Generalized Hypergeometric Functions & Meijer G -Function - 16.4 Argument Unity

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
16.4.E1 a q + b q = a q + 1 + 1 subscript π‘Ž π‘ž subscript 𝑏 π‘ž subscript π‘Ž π‘ž 1 1 {\displaystyle{\displaystyle a_{q}+b_{q}=a_{q+1}+1}}
a_{q}+b_{q} = a_{q+1}+1		 

a[q]+ b[q] = a[q + 1]+ 1
 
Subscript[a, q]+ Subscript[b, q] == Subscript[a, q + 1]+ 1
 Skipped - no semantic math Skipped - no semantic math - -
16.4.E3 F 2 3 ⁑ ( - n , a , b c , d ; 1 ) = ( c - a ) n ⁒ ( c - b ) n ( c ) n ⁒ ( c - a - b ) n Gauss-hypergeometric-pFq 3 2 𝑛 π‘Ž 𝑏 𝑐 𝑑 1 Pochhammer 𝑐 π‘Ž 𝑛 Pochhammer 𝑐 𝑏 𝑛 Pochhammer 𝑐 𝑛 Pochhammer 𝑐 π‘Ž 𝑏 𝑛 {\displaystyle{\displaystyle{{}_{3}F_{2}}\left({-n,a,b\atop c,d};1\right)=% \frac{{\left(c-a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}{\left% (c-a-b\right)_{n}}}}}
\genhyperF{3}{2}@@{-n,a,b}{c,d}{1} = \frac{\Pochhammersym{c-a}{n}\Pochhammersym{c-b}{n}}{\Pochhammersym{c}{n}\Pochhammersym{c-a-b}{n}}		 

hypergeom([- n , a , b], [c , d], 1) = (pochhammer(c - a, n)*pochhammer(c - b, n))/(pochhammer(c, n)*pochhammer(c - a - b, n))

HypergeometricPFQ[{- n , a , b}, {c , d}, 1] == Divide[Pochhammer[c - a, n]*Pochhammer[c - b, n],Pochhammer[c, n]*Pochhammer[c - a - b, n]]
Failure Failure
Failed [281 / 300]
Result: 2.299038106-.7499999997*I
Test Values: {a = -3/2, b = -3/2, c = -3/2, d = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 3.872595264-1.774519052*I
Test Values: {a = -3/2, b = -3/2, c = -3/2, d = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [281 / 300]
Result: Complex[2.299038105676658, -0.7499999999999998]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1]}


Result: Complex[3.872595264191645, -1.7745190528383286]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2]}

... skip entries to safe data
16.4.E4 F 2 3 ⁑ ( a , b , c a - b + 1 , a - c + 1 ; 1 ) = Ξ“ ⁑ ( 1 2 ⁒ a + 1 ) ⁒ Ξ“ ⁑ ( a - b + 1 ) ⁒ Ξ“ ⁑ ( a - c + 1 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ a - b - c + 1 ) Ξ“ ⁑ ( a + 1 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ a - b + 1 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ a - c + 1 ) ⁒ Ξ“ ⁑ ( a - b - c + 1 ) Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 𝑐 π‘Ž 𝑏 1 π‘Ž 𝑐 1 1 Euler-Gamma 1 2 π‘Ž 1 Euler-Gamma π‘Ž 𝑏 1 Euler-Gamma π‘Ž 𝑐 1 Euler-Gamma 1 2 π‘Ž 𝑏 𝑐 1 Euler-Gamma π‘Ž 1 Euler-Gamma 1 2 π‘Ž 𝑏 1 Euler-Gamma 1 2 π‘Ž 𝑐 1 Euler-Gamma π‘Ž 𝑏 𝑐 1 {\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,b,c\atop a-b+1,a-c+1};1% \right)=\frac{\Gamma\left(\frac{1}{2}a+1\right)\Gamma\left(a-b+1\right)\Gamma% \left(a-c+1\right)\Gamma\left(\frac{1}{2}a-b-c+1\right)}{\Gamma\left(a+1\right% )\Gamma\left(\frac{1}{2}a-b+1\right)\Gamma\left(\frac{1}{2}a-c+1\right)\Gamma% \left(a-b-c+1\right)}}}
\genhyperF{3}{2}@@{a,b,c}{a-b+1,a-c+1}{1} = \frac{\EulerGamma@{\frac{1}{2}a+1}\EulerGamma@{a-b+1}\EulerGamma@{a-c+1}\EulerGamma@{\frac{1}{2}a-b-c+1}}{\EulerGamma@{a+1}\EulerGamma@{\frac{1}{2}a-b+1}\EulerGamma@{\frac{1}{2}a-c+1}\EulerGamma@{a-b-c+1}}		 β„œ ⁑ ( 1 2 ⁒ a + 1 ) > 0 , β„œ ⁑ ( a - b + 1 ) > 0 , β„œ ⁑ ( a - c + 1 ) > 0 , β„œ ⁑ ( 1 2 ⁒ a - b - c + 1 ) > 0 , β„œ ⁑ ( a + 1 ) > 0 , β„œ ⁑ ( 1 2 ⁒ a - b + 1 ) > 0 , β„œ ⁑ ( 1 2 ⁒ a - c + 1 ) > 0 , β„œ ⁑ ( a - b - c + 1 ) > 0 formulae-sequence 1 2 π‘Ž 1 0 formulae-sequence π‘Ž 𝑏 1 0 formulae-sequence π‘Ž 𝑐 1 0 formulae-sequence 1 2 π‘Ž 𝑏 𝑐 1 0 formulae-sequence π‘Ž 1 0 formulae-sequence 1 2 π‘Ž 𝑏 1 0 formulae-sequence 1 2 π‘Ž 𝑐 1 0 π‘Ž 𝑏 𝑐 1 0 {\displaystyle{\displaystyle\Re(\frac{1}{2}a+1)>0,\Re(a-b+1)>0,\Re(a-c+1)>0,% \Re(\frac{1}{2}a-b-c+1)>0,\Re(a+1)>0,\Re(\frac{1}{2}a-b+1)>0,\Re(\frac{1}{2}a-% c+1)>0,\Re(a-b-c+1)>0}} 
hypergeom([a , b , c], [a - b + 1 , a - c + 1], 1) = (GAMMA((1)/(2)*a + 1)*GAMMA(a - b + 1)*GAMMA(a - c + 1)*GAMMA((1)/(2)*a - b - c + 1))/(GAMMA(a + 1)*GAMMA((1)/(2)*a - b + 1)*GAMMA((1)/(2)*a - c + 1)*GAMMA(a - b - c + 1))

