Hypergeometric Function - 15.8 Transformations of Variable

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
15.8.E1 𝐅 ⁑ ( a , b c ; z ) = ( 1 - z ) - a ⁒ 𝐅 ⁑ ( a , c - b c ; z z - 1 ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript 1 𝑧 π‘Ž scaled-hypergeometric-bold-F π‘Ž 𝑐 𝑏 𝑐 𝑧 𝑧 1 {\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}}		 | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 πœ‹ {\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 ( 1 - z ) - a ⁒ 𝐅 ⁑ ( a , c - b c ; z z - 1 ) = ( 1 - z ) - b ⁒ 𝐅 ⁑ ( c - a , b c ; z z - 1 ) superscript 1 𝑧 π‘Ž scaled-hypergeometric-bold-F π‘Ž 𝑐 𝑏 𝑐 𝑧 𝑧 1 superscript 1 𝑧 𝑏 scaled-hypergeometric-bold-F 𝑐 π‘Ž 𝑏 𝑐 𝑧 𝑧 1 {\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}}		 | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 πœ‹ {\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 ( 1 - z ) - b ⁒ 𝐅 ⁑ ( c - a , b c ; z z - 1 ) = ( 1 - z ) c - a - b ⁒ 𝐅 ⁑ ( c - a , c - b c ; z ) superscript 1 𝑧 𝑏 scaled-hypergeometric-bold-F 𝑐 π‘Ž 𝑏 𝑐 𝑧 𝑧 1 superscript 1 𝑧 𝑐 π‘Ž 𝑏 scaled-hypergeometric-bold-F 𝑐 π‘Ž 𝑐 𝑏 𝑐 𝑧 {\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}		 | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 πœ‹ {\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 sin ⁑ ( Ο€ ⁒ ( b - a ) ) Ο€ ⁒ 𝐅 ⁑ ( a , b c ; z ) = ( - z ) - a Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c - a ) ⁒ 𝐅 ⁑ ( a , a - c + 1 a - b + 1 ; 1 z ) - ( - z ) - b Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( c - b ) ⁒ 𝐅 ⁑ ( b , b - c + 1 b - a + 1 ; 1 z ) πœ‹ 𝑏 π‘Ž πœ‹ scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript 𝑧 π‘Ž Euler-Gamma 𝑏 Euler-Gamma 𝑐 π‘Ž scaled-hypergeometric-bold-F π‘Ž π‘Ž 𝑐 1 π‘Ž 𝑏 1 1 𝑧 superscript 𝑧 𝑏 Euler-Gamma π‘Ž Euler-Gamma 𝑐 𝑏 scaled-hypergeometric-bold-F 𝑏 𝑏 𝑐 1 𝑏 π‘Ž 1 1 𝑧 {\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}}		 | ph ⁑ ( - z ) | < Ο€ , β„œ ⁑ b > 0 , β„œ ⁑ ( c - a ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( c - b ) > 0 formulae-sequence phase 𝑧 πœ‹ formulae-sequence 𝑏 0 formulae-sequence 𝑐 π‘Ž 0 formulae-sequence π‘Ž 0 𝑐 𝑏 0 {\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 sin ⁑ ( Ο€ ⁒ ( b - a ) ) Ο€ ⁒ 𝐅 ⁑ ( a , b c ; z ) = ( 1 - z ) - a Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c - a ) ⁒ 𝐅 ⁑ ( a , c - b a - b + 1 ; 1 1 - z ) - ( 1 - z ) - b Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( c - b ) ⁒ 𝐅 ⁑ ( b , c - a b - a + 1 ; 1 1 - z ) πœ‹ 𝑏 π‘Ž πœ‹ scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript 1 𝑧 π‘Ž Euler-Gamma 𝑏 Euler-Gamma 𝑐 π‘Ž scaled-hypergeometric-bold-F π‘Ž 𝑐 𝑏 π‘Ž 𝑏 1 1 1 𝑧 superscript 1 𝑧 𝑏 Euler-Gamma π‘Ž Euler-Gamma 𝑐 𝑏 scaled-hypergeometric-bold-F 𝑏 𝑐 π‘Ž 𝑏 π‘Ž 1 1 1 𝑧 {\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}}		 | ph ⁑ ( - z ) | < Ο€ , β„œ ⁑ b > 0 , β„œ ⁑ ( c - a ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( c - b ) > 0 formulae-sequence phase 𝑧 πœ‹ formulae-sequence 𝑏 0 formulae-sequence 𝑐 π‘Ž 0 formulae-sequence π‘Ž 0 𝑐 𝑏 0 {\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 sin ⁑ ( Ο€ ⁒ ( c - a - b ) ) Ο€ ⁒ 𝐅 ⁑ ( a , b c ; z ) = 1 Ξ“ ⁑ ( c - a ) ⁒ Ξ“ ⁑ ( c - b ) ⁒ 𝐅 ⁑ ( a , b a + b - c + 1 ; 1 - z ) - ( 1 - z ) c - a - b Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ 𝐅 ⁑ ( c - a , c - b c - a - b + 1 ; 1 - z ) πœ‹ 𝑐 π‘Ž 𝑏 πœ‹ scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 1 Euler-Gamma 𝑐 π‘Ž Euler-Gamma 𝑐 𝑏 scaled-hypergeometric-bold-F π‘Ž 𝑏 π‘Ž 𝑏 𝑐 1 1 𝑧 superscript 1 𝑧 𝑐 π‘Ž 𝑏 Euler-Gamma π‘Ž Euler-Gamma 𝑏 scaled-hypergeometric-bold-F 𝑐 π‘Ž 𝑐 𝑏 𝑐 π‘Ž 𝑏 1 1 𝑧 {\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}		 | ph ⁑ z | < Ο€ , | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( c - a ) > 0 , β„œ ⁑ ( c - b ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ b > 0 , | z | < 1 , | ( 1 - z ) | < 1 , β„œ ⁑ ( c + s ) > 0 , β„œ ⁑ ( ( a + b - c + 1 ) + s ) > 0 , β„œ ⁑ ( ( c - a - b + 1 ) + s ) > 0 formulae-sequence phase 𝑧 πœ‹ formulae-sequence phase 1 𝑧 πœ‹ formulae-sequence 𝑐 π‘Ž 0 formulae-sequence 𝑐 𝑏 0 formulae-sequence π‘Ž 0 formulae-sequence 𝑏 0 formulae-sequence 𝑧 1 formulae-sequence 1 𝑧 1 formulae-sequence 𝑐 𝑠 0 formulae-sequence π‘Ž 𝑏 𝑐 1 𝑠 0 𝑐 π‘Ž 𝑏 1 𝑠 0 {\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 sin ⁑ ( Ο€ ⁒ ( c - a - b ) ) Ο€ ⁒ 𝐅 ⁑ ( a , b c ; z ) = z - a Ξ“ ⁑ ( c - a ) ⁒ Ξ“ ⁑ ( c - b ) ⁒ 𝐅 ⁑ ( a , a - c + 1 a + b - c + 1 ; 1 - 1 z ) - ( 1 - z ) c - a - b ⁒ z a - c Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ 𝐅 ⁑ ( c - a , 1 - a c - a - b + 1 ; 1 - 1 z ) πœ‹ 𝑐 π‘Ž 𝑏 πœ‹ scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript 𝑧 π‘Ž Euler-Gamma 𝑐 π‘Ž Euler-Gamma 𝑐 𝑏 scaled-hypergeometric-bold-F π‘Ž π‘Ž 𝑐 1 π‘Ž 𝑏 𝑐 1 1 1 𝑧 superscript 1 𝑧 𝑐 π‘Ž 𝑏 superscript 𝑧 π‘Ž 𝑐 Euler-Gamma π‘Ž Euler-Gamma 𝑏 scaled-hypergeometric-bold-F 𝑐 π‘Ž 1 π‘Ž 𝑐 π‘Ž 𝑏 1 1 1 𝑧 {\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}}		 | ph ⁑ z | < Ο€ , | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( c - a ) > 0 , β„œ ⁑ ( c - b ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ b > 0 formulae-sequence phase 𝑧 πœ‹ formulae-sequence phase 1 𝑧 πœ‹ formulae-sequence 𝑐 π‘Ž 0 formulae-sequence 𝑐 𝑏 0 formulae-sequence π‘Ž 0 𝑏 0 {\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 F ⁑ ( - m , b c ; z ) = ( b ) m ( c ) m ⁒ ( - z ) m ⁒ F ⁑ ( - m , 1 - c - m 1 - b - m ; 1 z ) Gauss-hypergeometric-F π‘š 𝑏 𝑐 𝑧 subscript 𝑏 π‘š subscript 𝑐 π‘š superscript 𝑧 π‘š Gauss-hypergeometric-F π‘š 1 𝑐 π‘š 1 𝑏 π‘š 1 𝑧 {\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 ( b ) m ( c ) m ⁒ ( - z ) m ⁒ F ⁑ ( - m , 1 - c - m 1 - b - m ; 1 z ) = ( b ) m ( c ) m ⁒ ( 1 - z ) m ⁒ F ⁑ ( - m , c - b 1 - b - m ; 1 1 - z ) subscript 𝑏 π‘š subscript 𝑐 π‘š superscript 𝑧 π‘š Gauss-hypergeometric-F π‘š 1 𝑐 π‘š 1 𝑏 π‘š 1 𝑧 subscript 𝑏 π‘š subscript 𝑐 π‘š superscript 1 𝑧 π‘š Gauss-hypergeometric-F π‘š 𝑐 𝑏 1 𝑏 π‘š 1 1 𝑧 {\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 F ⁑ ( - m , b c ; z ) = ( c - b ) m ( c ) m ⁒ F ⁑ ( - m , b b - c - m + 1 ; 1 - z ) Gauss-hypergeometric-F π‘š 𝑏 𝑐 𝑧 subscript 𝑐 𝑏 π‘š subscript 𝑐 π‘š Gauss-hypergeometric-F π‘š 𝑏 𝑏 𝑐 π‘š 1 1 𝑧 {\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 ( c - b ) m ( c ) m ⁒ F ⁑ ( - m , b b - c - m + 1 ; 1 - z ) = ( c - b ) m ( c ) m ⁒ z m ⁒ F ⁑ ( - m , 1 - c - m b - c - m + 1 ; 1 - 1 z ) subscript 𝑐 𝑏 π‘š subscript 𝑐 π‘š Gauss-hypergeometric-F π‘š 𝑏 𝑏 𝑐 π‘š 1 1 𝑧 subscript 𝑐 𝑏 π‘š subscript 𝑐 π‘š superscript 𝑧 π‘š Gauss-hypergeometric-F π‘š 1 𝑐 π‘š 𝑏 𝑐 π‘š 1 1 1 𝑧 {\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 𝐅 ⁑ ( a , a + m c ; z ) = ( - z ) - a Ξ“ ⁑ ( a + m ) ⁒ βˆ‘ k = 0 m - 1 ( a ) k ⁒ ( m - k - 1 ) ! k ! ⁒ Ξ“ ⁑ ( c - a - k ) ⁒ z - k + ( - z ) - a Ξ“ ⁑ ( a ) ⁒ βˆ‘ k = 0 ∞ ( a + m ) k k ! ⁒ ( k + m ) ! ⁒ Ξ“ ⁑ ( c - a - k - m ) ⁒ ( - 1 ) k ⁒ z - k - m ⁒ ( ln ⁑ ( - z ) + ψ ⁑ ( k + 1 ) + ψ ⁑ ( k + m + 1 ) - ψ ⁑ ( a + k + m ) - ψ ⁑ ( c - a - k - m ) ) scaled-hypergeometric-bold-F π‘Ž π‘Ž π‘š 𝑐 𝑧 superscript 𝑧 π‘Ž Euler-Gamma π‘Ž π‘š superscript subscript π‘˜ 0 π‘š 1 subscript π‘Ž π‘˜ π‘š π‘˜ 1 π‘˜ Euler-Gamma 𝑐 π‘Ž π‘˜ superscript 𝑧 π‘˜ superscript 𝑧 π‘Ž Euler-Gamma π‘Ž superscript subscript π‘˜ 0 subscript π‘Ž π‘š π‘˜ π‘˜ π‘˜ π‘š Euler-Gamma 𝑐 π‘Ž π‘˜ π‘š superscript 1 π‘˜ superscript 𝑧 π‘˜ π‘š 𝑧 digamma π‘˜ 1 digamma π‘˜ π‘š 1 digamma π‘Ž π‘˜ π‘š digamma 𝑐 π‘Ž π‘˜ π‘š {\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)		 | z | > 1 , | ph ⁑ ( - z ) | < Ο€ , β„œ ⁑ ( a + m ) > 0 , β„œ ⁑ ( c - a - k ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( c - a - k - m ) > 0 , | z | < 1 , β„œ ⁑ ( c + s ) > 0 formulae-sequence 𝑧 1 formulae-sequence phase 𝑧 πœ‹ formulae-sequence π‘Ž π‘š 0 formulae-sequence 𝑐 π‘Ž π‘˜ 0 formulae-sequence π‘Ž 0 formulae-sequence 𝑐 π‘Ž π‘˜ π‘š 0 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\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 𝐅 ⁑ ( a , a + m c ; z ) = ( 1 - z ) - a Ξ“ ⁑ ( a + m ) ⁒ Ξ“ ⁑ ( c - a ) ⁒ βˆ‘ k = 0 m - 1 ( a ) k ⁒ ( c - a - m ) k ⁒ ( m - k - 1 ) ! k ! ⁒ ( z - 1 ) - k + ( - 1 ) m ⁒ ( 1 - z ) - a - m Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( c - a - m ) ⁒ βˆ‘ k = 0 ∞ ( a + m ) k ⁒ ( c - a ) k k ! ⁒ ( k + m ) ! ⁒ ( 1 - z ) - k ⁒ ( ln ⁑ ( 1 - z ) + ψ ⁑ ( k + 1 ) + ψ ⁑ ( k + m + 1 ) - ψ ⁑ ( a + k + m ) - ψ ⁑ ( c - a + k ) ) scaled-hypergeometric-bold-F π‘Ž π‘Ž π‘š 𝑐 𝑧 superscript 1 𝑧 π‘Ž Euler-Gamma π‘Ž π‘š Euler-Gamma 𝑐 π‘Ž superscript subscript π‘˜ 0 π‘š 1 subscript π‘Ž π‘˜ subscript 𝑐 π‘Ž π‘š π‘˜ π‘š π‘˜ 1 π‘˜ superscript 𝑧 1 π‘˜ superscript 1 π‘š superscript 1 𝑧 π‘Ž π‘š Euler-Gamma π‘Ž Euler-Gamma 𝑐 π‘Ž π‘š superscript subscript π‘˜ 0 subscript π‘Ž π‘š π‘˜ subscript 𝑐 π‘Ž π‘˜ π‘˜ π‘˜ π‘š superscript 1 𝑧 π‘˜ 1 𝑧 digamma π‘˜ 1 digamma π‘˜ π‘š 1 digamma π‘Ž π‘˜ π‘š digamma 𝑐 π‘Ž π‘˜ {\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})		 | z - 1 | > 1 , | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( a + m ) > 0 , β„œ ⁑ ( c - a ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( c - a - m ) > 0 , | z | < 1 , β„œ ⁑ ( c + s ) > 0 formulae-sequence 𝑧 1 1 formulae-sequence phase 1 𝑧 πœ‹ formulae-sequence π‘Ž π‘š 0 formulae-sequence 𝑐 π‘Ž 0 formulae-sequence π‘Ž 0 formulae-sequence 𝑐 π‘Ž π‘š 0 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\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 𝐅 ⁑ ( a , b a + b + m ; z ) = 1 Ξ“ ⁑ ( a + m ) ⁒ Ξ“ ⁑ ( b + m ) ⁒ βˆ‘ k = 0 m - 1 ( a ) k ⁒ ( b ) k ⁒ ( m - k - 1 ) ! k ! ⁒ ( z - 1 ) k - ( z - 1 ) m Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ βˆ‘ k = 0 ∞ ( a + m ) k ⁒ ( b + m ) k k ! ⁒ ( k + m ) ! ⁒ ( 1 - z ) k ⁒ ( ln ⁑ ( 1 - z ) - ψ ⁑ ( k + 1 ) - ψ ⁑ ( k + m + 1 ) + ψ ⁑ ( a + k + m ) + ψ ⁑ ( b + k + m ) ) scaled-hypergeometric-bold-F π‘Ž 𝑏 π‘Ž 𝑏 π‘š 𝑧 1 Euler-Gamma π‘Ž π‘š Euler-Gamma 𝑏 π‘š superscript subscript π‘˜ 0 π‘š 1 subscript π‘Ž π‘˜ subscript 𝑏 π‘˜ π‘š π‘˜ 1 π‘˜ superscript 𝑧 1 π‘˜ superscript 𝑧 1 π‘š Euler-Gamma π‘Ž Euler-Gamma 𝑏 superscript subscript π‘˜ 0 subscript π‘Ž π‘š π‘˜ subscript 𝑏 π‘š π‘˜ π‘˜ π‘˜ π‘š superscript 1 𝑧 π‘˜ 1 𝑧 digamma π‘˜ 1 digamma π‘˜ π‘š 1 digamma π‘Ž π‘˜ π‘š digamma 𝑏 π‘˜ π‘š {\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)		 | z - 1 | < 1 , | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( a + m ) > 0 , β„œ ⁑ ( b + m ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ b > 0 , | z | < 1 , β„œ ⁑ ( ( a + b + m ) + s ) > 0 formulae-sequence 𝑧 1 1 formulae-sequence phase 1 𝑧 πœ‹ formulae-sequence π‘Ž π‘š 0 formulae-sequence 𝑏 π‘š 0 formulae-sequence π‘Ž 0 formulae-sequence 𝑏 0 formulae-sequence 𝑧 1 π‘Ž 𝑏 π‘š 𝑠 0 {\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 𝐅 ⁑ ( a , b a + b + m ; z ) = z - a Ξ“ ⁑ ( a + m ) ⁒ βˆ‘ k = 0 m - 1 ( a ) k ⁒ ( m - k - 1 ) ! k ! ⁒ Ξ“ ⁑ ( b + m - k ) ⁒ ( 1 - 1 z ) k - z - a Ξ“ ⁑ ( a ) ⁒ βˆ‘ k = 0 ∞ ( a + m ) k k ! ⁒ ( k + m ) ! ⁒ Ξ“ ⁑ ( b - k ) ⁒ ( - 1 ) k ⁒ ( 1 - 1 z ) k + m ⁒ ( ln ⁑ ( 1 - z z ) - ψ ⁑ ( k + 1 ) - ψ ⁑ ( k + m + 1 ) + ψ ⁑ ( a + k + m ) + ψ ⁑ ( b - k ) ) scaled-hypergeometric-bold-F π‘Ž 𝑏 π‘Ž 𝑏 π‘š 𝑧 superscript 𝑧 π‘Ž Euler-Gamma π‘Ž π‘š superscript subscript π‘˜ 0 π‘š 1 subscript π‘Ž π‘˜ π‘š π‘˜ 1 π‘˜ Euler-Gamma 𝑏 π‘š π‘˜ superscript 1 1 𝑧 π‘˜ superscript 𝑧 π‘Ž Euler-Gamma π‘Ž superscript subscript π‘˜ 0 subscript π‘Ž π‘š π‘˜ π‘˜ π‘˜ π‘š Euler-Gamma 𝑏 π‘˜ superscript 1 π‘˜ superscript 1 1 𝑧 π‘˜ π‘š 1 𝑧 𝑧 digamma π‘˜ 1 digamma π‘˜ π‘š 1 digamma π‘Ž π‘˜ π‘š digamma 𝑏 π‘˜ {\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)		 β„œ ⁑ z > 1 2 , | ph ⁑ z | < Ο€ , | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( a + m ) > 0 , β„œ ⁑ ( b + m - k ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( b - k ) > 0 , | z | < 1 , β„œ ⁑ ( ( a + b + m ) + s ) > 0 formulae-sequence 𝑧 1 2 formulae-sequence phase 𝑧 πœ‹ formulae-sequence phase 1 𝑧 πœ‹ formulae-sequence π‘Ž π‘š 0 formulae-sequence 𝑏 π‘š π‘˜ 0 formulae-sequence π‘Ž 0 formulae-sequence 𝑏 π‘˜ 0 formulae-sequence 𝑧 1 π‘Ž 𝑏 π‘š 𝑠 0 {\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 F ⁑ ( a , b 2 ⁒ b ; z ) = ( 1 - 1 2 ⁒ z ) - a ⁒ F ⁑ ( 1 2 ⁒ a , 1 2 ⁒ a + 1 2 b + 1 2 ; ( z 2 - z ) 2 ) Gauss-hypergeometric-F π‘Ž 𝑏 2 𝑏 𝑧 superscript 1 1 2 𝑧 π‘Ž Gauss-hypergeometric-F 1 2 π‘Ž 1 2 π‘Ž 1 2 𝑏 1 2 superscript 𝑧 2 𝑧 2 {\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}}		 | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 πœ‹ {\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 F ⁑ ( a , b 2 ⁒ b ; z ) = ( 1 - z ) - a / 2 ⁒ F ⁑ ( 1 2 ⁒ a , b - 1 2 ⁒ a b + 1 2 ; z 2 4 ⁒ z - 4 ) Gauss-hypergeometric-F π‘Ž 𝑏 2 𝑏 𝑧 superscript 1 𝑧 π‘Ž 2 Gauss-hypergeometric-F 1 2 π‘Ž 𝑏 1 2 π‘Ž 𝑏 1 2 superscript 𝑧 2 4 𝑧 4 {\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}}		 | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 πœ‹ {\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 F ⁑ ( a , b a - b + 1 ; z ) = ( 1 + z ) - a ⁒ F ⁑ ( 1 2 ⁒ a , 1 2 ⁒ a + 1 2 a - b + 1 ; 4 ⁒ z ( 1 + z ) 2 ) Gauss-hypergeometric-F π‘Ž 𝑏 π‘Ž 𝑏 1 𝑧 superscript 1 𝑧 π‘Ž Gauss-hypergeometric-F 1 2 π‘Ž 1 2 π‘Ž 1 2 π‘Ž 𝑏 1 4 𝑧 superscript 1 𝑧 2 {\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}}}		 | z | < 1 𝑧 1 {\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 F ⁑ ( a , b a - b + 1 ; z ) = ( 1 - z ) - a ⁒ F ⁑ ( 1 2 ⁒ a , 1 2 ⁒ a - b + 1 2 a - b + 1 ; - 4 ⁒ z ( 1 - z ) 2 ) Gauss-hypergeometric-F π‘Ž 𝑏 π‘Ž 𝑏 1 𝑧 superscript 1 𝑧 π‘Ž Gauss-hypergeometric-F 1 2 π‘Ž 1 2 π‘Ž 𝑏 1 2 π‘Ž 𝑏 1 4 𝑧 superscript 1 𝑧 2 {\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}}}		 | z | < 1 𝑧 1 {\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 F ⁑ ( a , b 1 2 ⁒ ( a + b + 1 ) ; z ) = ( 1 - 2 ⁒ z ) - a ⁒ F ⁑ ( 1 2 ⁒ a , 1 2 ⁒ a + 1 2 1 2 ⁒ ( a + b + 1 ) ; 4 ⁒ z ⁒ ( z - 1 ) ( 1 - 2 ⁒ z ) 2 ) Gauss-hypergeometric-F π‘Ž 𝑏 1 2 π‘Ž 𝑏 1 𝑧 superscript 1 2 𝑧 π‘Ž Gauss-hypergeometric-F 1 2 π‘Ž 1 2 π‘Ž 1 2 1 2 π‘Ž 𝑏 1 4 𝑧 𝑧 1 superscript 1 2 𝑧 2 {\displaystyle{\displaystyle F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)=(1-2% z)^{-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 F ⁑ ( a , b 1 2 ⁒ ( a + b + 1 ) ; z ) = F ⁑ ( 1 2 ⁒ a , 1 2 ⁒ b 1 2 ⁒ ( a + b + 1 ) ; 4 ⁒ z ⁒ ( 1 - z ) ) Gauss-hypergeometric-F π‘Ž 𝑏 1 2 π‘Ž 𝑏 1 𝑧 Gauss-hypergeometric-F 1 2 π‘Ž 1 2 𝑏 1 2 π‘Ž 𝑏 1 4 𝑧 1 𝑧 {\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 F ⁑ ( a , 1 - a c ; z ) = ( 1 - 2 ⁒ z ) 1 - a - c ⁒ ( 1 - z ) c - 1 ⁒ F ⁑ ( 1 2 ⁒ ( a + c ) , 1 2 ⁒ ( a + c - 1 ) c ; 4 ⁒ z ⁒ ( z - 1 ) ( 1 - 2 ⁒ z ) 2 ) Gauss-hypergeometric-F π‘Ž 1 π‘Ž 𝑐 𝑧 superscript 1 2 𝑧 1 π‘Ž 𝑐 superscript 1 𝑧 𝑐 1 Gauss-hypergeometric-F 1 2 π‘Ž 𝑐 1 2 π‘Ž 𝑐 1 𝑐 4 𝑧 𝑧 1 superscript 1 2 𝑧 2 {\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)}{(1-2% z)^{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 F ⁑ ( a , 1 - a c ; z ) = ( 1 - z ) c - 1 ⁒ F ⁑ ( 1 2 ⁒ ( c - a ) , 1 2 ⁒ ( a + c - 1 ) c ; 4 ⁒ z ⁒ ( 1 - z ) ) Gauss-hypergeometric-F π‘Ž 1 π‘Ž 𝑐 𝑧 superscript 1 𝑧 𝑐 1 Gauss-hypergeometric-F 1 2 𝑐 π‘Ž 1 2 π‘Ž 𝑐 1 𝑐 4 𝑧 1 𝑧 {\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 F ⁑ ( a , b a - b + 1 ; z ) = ( 1 + z ) - 2 ⁒ a ⁒ F ⁑ ( a , a - b + 1 2 2 ⁒ a - 2 ⁒ b + 1 ; 4 ⁒ z ( 1 + z ) 2 ) Gauss-hypergeometric-F π‘Ž 𝑏 π‘Ž 𝑏 1 𝑧 superscript 1 𝑧 2 π‘Ž Gauss-hypergeometric-F π‘Ž π‘Ž 𝑏 1 2 2 π‘Ž 2 𝑏 1 4 𝑧 superscript 1 𝑧 2 {\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}}}		 | ph ⁑ z | < Ο€ , | z | < 1 formulae-sequence phase 𝑧 πœ‹ 𝑧 1 {\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 F ⁑ ( a , b 1 2 ⁒ ( a + b + 1 ) ; z ) = ( 1 - z - 1 - 1 1 - z - 1 + 1 ) a ⁒ F ⁑ ( a , 1 2 ⁒ ( a + b ) a + b ; 4 ⁒ 1 - z - 1 ( 1 - z - 1 + 1 ) 2 ) Gauss-hypergeometric-F π‘Ž 𝑏 1 2 π‘Ž 𝑏 1 𝑧 superscript 1 superscript 𝑧 1 1 1 superscript 𝑧 1 1 π‘Ž Gauss-hypergeometric-F π‘Ž 1 2 π‘Ž 𝑏 π‘Ž 𝑏 4 1 superscript 𝑧 1 superscript 1 superscript 𝑧 1 1 2 {\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}}}		 | ph ⁑ ( - z ) | < Ο€ phase 𝑧 πœ‹ {\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 F ⁑ ( a , 1 - a c ; z ) = ( 1 - z - 1 - 1 ) 1 - a ⁒ ( 1 - z - 1 + 1 ) a - 2 ⁒ c + 1 ⁒ ( 1 - z - 1 ) c - 1 ⁒ F ⁑ ( c - a , c - 1 2 2 ⁒ c - 1 ; 4 ⁒ 1 - z - 1 ( 1 - z - 1 + 1 ) 2 ) Gauss-hypergeometric-F π‘Ž 1 π‘Ž 𝑐 𝑧 superscript 1 superscript 𝑧 1 1 1 π‘Ž superscript 1 superscript 𝑧 1 1 π‘Ž 2 𝑐 1 superscript 1 superscript 𝑧 1 𝑐 1 Gauss-hypergeometric-F 𝑐 π‘Ž 𝑐 1 2 2 𝑐 1 4 1 superscript 𝑧 1 superscript 1 superscript 𝑧 1 1 2 {\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}}}		 | ph ⁑ ( - z ) | < Ο€ phase 𝑧 πœ‹ {\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 F ⁑ ( a , b a - b + 1 ; z ) = ( 1 - z ) - a ⁒ Ξ“ ⁑ ( a - b + 1 ) ⁒ Ξ“ ⁑ ( 1 2 ) Ξ“ ⁑ ( 1 2 ⁒ a + 1 2 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ a - b + 1 ) ⁒ F ⁑ ( 1 2 ⁒ a , 1 2 ⁒ a - b + 1 2 1 2 ; ( z + 1 z - 1 ) 2 ) + ( 1 + z ) ⁒ ( 1 - z ) - a - 1 ⁒ Ξ“ ⁑ ( a - b + 1 ) ⁒ Ξ“ ⁑ ( - 1 2 ) Ξ“ ⁑ ( 1 2 ⁒ a ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ a - b + 1 2 ) ⁒ F ⁑ ( 1 2 ⁒ a + 1 2 , 1 2 ⁒ a - b + 1 3 2 ; ( z + 1 z - 1 ) 2 ) Gauss-hypergeometric-F π‘Ž 𝑏 π‘Ž 𝑏 1 𝑧 superscript 1 𝑧 π‘Ž Euler-Gamma π‘Ž 𝑏 1 Euler-Gamma 1 2 Euler-Gamma 1 2 π‘Ž 1 2 Euler-Gamma 1 2 π‘Ž 𝑏 1 Gauss-hypergeometric-F 1 2 π‘Ž 1 2 π‘Ž 𝑏 1 2 1 2 superscript 𝑧 1 𝑧 1 2 1 𝑧 superscript 1 𝑧 π‘Ž 1 Euler-Gamma π‘Ž 𝑏 1 Euler-Gamma 1 2 Euler-Gamma 1 2 π‘Ž Euler-Gamma 1 2 π‘Ž 𝑏 1 2 Gauss-hypergeometric-F 1 2 π‘Ž 1 2 1 2 π‘Ž 𝑏 1 3 2 superscript 𝑧 1 𝑧 1 2 {\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}}		 | ph ⁑ ( - z ) | < Ο€ , β„œ ⁑ ( a - b + 1 ) > 0 , β„œ ⁑ ( 1 2 ⁒ a + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ a - b + 1 ) > 0 , β„œ ⁑ ( 1 2 ⁒ a ) > 0 , β„œ ⁑ ( 1 2 ⁒ a - b + 1 2 ) > 0 formulae-sequence phase 𝑧 πœ‹ formulae-sequence π‘Ž 𝑏 1 0 formulae-sequence 1 2 π‘Ž 1 2 0 formulae-sequence 1 2 π‘Ž 𝑏 1 0 formulae-sequence 1 2 π‘Ž 0 1 2 π‘Ž 𝑏 1 2 0 {\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 F ⁑ ( a , b 1 2 ⁒ ( a + b + 1 ) ; z ) = Ξ“ ⁑ ( 1 2 ⁒ ( a + b + 1 ) ) ⁒ Ξ“ ⁑ ( 1 2 ) Ξ“ ⁑ ( 1 2 ⁒ a + 1 2 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ b + 1 2 ) ⁒ F ⁑ ( 1 2 ⁒ a , 1 2 ⁒ b 1 2 ; ( 1 - 2 ⁒ z ) 2 ) + ( 1 - 2 ⁒ z ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ ( a + b + 1 ) ) ⁒ Ξ“ ⁑ ( - 1 2 ) Ξ“ ⁑ ( 1 2 ⁒ a ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ b ) ⁒ F ⁑ ( 1 2 ⁒ a + 1 2 , 1 2 ⁒ b + 1 2 3 2 ; ( 1 - 2 ⁒ z ) 2 ) Gauss-hypergeometric-F π‘Ž 𝑏 1 2 π‘Ž 𝑏 1 𝑧 Euler-Gamma 1 2 π‘Ž 𝑏 1 Euler-Gamma 1 2 Euler-Gamma 1 2 π‘Ž 1 2 Euler-Gamma 1 2 𝑏 1 2 Gauss-hypergeometric-F 1 2 π‘Ž 1 2 𝑏 1 2 superscript 1 2 𝑧 2 1 2 𝑧 Euler-Gamma 1 2 π‘Ž 𝑏 1 Euler-Gamma 1 2 Euler-Gamma 1 2 π‘Ž Euler-Gamma 1 2 𝑏 Gauss-hypergeometric-F 1 2 π‘Ž 1 2 1 2 𝑏 1 2 3 2 superscript 1 2 𝑧 2 {\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}}		 | ph ⁑ z | < Ο€ , | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( 1 2 ⁒ ( a + b + 1 ) ) > 0 , β„œ ⁑ ( 1 2 ⁒ a + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ b + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ a ) > 0 , β„œ ⁑ ( 1 2 ⁒ b ) > 0 formulae-sequence phase 𝑧 πœ‹ formulae-sequence phase 1 𝑧 πœ‹ formulae-sequence 