)*(Exp[+ Pi*I*(a + b -(Divide[1,2]))]*LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[1,2]), 3, -Sqrt[z]]+ LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[1,2]), 3, Sqrt[z]])	Failure	Failure	Error	Failed [10 / 36] Result: Complex[-0.8582540688970105, -2.787267603366778] Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]]} Result: Complex[-0.09762832897349609, -0.474497895465574] Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]]} ... skip entries to safe data
15.9.E22	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}};z\right)=\frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left(b+\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{-\pi\mathrm{i}(a+b-(\ifrac{1}{2}))}P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(-\sqrt{z}\right)+P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{z}\right)\right)}}$ \hyperOlverF@@{a}{b}{\tfrac{1}{2}}{z} = \frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\EulerGamma@{a+\tfrac{1}{2}}\EulerGamma@{b+\tfrac{1}{2}}\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{-\pi\iunit(a+b-(\ifrac{1}{2}))}\assLegendreP[-a-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{-\sqrt{z}}+\assLegendreP[-a-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{\sqrt{z}}\right)	${\displaystyle{\displaystyle a\neq-\frac{1}{2},b\neq-\frac{1}{2},0<|\operatorname{ph}z|,|\operatorname{ph}z|<\pi,\Re(a+\tfrac{1}{2})>0,\Re(b+\tfrac{1}{2})>0,|z|<1,\Re((\tfrac{1}{2})+s)>0}}$	hypergeom([a, b], [(1)/(2)], z)/GAMMA((1)/(2)) = ((2)^(a + b -((3)/(2))))/(Pi)*GAMMA(a +(1)/(2))*GAMMA(b +(1)/(2))*(z - 1)^((- a - b +((1)/(2)))/2)*(exp(- Pi*I*(a + b -((1)/(2))))*LegendreP(a - b -((1)/(2)), - a - b +((1)/(2)), -sqrt(z))+ LegendreP(a - b -((1)/(2)), - a - b +((1)/(2)), sqrt(z)))	Hypergeometric2F1Regularized[a, b, Divide[1,2], z] == Divide[(2)^(a + b -(Divide[3,2])),Pi]*Gamma[a +Divide[1,2]]*Gamma[b +Divide[1,2]]*(z - 1)^((- a - b +(Divide[1,2]))/2)*(Exp[- Pi*I*(a + b -(Divide[1,2]))]*LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[1,2]), 3, -Sqrt[z]]+ LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[1,2]), 3, Sqrt[z]])	Failure	Failure	Error	Failed [10 / 36] Result: Complex[1.7877768256534143, 6.989426464541403] Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.26682868759795453, 0.7163138167399228] Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.9.E23	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{3}{2}};z\right)=\frac{2^{a+b-(\ifrac{5}{2})}}{\pi\sqrt{z}}\Gamma\left(a-\tfrac{1}{2}\right)\Gamma\left(b-\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{3}{2}))/2}\*\left(e^{+\pi\mathrm{i}(a+b-(\ifrac{3}{2}))}P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}\left(-\sqrt{z}\right)-P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{z}\right)\right)}}$ \hyperOlverF@@{a}{b}{\tfrac{3}{2}}{z} = \frac{2^{a+b-(\ifrac{5}{2})}}{\pi\sqrt{z}}\EulerGamma@{a-\tfrac{1}{2}}\EulerGamma@{b-\tfrac{1}{2}}\*(z-1)^{(-a-b+(\ifrac{3}{2}))/2}\*\left(e^{+\pi\iunit(a+b-(\ifrac{3}{2}))}\assLegendreP[-a-b+(\ifrac{3}{2})]{a-b-(\ifrac{1}{2})}@{-\sqrt{z}}-\assLegendreP[-a-b+(\ifrac{3}{2})]{a-b-(\ifrac{1}{2})}@{\sqrt{z}}\right)	