HypergeometricPFQ[{a , b , c}, {a - b + 1 , a - c + 1}, 1] == Divide[Gamma[Divide[1,2]*a + 1]*Gamma[a - b + 1]*Gamma[a - c + 1]*Gamma[Divide[1,2]*a - b - c + 1],Gamma[a + 1]*Gamma[Divide[1,2]*a - b + 1]*Gamma[Divide[1,2]*a - c + 1]*Gamma[a - b - c + 1]]
Successful Successful - Successful [Tested: 69]
16.4.E6 F 2 3 ⁑ ( a , b , c 1 2 ⁒ ( a + b + 1 ) , 2 ⁒ c ; 1 ) = Ξ“ ⁑ ( 1 2 ) ⁒ Ξ“ ⁑ ( c + 1 2 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ ( a + b + 1 ) ) ⁒ Ξ“ ⁑ ( c + 1 2 ⁒ ( 1 - a - b ) ) Ξ“ ⁑ ( 1 2 ⁒ ( a + 1 ) ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ ( b + 1 ) ) ⁒ Ξ“ ⁑ ( c + 1 2 ⁒ ( 1 - a ) ) ⁒ Ξ“ ⁑ ( c + 1 2 ⁒ ( 1 - b ) ) Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 𝑐 1 2 π‘Ž 𝑏 1 2 𝑐 1 Euler-Gamma 1 2 Euler-Gamma 𝑐 1 2 Euler-Gamma 1 2 π‘Ž 𝑏 1 Euler-Gamma 𝑐 1 2 1 π‘Ž 𝑏 Euler-Gamma 1 2 π‘Ž 1 Euler-Gamma 1 2 𝑏 1 Euler-Gamma 𝑐 1 2 1 π‘Ž Euler-Gamma 𝑐 1 2 1 𝑏 {\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,b,c\atop\frac{1}{2}(a+b+1),2% c};1\right)=\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(c+\frac{1}{2}% \right)\Gamma\left(\frac{1}{2}(a+b+1)\right)\Gamma\left(c+\frac{1}{2}(1-a-b)% \right)}{\Gamma\left(\frac{1}{2}(a+1)\right)\Gamma\left(\frac{1}{2}(b+1)\right% )\Gamma\left(c+\frac{1}{2}(1-a)\right)\Gamma\left(c+\frac{1}{2}(1-b)\right)}}}
\genhyperF{3}{2}@@{a,b,c}{\frac{1}{2}(a+b+1),2c}{1} = \frac{\EulerGamma@{\frac{1}{2}}\EulerGamma@{c+\frac{1}{2}}\EulerGamma@{\frac{1}{2}(a+b+1)}\EulerGamma@{c+\frac{1}{2}(1-a-b)}}{\EulerGamma@{\frac{1}{2}(a+1)}\EulerGamma@{\frac{1}{2}(b+1)}\EulerGamma@{c+\frac{1}{2}(1-a)}\EulerGamma@{c+\frac{1}{2}(1-b)}}		 β„œ ⁑ ( c + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ ( a + b + 1 ) ) > 0 , β„œ ⁑ ( c + 1 2 ⁒ ( 1 - a - b ) ) > 0 , β„œ ⁑ ( 1 2 ⁒ ( a + 1 ) ) > 0 , β„œ ⁑ ( 1 2 ⁒ ( b + 1 ) ) > 0 , β„œ ⁑ ( c + 1 2 ⁒ ( 1 - a ) ) > 0 , β„œ ⁑ ( c + 1 2 ⁒ ( 1 - b ) ) > 0 formulae-sequence 𝑐 1 2 0 formulae-sequence 1 2 π‘Ž 𝑏 1 0 formulae-sequence 𝑐 1 2 1 π‘Ž 𝑏 0 formulae-sequence 1 2 π‘Ž 1 0 formulae-sequence 1 2 𝑏 1 0 formulae-sequence 𝑐 1 2 1 π‘Ž 0 𝑐 1 2 1 𝑏 0 {\displaystyle{\displaystyle\Re(c+\frac{1}{2})>0,\Re(\frac{1}{2}(a+b+1))>0,\Re% (c+\frac{1}{2}(1-a-b))>0,\Re(\frac{1}{2}(a+1))>0,\Re(\frac{1}{2}(b+1))>0,\Re(c% +\frac{1}{2}(1-a))>0,\Re(c+\frac{1}{2}(1-b))>0}} 
hypergeom([a , b , c], [(1)/(2)*(a + b + 1), 2*c], 1) = (GAMMA((1)/(2))*GAMMA(c +(1)/(2))*GAMMA((1)/(2)*(a + b + 1))*GAMMA(c +(1)/(2)*(1 - a - b)))/(GAMMA((1)/(2)*(a + 1))*GAMMA((1)/(2)*(b + 1))*GAMMA(c +(1)/(2)*(1 - a))*GAMMA(c +(1)/(2)*(1 - b)))