1 2 π‘Ž 𝑏 1 0 formulae-sequence 1 2 π‘Ž 1 2 0 formulae-sequence 1 2 𝑏 1 2 0 formulae-sequence 1 2 π‘Ž 0 1 2 𝑏 0 {\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 F ⁑ ( a , 1 - a c ; z ) = ( 1 - z ) c - 1 ⁒ Ξ“ ⁑ ( c ) ⁒ Ξ“ ⁑ ( 1 2 ) Ξ“ ⁑ ( 1 2 ⁒ ( c - a + 1 ) ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ c + 1 2 ⁒ a ) ⁒ F ⁑ ( 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 ⁒ Ξ“ ⁑ ( c ) ⁒ Ξ“ ⁑ ( - 1 2 ) Ξ“ ⁑ ( 1 2 ⁒ c - 1 2 ⁒ a ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ ( c + a - 1 ) ) ⁒ F ⁑ ( 1 2 ⁒ c - 1 2 ⁒ a + 1 2 , 1 2 ⁒ c + 1 2 ⁒ a 3 2 ; ( 1 - 2 ⁒ z ) 2 ) Gauss-hypergeometric-F π‘Ž 1 π‘Ž 𝑐 𝑧 superscript 1 𝑧 𝑐 1 Euler-Gamma 𝑐 Euler-Gamma 1 2 Euler-Gamma 1 2 𝑐 π‘Ž 1 Euler-Gamma 1 2 𝑐 1 2 π‘Ž Gauss-hypergeometric-F 1 2 𝑐 1 2 π‘Ž 1 2 𝑐 1 2 π‘Ž 1 2 1 2 superscript 1 2 𝑧 2 1 2 𝑧 superscript 1 𝑧 𝑐 1 Euler-Gamma 𝑐 Euler-Gamma 1 2 Euler-Gamma 1 2 𝑐 1 2 π‘Ž Euler-Gamma 1 2 𝑐 π‘Ž 1 Gauss-hypergeometric-F 1 2 𝑐 1 2 π‘Ž 1 2 1 2 𝑐 1 2 π‘Ž 3 2 superscript 1 2 𝑧 2 {\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}}		 | ph ⁑ z | < Ο€ , | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ c > 0 , β„œ ⁑ ( 1 2 ⁒ ( c - a + 1 ) ) > 0 , β„œ ⁑ ( 1 2 ⁒ c + 1 2 ⁒ a ) > 0 , β„œ ⁑ ( 1 2 ⁒ c - 1 2 ⁒ a ) > 0 , β„œ ⁑ ( 1 2 ⁒ ( c + a - 1 ) ) > 0 formulae-sequence phase 𝑧 πœ‹ formulae-sequence phase 1 𝑧 πœ‹ formulae-sequence 𝑐 0 formulae-sequence 1 2 𝑐 π‘Ž 1 0 formulae-sequence 1 2 𝑐 1 2 π‘Ž 0 formulae-sequence 1 2 𝑐 1 2 π‘Ž 0 1 2 𝑐 π‘Ž 1 0 {\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 2 ⁒ Ξ“ ⁑ ( 1 2 ) ⁒ Ξ“ ⁑ ( a + b + 1 2 ) Ξ“ ⁑ ( a + 1 2 ) ⁒ Ξ“ ⁑ ( b + 1 2 ) ⁒ F ⁑ ( a , b ; 1 2 ; z ) = F ⁑ ( 2 ⁒ a , 2 ⁒ b ; a + b + 1 2 ; 1 2 - 1 2 ⁒ z ) + F ⁑ ( 2 ⁒ a , 2 ⁒ b ; a + b + 1 2 ; 1 2 + 1 2 ⁒ z ) 2 Euler-Gamma 1 2 Euler-Gamma π‘Ž 𝑏 1 2 Euler-Gamma π‘Ž 1 2 Euler-Gamma 𝑏 1 2 Gauss-hypergeometric-F π‘Ž 𝑏 1 2 𝑧 Gauss-hypergeometric-F 2 π‘Ž 2 𝑏 π‘Ž 𝑏 1 2 1 2 1 2 𝑧 Gauss-hypergeometric-F 2 π‘Ž 2 𝑏 π‘Ž 𝑏 1 2 1 2 1 2 𝑧 {\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}}		 | ph ⁑ z | < Ο€ , | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( a + b + 1 2 ) > 0 , β„œ ⁑ ( a + 1 2 ) > 0 , β„œ ⁑ ( b + 1 2 ) > 0 formulae-sequence phase 𝑧 πœ‹ formulae-sequence phase 1 𝑧 πœ‹ formulae-sequence π‘Ž 𝑏 1 2 0 formulae-sequence π‘Ž 1 2 0 𝑏 1 2 0 {\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 2 ⁒ z ⁒ Ξ“ ⁑ ( - 1 2 ) ⁒ Ξ“ ⁑ ( a + b - 1 2 ) Ξ“ ⁑ ( a - 1 2 ) ⁒ Ξ“ ⁑ ( b - 1 2 ) ⁒ F ⁑ ( a , b ; 3 2 ; z ) = F ⁑ ( 2 ⁒ a - 1 , 2 ⁒ b - 1 ; a + b - 1 2 ; 1 2 - 1 2 ⁒ z ) - F ⁑ ( 2 ⁒ a - 1 , 2 ⁒ b - 1 ; a + b - 1 2 ; 1 2 + 1 2 ⁒ z ) 2 𝑧 Euler-Gamma 1 2 Euler-Gamma π‘Ž 𝑏 1 2 Euler-Gamma π‘Ž 1 2 Euler-Gamma 𝑏 1 2 Gauss-hypergeometric-F π‘Ž 𝑏 3 2 𝑧 Gauss-hypergeometric-F 2 π‘Ž 1 2 𝑏 1 π‘Ž 𝑏 1 2 1 2 1 2 𝑧 Gauss-hypergeometric-F 2 π‘Ž 1 2 𝑏 1 π‘Ž 𝑏 1 2 1 2 1 2 𝑧 {\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}}		 | ph ⁑ z | < Ο€ , | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( a + b - 1 2 ) > 0 , β„œ ⁑ ( a - 1 2 ) > 0 , β„œ ⁑ ( b - 1 2 ) > 0 formulae-sequence phase 𝑧 πœ‹ formulae-sequence phase 1 𝑧 πœ‹ formulae-sequence π‘Ž 𝑏 1 2 0 formulae-sequence π‘Ž 1 2 0 𝑏 1 2 0 {\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 F ⁑ ( a , 1 3 ⁒ a + 1 3 2 3 ⁒ a + 2 3 ; z ) = ( 1 + z ) - 2 ⁒ a ⁒ F ⁑ ( a , 2 3 ⁒ a + 1 6 4 3 ⁒ a + 1 3 ; 4 ⁒ z ( 1 + z ) 2 ) Gauss-hypergeometric-F π‘Ž 1 3 π‘Ž 1 3 2 3 π‘Ž 2 3 𝑧 superscript 1 𝑧 2 π‘Ž Gauss-hypergeometric-F π‘Ž 2 3 π‘Ž 1 6 4 3 π‘Ž 1 3 4 𝑧 superscript 1 𝑧 2 {\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 ( 1 - 1 2 ⁒ z ) - a ⁒ F ⁑ ( 1 2 ⁒ a , 1 2 ⁒ a + 1 2 1 3 ⁒ a + 5 6 ; ( z 2 - z ) 2 ) = F ⁑ ( a , 1 3 ⁒ a + 1 3 2 3 ⁒ a + 2 3 ; z ) superscript 1 1 2 𝑧 π‘Ž Gauss-hypergeometric-F 1 2 π‘Ž 1 2 π‘Ž 1 2 1 3 π‘Ž 5 6 superscript 𝑧 2 𝑧 2 Gauss-hypergeometric-F π‘Ž 1 3 π‘Ž 1 3 2 3 π‘Ž 2 3 𝑧 {\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 F ⁑ ( a , 1 3 ⁒ a + 1 3 2 3 ⁒ a + 2 3 ; z ) = ( 1 + z ) - a ⁒ F ⁑ ( 1 2 ⁒ a , 1 2 ⁒ a + 1 2 2 3 ⁒ a + 2 3 ; 4 ⁒ z ( 1 + z ) 2 ) Gauss-hypergeometric-F π‘Ž 1 3 π‘Ž 1 3 2 3 π‘Ž 2 3 𝑧 superscript 1 𝑧 π‘Ž Gauss-hypergeometric-F 1 2 π‘Ž 1 2 π‘Ž 1 2 2 3 π‘Ž 2 3 4 𝑧 superscript 1 𝑧 2 {\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 F ⁑ ( 3 ⁒ a , 3 ⁒ a + 1 2 4 ⁒ a + 2 3 ; z ) = ( 1 - 9 8 ⁒ z ) - 2 ⁒ a ⁒ F ⁑ ( a , a + 1 2 2 ⁒ a + 5 6 ; 27 ⁒ z 2 ⁒ ( z - 1 ) ( 9 ⁒ z - 8 ) 2 ) Gauss-hypergeometric-F 3 π‘Ž 3 π‘Ž 1 2 4 π‘Ž 2 3 𝑧 superscript 1 9 8 𝑧 2 π‘Ž Gauss-hypergeometric-F π‘Ž π‘Ž 1 2 2 π‘Ž 5 6 27 superscript 𝑧 2 𝑧 1 superscript 9 𝑧 8 2 {\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 ( 1 - z 3 ) a ( - z ) 3 ⁒ a ⁒ ( 1 Ξ“ ⁑ ( a + 2 3 ) ⁒ Ξ“ ⁑ ( 2 3 ) ⁒ F ⁑ ( a , a + 1 3 2 3 ; z - 3 ) + e 1 3 ⁒ Ο€ ⁒ i z ⁒ Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( 4 3 ) ⁒ F ⁑ ( a + 1 3 , a + 2 3 4 3 ; z - 3 ) ) = 3 3 2 ⁒ a + 1 2 ⁒ e 1 2 ⁒ a ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( a + 1 3 ) ⁒ ( 1 - ΞΆ ) a 2 ⁒ Ο€ ⁒ Ξ“ ⁑ ( 2 ⁒ a + 2 3 ) ⁒ ( - ΞΆ ) 2 ⁒ a ⁒ F ⁑ ( a + 1 3 , 3 ⁒ a 2 ⁒ a + 2 3 ; ΞΆ - 1 ) superscript 1 superscript 𝑧 3 π‘Ž superscript 𝑧 3 π‘Ž 1 Euler-Gamma π‘Ž 2 3 Euler-Gamma 2 3 Gauss-hypergeometric-F π‘Ž π‘Ž 1 3 2 3 superscript 𝑧 3 superscript 𝑒 1 3 πœ‹ imaginary-unit 𝑧 Euler-Gamma π‘Ž Euler-Gamma 4 3 Gauss-hypergeometric-F π‘Ž 1 3 π‘Ž 2 3 4 3 superscript 𝑧 3 superscript 3 3 2 π‘Ž 1 2 superscript 𝑒 1 2 π‘Ž πœ‹ imaginary-unit Euler-Gamma π‘Ž 1 3 superscript 1 𝜁 π‘Ž 2 πœ‹ Euler-Gamma 2 π‘Ž 2 3 superscript 𝜁 2 π‘Ž Gauss-hypergeometric-F π‘Ž 1 3 3 π‘Ž 2 π‘Ž 2 3 superscript 𝜁 1 {\displaystyle{\displaystyle\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3% a}}\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}}		 | z | > 1 , | ph ⁑ ( - z ) | < 1 3 ⁒ Ο€ , β„œ ⁑ ( a + 2 3 ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ ( a + 1 3 ) > 0 , β„œ ⁑ ( 2 ⁒ a + 2 3 ) > 0 formulae-sequence 𝑧 1 formulae-sequence phase 𝑧 1 3 πœ‹ formulae-sequence π‘Ž 2 3 0 formulae-sequence π‘Ž 0 formulae-sequence π‘Ž 1 3 0 2 π‘Ž 2 3 0 {\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 F ⁑ ( 1 3 , 2 3 1 ; 1 - ( 1 - z 1 + 2 ⁒ z ) 3 ) = ( 1 + 2 ⁒ z ) ⁒ F ⁑ ( 1 3 , 2 3 1 ; z 3 ) Gauss-hypergeometric-F 1 3 2 3 1 1 superscript 1 𝑧 1 2 𝑧 3 1 2 𝑧 Gauss-hypergeometric-F 1 3 2 3 1 superscript 𝑧 3 {\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
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