${\displaystyle{\displaystyle a\neq\frac{1}{2},b\neq\frac{1}{2},0<|\operatorname{ph}z|,|\operatorname{ph}z|<\pi,\Re(a-\tfrac{1}{2})>0,\Re(b-\tfrac{1}{2})>0,|z|<1,\Re((\tfrac{3}{2})+s)>0}}$	hypergeom([a, b], [(3)/(2)], z)/GAMMA((3)/(2)) = ((2)^(a + b -((5)/(2))))/(Pi*sqrt(z))*GAMMA(a -(1)/(2))*GAMMA(b -(1)/(2))*(z - 1)^((- a - b +((3)/(2)))/2)*(exp(+ Pi*I*(a + b -((3)/(2))))*LegendreP(a - b -((1)/(2)), - a - b +((3)/(2)), -sqrt(z))- LegendreP(a - b -((1)/(2)), - a - b +((3)/(2)), sqrt(z)))	Hypergeometric2F1Regularized[a, b, Divide[3,2], z] == Divide[(2)^(a + b -(Divide[5,2])),Pi*Sqrt[z]]*Gamma[a -Divide[1,2]]*Gamma[b -Divide[1,2]]*(z - 1)^((- a - b +(Divide[3,2]))/2)*(Exp[+ Pi*I*(a + b -(Divide[3,2]))]*LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[3,2]), 3, -Sqrt[z]]- LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[3,2]), 3, Sqrt[z]])	Failure	Failure	Error	Failed [4 / 16] Result: Complex[2.2779820596001903, -1.628954540775632] Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]]} Result: Complex[0.907830443893564, 0.19750251034857133] Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]]} ... skip entries to safe data
15.9.E23	${\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{3}{2}};z\right)=\frac{2^{a+b-(\ifrac{5}{2})}}{\pi\sqrt{z}}\Gamma\left(a-\tfrac{1}{2}\right)\Gamma\left(b-\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{3}{2}))/2}\*\left(e^{-\pi\mathrm{i}(a+b-(\ifrac{3}{2}))}P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}\left(-\sqrt{z}\right)-P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{z}\right)\right)}}$ \hyperOlverF@@{a}{b}{\tfrac{3}{2}}{z} = \frac{2^{a+b-(\ifrac{5}{2})}}{\pi\sqrt{z}}\EulerGamma@{a-\tfrac{1}{2}}\EulerGamma@{b-\tfrac{1}{2}}\*(z-1)^{(-a-b+(\ifrac{3}{2}))/2}\*\left(e^{-\pi\iunit(a+b-(\ifrac{3}{2}))}\assLegendreP[-a-b+(\ifrac{3}{2})]{a-b-(\ifrac{1}{2})}@{-\sqrt{z}}-\assLegendreP[-a-b+(\ifrac{3}{2})]{a-b-(\ifrac{1}{2})}@{\sqrt{z}}\right)	${\displaystyle{\displaystyle a\neq\frac{1}{2},b\neq\frac{1}{2},0<|\operatorname{ph}z|,|\operatorname{ph}z|<\pi,\Re(a-\tfrac{1}{2})>0,\Re(b-\tfrac{1}{2})>0,|z|<1,\Re((\tfrac{3}{2})+s)>0}}$	hypergeom([a, b], [(3)/(2)], z)/GAMMA((3)/(2)) = ((2)^(a + b -((5)/(2))))/(Pi*sqrt(z))*GAMMA(a -(1)/(2))*GAMMA(b -(1)/(2))*(z - 1)^((- a - b +((3)/(2)))/2)*(exp(- Pi*I*(a + b -((3)/(2))))*LegendreP(a - b -((1)/(2)), - a - b +((3)/(2)), -sqrt(z))- LegendreP(a - b -((1)/(2)), - a - b +((3)/(2)), sqrt(z)))	Hypergeometric2F1Regularized[a, b, Divide[3,2], z] == Divide[(2)^(a + b -(Divide[5,2])),Pi*Sqrt[z]]*Gamma[a -Divide[1,2]]*Gamma[b -Divide[1,2]]*(z - 1)^((- a - b +(Divide[3,2]))/2)*(Exp[- Pi*I*(a + b -(Divide[3,2]))]*LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[3,2]), 3, -Sqrt[z]]- LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[3,2]), 3, Sqrt[z]])	Failure	Failure	Error	Failed [4 / 16] Result: Complex[4.158519870861856, 2.5132294016879406] Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[1.2196744558627868, 0.17160454696174166] Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data