HypergeometricPFQ[{a , b , c}, {Divide[1,2]*(a + b + 1), 2*c}, 1] == Divide[Gamma[Divide[1,2]]*Gamma[c +Divide[1,2]]*Gamma[Divide[1,2]*(a + b + 1)]*Gamma[c +Divide[1,2]*(1 - a - b)],Gamma[Divide[1,2]*(a + 1)]*Gamma[Divide[1,2]*(b + 1)]*Gamma[c +Divide[1,2]*(1 - a)]*Gamma[c +Divide[1,2]*(1 - b)]]
Successful Failure - Skipped - Because timed out
16.4.E7 F 2 3 ⁑ ( a , 1 - a , c d , 2 ⁒ c - d + 1 ; 1 ) = Ο€ ⁒ Ξ“ ⁑ ( d ) ⁒ Ξ“ ⁑ ( 2 ⁒ c - d + 1 ) ⁒ 2 1 - 2 ⁒ c Ξ“ ⁑ ( c + 1 2 ⁒ ( a - d + 1 ) ) ⁒ Ξ“ ⁑ ( c + 1 - 1 2 ⁒ ( a + d ) ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ ( a + d ) ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ ( d - a + 1 ) ) Gauss-hypergeometric-pFq 3 2 π‘Ž 1 π‘Ž 𝑐 𝑑 2 𝑐 𝑑 1 1 πœ‹ Euler-Gamma 𝑑 Euler-Gamma 2 𝑐 𝑑 1 superscript 2 1 2 𝑐 Euler-Gamma 𝑐 1 2 π‘Ž 𝑑 1 Euler-Gamma 𝑐 1 1 2 π‘Ž 𝑑 Euler-Gamma 1 2 π‘Ž 𝑑 Euler-Gamma 1 2 𝑑 π‘Ž 1 {\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,1-a,c\atop d,2c-d+1};1\right% )=\frac{\pi\Gamma\left(d\right)\Gamma\left(2c-d+1\right)2^{1-2c}}{\Gamma\left(% c+\frac{1}{2}(a-d+1)\right)\Gamma\left(c+1-\frac{1}{2}(a+d)\right)\Gamma\left(% \frac{1}{2}(a+d)\right)\Gamma\left(\frac{1}{2}(d-a+1)\right)}}}
\genhyperF{3}{2}@@{a,1-a,c}{d,2c-d+1}{1} = \frac{\pi\EulerGamma@{d}\EulerGamma@{2c-d+1}2^{1-2c}}{\EulerGamma@{c+\frac{1}{2}(a-d+1)}\EulerGamma@{c+1-\frac{1}{2}(a+d)}\EulerGamma@{\frac{1}{2}(a+d)}\EulerGamma@{\frac{1}{2}(d-a+1)}}		 β„œ ⁑ d > 0 , β„œ ⁑ ( 2 ⁒ c - d + 1 ) > 0 , β„œ ⁑ ( c + 1 2 ⁒ ( a - d + 1 ) ) > 0 , β„œ ⁑ ( c + 1 - 1 2 ⁒ ( a + d ) ) > 0 , β„œ ⁑ ( 1 2 ⁒ ( a + d ) ) > 0 , β„œ ⁑ ( 1 2 ⁒ ( d - a + 1 ) ) > 0 formulae-sequence 𝑑 0 formulae-sequence 2 𝑐 𝑑 1 0 formulae-sequence 𝑐 1 2 π‘Ž 𝑑 1 0 formulae-sequence 𝑐 1 1 2 π‘Ž 𝑑 0 formulae-sequence 1 2 π‘Ž 𝑑 0 1 2 𝑑 π‘Ž 1 0 {\displaystyle{\displaystyle\Re d>0,\Re(2c-d+1)>0,\Re(c+\frac{1}{2}(a-d+1))>0,% \Re(c+1-\frac{1}{2}(a+d))>0,\Re(\frac{1}{2}(a+d))>0,\Re(\frac{1}{2}(d-a+1))>0}} 
hypergeom([a , 1 - a , c], [d , 2*c - d + 1], 1) = (Pi*GAMMA(d)*GAMMA(2*c - d + 1)*(2)^(1 - 2*c))/(GAMMA(c +(1)/(2)*(a - d + 1))*GAMMA(c + 1 -(1)/(2)*(a + d))*GAMMA((1)/(2)*(a + d))*GAMMA((1)/(2)*(d - a + 1)))

HypergeometricPFQ[{a , 1 - a , c}, {d , 2*c - d + 1}, 1] == Divide[Pi*Gamma[d]*Gamma[2*c - d + 1]*(2)^(1 - 2*c),Gamma[c +Divide[1,2]*(a - d + 1)]*Gamma[c + 1 -Divide[1,2]*(a + d)]*Gamma[Divide[1,2]*(a + d)]*Gamma[Divide[1,2]*(d - a + 1)]]
Successful Successful - Successful [Tested: 40]
16.4.E8 F 2 3 ⁑ ( - n , a , 1 - a d , 1 - d - 2 ⁒ n ; 1 ) = ( 1 2 ⁒ ( a + d ) ) n ⁒ ( 1 2 ⁒ ( d - a + 1 ) ) n ( 1 2 ⁒ d ) n ⁒ ( 1 2 ⁒ ( d + 1 ) ) n Gauss-hypergeometric-pFq 3 2 𝑛 π‘Ž 1 π‘Ž 𝑑 1 𝑑 2 𝑛 1 Pochhammer 1 2 π‘Ž 𝑑 𝑛 Pochhammer 1 2 𝑑 π‘Ž 1 𝑛 Pochhammer 1 2 𝑑 𝑛 Pochhammer 1 2 𝑑 1 𝑛 {\displaystyle{\displaystyle{{}_{3}F_{2}}\left({-n,a,1-a\atop d,1-d-2n};1% \right)=\frac{{\left(\frac{1}{2}(a+d)\right)_{n}}{\left(\frac{1}{2}(d-a+1)% \right)_{n}}}{{\left(\frac{1}{2}d\right)_{n}}{\left(\frac{1}{2}(d+1)\right)_{n% }}}}}
\genhyperF{3}{2}@@{-n,a,1-a}{d,1-d-2n}{1} = \frac{\Pochhammersym{\frac{1}{2}(a+d)}{n}\Pochhammersym{\frac{1}{2}(d-a+1)}{n}}{\Pochhammersym{\frac{1}{2}d}{n}\Pochhammersym{\frac{1}{2}(d+1)}{n}}		 

hypergeom([- n , a , 1 - a], [d , 1 - d - 2*n], 1) = (pochhammer((1)/(2)*(a + d), n)*pochhammer((1)/(2)*(d - a + 1), n))/(pochhammer((1)/(2)*d, n)*pochhammer((1)/(2)*(d + 1), n))

HypergeometricPFQ[{- n , a , 1 - a}, {d , 1 - d - 2*n}, 1] == Divide[Pochhammer[Divide[1,2]*(a + d), n]*Pochhammer[Divide[1,2]*(d - a + 1), n],Pochhammer[Divide[1,2]*d, n]*Pochhammer[Divide[1,2]*(d + 1), n]]
Failure Failure Manual Skip!
Failed [112 / 180]
Result: Complex[-0.5976759376684342, 0.11432617133831768]
Test Values: {Rule[a, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1]}


Result: Complex[-0.4201764035832656, 0.019572796644155455]
Test Values: {Rule[a, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2]}

... skip entries to safe data
16.4.E9 F 4 5 ⁑ ( a , 1 2 ⁒ a + 1 , b , c , d 1 2 ⁒ a , a - b + 1 , a - c + 1 , a - d + 1 ; 1 ) = Ξ“ ⁑ ( a - b + 1 ) ⁒ Ξ“ ⁑ ( a - c + 1 ) ⁒ Ξ“ ⁑ ( a - d + 1 ) ⁒ Ξ“ ⁑ ( a - b - c - d + 1 ) Ξ“ ⁑ ( a + 1 ) ⁒ Ξ“ ⁑ ( a - b - c + 1 ) ⁒ Ξ“ ⁑ ( a - b - d + 1 ) ⁒ Ξ“ ⁑ ( a - c - d + 1 ) Gauss-hypergeometric-pFq 5 4 π‘Ž 1 2 π‘Ž 1 𝑏 𝑐 𝑑 1 2 π‘Ž π‘Ž 𝑏 1 π‘Ž 𝑐 1 π‘Ž 𝑑 1 1 Euler-Gamma π‘Ž 𝑏 1 Euler-Gamma π‘Ž 𝑐 1 Euler-Gamma π‘Ž 𝑑 1 Euler-Gamma π‘Ž 𝑏 𝑐 𝑑 1 Euler-Gamma π‘Ž 1 Euler-Gamma π‘Ž 𝑏 𝑐 1 Euler-Gamma π‘Ž 𝑏 𝑑 1 Euler-Gamma π‘Ž 𝑐 𝑑 1 {\displaystyle{\displaystyle{{}_{5}F_{4}}\left({a,\frac{1}{2}a+1,b,c,d\atop% \frac{1}{2}a,a-b+1,a-c+1,a-d+1};1\right)=\frac{\Gamma\left(a-b+1\right)\Gamma% \left(a-c+1\right)\Gamma\left(a-d+1\right)\Gamma\left(a-b-c-d+1\right)}{\Gamma% \left(a+1\right)\Gamma\left(a-b-c+1\right)\Gamma\left(a-b-d+1\right)\Gamma% \left(a-c-d+1\right)}}}
\genhyperF{5}{4}@@{a,\frac{1}{2}a+1,b,c,d}{\frac{1}{2}a,a-b+1,a-c+1,a-d+1}{1} = \frac{\EulerGamma@{a-b+1}\EulerGamma@{a-c+1}\EulerGamma@{a-d+1}\EulerGamma@{a-b-c-d+1}}{\EulerGamma@{a+1}\EulerGamma@{a-b-c+1}\EulerGamma@{a-b-d+1}\EulerGamma@{a-c-d+1}}		 β„œ ⁑ ( a - b + 1 ) > 0 , β„œ ⁑ ( a - c + 1 ) > 0 , β„œ ⁑ ( a - d + 1 ) > 0 , β„œ ⁑ ( a - b - c - d + 1 ) > 0 , β„œ ⁑ ( a + 1 ) > 0 , β„œ ⁑ ( a - b - c + 1 ) > 0 , β„œ ⁑ ( a - b - d + 1 ) > 0 , β„œ ⁑ ( a - c - d + 1 ) > 0 formulae-sequence π‘Ž 𝑏 1 0 formulae-sequence π‘Ž 𝑐 1 0 formulae-sequence π‘Ž 𝑑 1 0 formulae-sequence π‘Ž 𝑏 𝑐 𝑑 1 0 formulae-sequence π‘Ž 1 0 formulae-sequence π‘Ž 𝑏 𝑐 1 0 formulae-sequence π‘Ž 𝑏 𝑑 1 0 π‘Ž 𝑐 𝑑 1 0 {\displaystyle{\displaystyle\Re(a-b+1)>0,\Re(a-c+1)>0,\Re(a-d+1)>0,\Re(a-b-c-d% +1)>0,\Re(a+1)>0,\Re(a-b-c+1)>0,\Re(a-b-d+1)>0,\Re(a-c-d+1)>0}} 
hypergeom([a ,(1)/(2)*a + 1 , b , c , d], [(1)/(2)*a , a - b + 1 , a - c + 1 , a - d + 1], 1) = (GAMMA(a - b + 1)*GAMMA(a - c + 1)*GAMMA(a - d + 1)*GAMMA(a - b - c - d + 1))/(GAMMA(a + 1)*GAMMA(a - b - c + 1)*GAMMA(a - b - d + 1)*GAMMA(a - c - d + 1))

HypergeometricPFQ[{a ,Divide[1,2]*a + 1 , b , c , d}, {Divide[1,2]*a , a - b + 1 , a - c + 1 , a - d + 1}, 1] == Divide[Gamma[a - b + 1]*Gamma[a - c + 1]*Gamma[a - d + 1]*Gamma[a - b - c - d + 1],Gamma[a + 1]*Gamma[a - b - c + 1]*Gamma[a - b - d + 1]*Gamma[a - c - d + 1]]
Failure Failure Successful [Tested: 300] Successful [Tested: 300]
16.4.E10 F 6 7 ⁑ ( a , 1 2 ⁒ a + 1 , b , c , d , f , - n 1 2 ⁒ a , a - b + 1 , a - c + 1 , a - d + 1 , a - f + 1 , a + n + 1 ; 1 ) = ( a + 1 ) n ⁒ ( a - b - c + 1 ) n ⁒ ( a - b - d + 1 ) n ⁒ ( a - c - d + 1 ) n ( a - b + 1 ) n ⁒ ( a - c + 1 ) n ⁒ ( a - d + 1 ) n ⁒ ( a - b - c - d + 1 ) n Gauss-hypergeometric-pFq 7 6 π‘Ž 1 2 π‘Ž 1 𝑏 𝑐 𝑑 𝑓 𝑛 1 2 π‘Ž π‘Ž 𝑏 1 π‘Ž 𝑐 1 π‘Ž 𝑑 1 π‘Ž 𝑓 1 π‘Ž 𝑛 1 1 Pochhammer π‘Ž 1 𝑛 Pochhammer π‘Ž 𝑏 𝑐 1 𝑛 Pochhammer π‘Ž 𝑏 𝑑 1 𝑛 Pochhammer π‘Ž 𝑐 𝑑 1 𝑛 Pochhammer π‘Ž 𝑏 1 𝑛 Pochhammer π‘Ž 𝑐 1 𝑛 Pochhammer π‘Ž 𝑑 1 𝑛 Pochhammer π‘Ž 𝑏 𝑐 𝑑 1 𝑛 {\displaystyle{\displaystyle{{}_{7}F_{6}}\left({a,\frac{1}{2}a+1,b,c,d,f,-n% \atop\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-f+1,a+n+1};1\right)=\frac{{\left(a+1% \right)_{n}}{\left(a-b-c+1\right)_{n}}{\left(a-b-d+1\right)_{n}}{\left(a-c-d+1% \right)_{n}}}{{\left(a-b+1\right)_{n}}{\left(a-c+1\right)_{n}}{\left(a-d+1% \right)_{n}}{\left(a-b-c-d+1\right)_{n}}}}}
\genhyperF{7}{6}@@{a,\frac{1}{2}a+1,b,c,d,f,-n}{\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-f+1,a+n+1}{1} = \frac{\Pochhammersym{a+1}{n}\Pochhammersym{a-b-c+1}{n}\Pochhammersym{a-b-d+1}{n}\Pochhammersym{a-c-d+1}{n}}{\Pochhammersym{a-b+1}{n}\Pochhammersym{a-c+1}{n}\Pochhammersym{a-d+1}{n}\Pochhammersym{a-b-c-d+1}{n}}		 

hypergeom([a ,(1)/(2)*a + 1 , b , c , d , f , - n], [(1)/(2)*a , a - b + 1 , a - c + 1 , a - d + 1 , a - f + 1 , a + n + 1], 1) = (pochhammer(a + 1, n)*pochhammer(a - b - c + 1, n)*pochhammer(a - b - d + 1, n)*pochhammer(a - c - d + 1, n))/(pochhammer(a - b + 1, n)*pochhammer(a - c + 1, n)*pochhammer(a - d + 1, n)*pochhammer(a - b - c - d + 1, n))

HypergeometricPFQ[{a ,Divide[1,2]*a + 1 , b , c , d , f , - n}, {Divide[1,2]*a , a - b + 1 , a - c + 1 , a - d + 1 , a - f + 1 , a + n + 1}, 1] == Divide[Pochhammer[a + 1, n]*Pochhammer[a - b - c + 1, n]*Pochhammer[a - b - d + 1, n]*Pochhammer[a - c - d + 1, n],Pochhammer[a - b + 1, n]*Pochhammer[a - c + 1, n]*Pochhammer[a - d + 1, n]*Pochhammer[a - b - c - d + 1, n]]
Failure Aborted
Failed [299 / 300]
Result: .2096832772+.6841105627e-1*I
Test Values: {a = -3/2, b = -3/2, c = -3/2, d = 1/2*3^(1/2)+1/2*I, f = 1/2*3^(1/2)+1/2*I, n = 3}


Result: .1072644549-.5307589441*I
Test Values: {a = -3/2, b = -3/2, c = -3/2, d = 1/2*3^(1/2)+1/2*I, f = -1/2+1/2*I*3^(1/2), n = 3}

... skip entries to safe data
 Skipped - Because timed out
16.4.E11 F 2 3 ⁑ ( a , b , c d , e ; 1 ) = Ξ“ ⁑ ( e ) ⁒ Ξ“ ⁑ ( d + e - a - b - c ) Ξ“ ⁑ ( e - a ) ⁒ Ξ“ ⁑ ( d + e - b - c ) ⁒ F 2 3 ⁑ ( a , d - b , d - c d , d + e - b - c ; 1 ) Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 𝑐 𝑑 𝑒 1 Euler-Gamma 𝑒 Euler-Gamma 𝑑 𝑒 π‘Ž 𝑏 𝑐 Euler-Gamma 𝑒 π‘Ž Euler-Gamma 𝑑 𝑒 𝑏 𝑐 Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑑 𝑏 𝑑 𝑐 𝑑 𝑑 𝑒 𝑏 𝑐 1 {\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,b,c\atop d,e};1\right)=\frac% {\Gamma\left(e\right)\Gamma\left(d+e-a-b-c\right)}{\Gamma\left(e-a\right)% \Gamma\left(d+e-b-c\right)}{{}_{3}F_{2}}\left({a,d-b,d-c\atop d,d+e-b-c};1% \right)}}
\genhyperF{3}{2}@@{a,b,c}{d,e}{1} = \frac{\EulerGamma@{e}\EulerGamma@{d+e-a-b-c}}{\EulerGamma@{e-a}\EulerGamma@{d+e-b-c}}\genhyperF{3}{2}@@{a,d-b,d-c}{d,d+e-b-c}{1}		 β„œ ⁑ e > 0 , β„œ ⁑ ( d + e - a - b - c ) > 0 , β„œ ⁑ ( e - a ) > 0 , β„œ ⁑ ( d + e - b - c ) > 0 formulae-sequence 𝑒 0 formulae-sequence 𝑑 𝑒 π‘Ž 𝑏 𝑐 0 formulae-sequence 𝑒 π‘Ž 0 𝑑 𝑒 𝑏 𝑐 0 {\displaystyle{\displaystyle\Re e>0,\Re(d+e-a-b-c)>0,\Re(e-a)>0,\Re(d+e-b-c)>0}} 
hypergeom([a , b , c], [d , e], 1) = (GAMMA(e)*GAMMA(d + e - a - b - c))/(GAMMA(e - a)*GAMMA(d + e - b - c))*hypergeom([a , d - b , d - c], [d , d + e - b - c], 1)

HypergeometricPFQ[{a , b , c}, {d , e}, 1] == Divide[Gamma[e]*Gamma[d + e - a - b - c],Gamma[e - a]*Gamma[d + e - b - c]]*HypergeometricPFQ[{a , d - b , d - c}, {d , d + e - b - c}, 1]
Failure Failure Skipped - Because timed out Skipped - Because timed out
16.4.E12 ( a - d ) ⁒ ( b - d ) ⁒ ( c - d ) ⁒ ( F 2 3 ⁑ ( a , b , c d + 1 , e ; 1 ) - F 2 3 ⁑ ( a , b , c d , e ; 1 ) ) + a ⁒ b ⁒ c ⁒ F 2 3 ⁑ ( a , b , c d , e ; 1 ) = d ⁒ ( d - 1 ) ⁒ ( a + b + c - d - e + 1 ) ⁒ ( F 2 3 ⁑ ( a , b , c d , e ; 1 ) - F 2 3 ⁑ ( a , b , c d - 1 , e ; 1 ) ) π‘Ž 𝑑 𝑏 𝑑 𝑐 𝑑 Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 𝑐 𝑑 1 𝑒 1 Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 𝑐 𝑑 𝑒 1 π‘Ž 𝑏 𝑐 Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 𝑐 𝑑 𝑒 1 𝑑 𝑑 1 π‘Ž 𝑏 𝑐 𝑑 𝑒 1 Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 𝑐 𝑑 𝑒 1 Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 𝑐 𝑑 1 𝑒 1 {\displaystyle{\displaystyle(a-d)(b-d)(c-d)\left({{}_{3}F_{2}}\left({a,b,c% \atop d+1,e};1\right)-{{}_{3}F_{2}}\left({a,b,c\atop d,e};1\right)\right)+abc{% {}_{3}F_{2}}\left({a,b,c\atop d,e};1\right)=d(d-1)(a+b+c-d-e+1)\left({{}_{3}F_% {2}}\left({a,b,c\atop d,e};1\right)-{{}_{3}F_{2}}\left({a,b,c\atop d-1,e};1% \right)\right)}}
(a-d)(b-d)(c-d)\left(\genhyperF{3}{2}@@{a,b,c}{d+1,e}{1}-\genhyperF{3}{2}@@{a,b,c}{d,e}{1}\right)+abc\genhyperF{3}{2}@@{a,b,c}{d,e}{1} = d(d-1)(a+b+c-d-e+1)\left(\genhyperF{3}{2}@@{a,b,c}{d,e}{1}-\genhyperF{3}{2}@@{a,b,c}{d-1,e}{1}\right)		 

(a - d)*(b - d)*(c - d)*(hypergeom([a , b , c], [d + 1 , e], 1)- hypergeom([a , b , c], [d , e], 1))+ a*b*c*hypergeom([a , b , c], [d , e], 1) = d*(d - 1)*(a + b + c - d - e + 1)*(hypergeom([a , b , c], [d , e], 1)- hypergeom([a , b , c], [d - 1 , e], 1))

(a - d)*(b - d)*(c - d)*(HypergeometricPFQ[{a , b , c}, {d + 1 , e}, 1]- HypergeometricPFQ[{a , b , c}, {d , e}, 1])+ a*b*c*HypergeometricPFQ[{a , b , c}, {d , e}, 1] == d*(d - 1)*(a + b + c - d - e + 1)*(HypergeometricPFQ[{a , b , c}, {d , e}, 1]- HypergeometricPFQ[{a , b , c}, {d - 1 , e}, 1])
Failure Failure Skipped - Because timed out Skipped - Because timed out
16.4.E13 F 2 3 ⁑ ( a , b , c d , e ; 1 ) = c ⁒ ( e - a ) d ⁒ e ⁒ F 2 3 ⁑ ( a , b + 1 , c + 1 d + 1 , e + 1 ; 1 ) + d - c d ⁒ F 2 3 ⁑ ( a , b + 1 , c d + 1 , e ; 1 ) Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 𝑐 𝑑 𝑒 1 𝑐 𝑒 π‘Ž 𝑑 𝑒 Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 1 𝑐 1 𝑑 1 𝑒 1 1 𝑑 𝑐 𝑑 Gauss-hypergeometric-pFq 3 2 π‘Ž 𝑏 1 𝑐 𝑑 1 𝑒 1 {\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a,b,c\atop d,e};1\right)=% \dfrac{c(e-a)}{de}{{}_{3}F_{2}}\left({a,b+1,c+1\atop d+1,e+1};1\right)+\dfrac{% d-c}{d}{{}_{3}F_{2}}\left({a,b+1,c\atop d+1,e};1\right)}}
\genhyperF{3}{2}@@{a,b,c}{d,e}{1} = \dfrac{c(e-a)}{de}\genhyperF{3}{2}@@{a,b+1,c+1}{d+1,e+1}{1}+\dfrac{d-c}{d}\genhyperF{3}{2}@@{a,b+1,c}{d+1,e}{1}		 

hypergeom([a , b , c], [d , e], 1) = (c*(e - a))/(d*e)*hypergeom([a , b + 1 , c + 1], [d + 1 , e + 1], 1)+(d - c)/(d)*hypergeom([a , b + 1 , c], [d + 1 , e], 1)

HypergeometricPFQ[{a , b , c}, {d , e}, 1] == Divide[c*(e - a),d*e]*HypergeometricPFQ[{a , b + 1 , c + 1}, {d + 1 , e + 1}, 1]+Divide[d - c,d]*HypergeometricPFQ[{a , b + 1 , c}, {d + 1 , e}, 1]
Failure Failure Skipped - Because timed out Skipped - Because timed out
16.4.E14 F 3 4 ⁑ ( - n , a , b , c d , e , f ; 1 ) = ( e - a ) n ⁒ ( f - a ) n ( e ) n ⁒ ( f ) n ⁒ F 3 4 ⁑ ( - n , a , d - b , d - c d , a - e - n + 1 , a - f - n + 1 ; 1 ) Gauss-hypergeometric-pFq 4 3 𝑛 π‘Ž 𝑏 𝑐 𝑑 𝑒 𝑓 1 Pochhammer 𝑒 π‘Ž 𝑛 Pochhammer 𝑓 π‘Ž 𝑛 Pochhammer 𝑒 𝑛 Pochhammer 𝑓 𝑛 Gauss-hypergeometric-pFq 4 3 𝑛 π‘Ž 𝑑 𝑏 𝑑 𝑐 𝑑 π‘Ž 𝑒 𝑛 1 π‘Ž 𝑓 𝑛 1 1 {\displaystyle{\displaystyle{{}_{4}F_{3}}\left({-n,a,b,c\atop d,e,f};1\right)=% \frac{{\left(e-a\right)_{n}}{\left(f-a\right)_{n}}}{{\left(e\right)_{n}}{\left% (f\right)_{n}}}{{}_{4}F_{3}}\left({-n,a,d-b,d-c\atop d,a-e-n+1,a-f-n+1};1% \right)}}
\genhyperF{4}{3}@@{-n,a,b,c}{d,e,f}{1} = \frac{\Pochhammersym{e-a}{n}\Pochhammersym{f-a}{n}}{\Pochhammersym{e}{n}\Pochhammersym{f}{n}}\genhyperF{4}{3}@@{-n,a,d-b,d-c}{d,a-e-n+1,a-f-n+1}{1}		 

hypergeom([- n , a , b , c], [d , e , f], 1) = (pochhammer(e - a, n)*pochhammer(f - a, n))/(pochhammer(e, n)*pochhammer(f, n))*hypergeom([- n , a , d - b , d - c], [d , a - e - n + 1 , a - f - n + 1], 1)

HypergeometricPFQ[{- n , a , b , c}, {d , e , f}, 1] == Divide[Pochhammer[e - a, n]*Pochhammer[f - a, n],Pochhammer[e, n]*Pochhammer[f, n]]*HypergeometricPFQ[{- n , a , d - b , d - c}, {d , a - e - n + 1 , a - f - n + 1}, 1]
Failure Failure Skipped - Because timed out
Failed [300 / 300]
Result: Complex[-7.272114317029979, 8.095671475544961]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[e, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1]}


Result: Complex[-18.740982240718687, 40.16393590217987]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[e, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2]}

... skip entries to safe data
16.4.E15 F 6 7 ⁑ ( a , 1 2 ⁒ a + 1 , b , c , d , e , f 1 2 ⁒ a , a - b + 1 , a - c + 1 , a - d + 1 , a - e + 1 , a - f + 1 ; 1 ) = Ξ“ ⁑ ( a - d + 1 ) ⁒ Ξ“ ⁑ ( a - e + 1 ) ⁒ Ξ“ ⁑ ( a - f + 1 ) ⁒ Ξ“ ⁑ ( a - d - e - f + 1 ) Ξ“ ⁑ ( a + 1 ) ⁒ Ξ“ ⁑ ( a - d - e + 1 ) ⁒ Ξ“ ⁑ ( a - d - f + 1 ) ⁒ Ξ“ ⁑ ( a - e - f + 1 ) ⁒ F 3 4 ⁑ ( a - b - c + 1 , d , e , f a - b + 1 , a - c + 1 , d + e + f - a ; 1 ) Gauss-hypergeometric-pFq 7 6 π‘Ž 1 2 π‘Ž 1 𝑏 𝑐 𝑑 𝑒 𝑓 1 2 π‘Ž π‘Ž 𝑏 1 π‘Ž 𝑐 1 π‘Ž 𝑑 1 π‘Ž 𝑒 1 π‘Ž 𝑓 1 1 Euler-Gamma π‘Ž 𝑑 1 Euler-Gamma π‘Ž 𝑒 1 Euler-Gamma π‘Ž 𝑓 1 Euler-Gamma π‘Ž 𝑑 𝑒 𝑓 1 Euler-Gamma π‘Ž 1 Euler-Gamma π‘Ž 𝑑 𝑒 1 Euler-Gamma π‘Ž 𝑑 𝑓 1 Euler-Gamma π‘Ž 𝑒 𝑓 1 Gauss-hypergeometric-pFq 4 3 π‘Ž 𝑏 𝑐 1 𝑑 𝑒 𝑓 π‘Ž 𝑏 1 π‘Ž 𝑐 1 𝑑 𝑒 𝑓 π‘Ž 1 {\displaystyle{\displaystyle{{}_{7}F_{6}}\left({a,\frac{1}{2}a+1,b,c,d,e,f% \atop\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-e+1,a-f+1};1\right)=\frac{\Gamma\left(a-% d+1\right)\Gamma\left(a-e+1\right)\Gamma\left(a-f+1\right)\Gamma\left(a-d-e-f+% 1\right)}{\Gamma\left(a+1\right)\Gamma\left(a-d-e+1\right)\Gamma\left(a-d-f+1% \right)\Gamma\left(a-e-f+1\right)}{{}_{4}F_{3}}\left({a-b-c+1,d,e,f\atop a-b+1% ,a-c+1,d+e+f-a};1\right)}}
\genhyperF{7}{6}@@{a,\frac{1}{2}a+1,b,c,d,e,f}{\frac{1}{2}a,a-b+1,a-c+1,a-d+1,a-e+1,a-f+1}{1} = \frac{\EulerGamma@{a-d+1}\EulerGamma@{a-e+1}\EulerGamma@{a-f+1}\EulerGamma@{a-d-e-f+1}}{\EulerGamma@{a+1}\EulerGamma@{a-d-e+1}\EulerGamma@{a-d-f+1}\EulerGamma@{a-e-f+1}}\genhyperF{4}{3}@@{a-b-c+1,d,e,f}{a-b+1,a-c+1,d+e+f-a}{1}		 β„œ ⁑ ( a - d + 1 ) > 0 , β„œ ⁑ ( a - e + 1 ) > 0 , β„œ ⁑ ( a - f + 1 ) > 0 , β„œ ⁑ ( a - d - e - f + 1 ) > 0 , β„œ ⁑ ( a + 1 ) > 0 , β„œ ⁑ ( a - d - e + 1 ) > 0 , β„œ ⁑ ( a - d - f + 1 ) > 0 , β„œ ⁑ ( a - e - f + 1 ) > 0 formulae-sequence π‘Ž 𝑑 1 0 formulae-sequence π‘Ž 𝑒 1 0 formulae-sequence π‘Ž 𝑓 1 0 formulae-sequence π‘Ž 𝑑 𝑒 𝑓 1 0 formulae-sequence π‘Ž 1 0 formulae-sequence π‘Ž 𝑑 𝑒 1 0 formulae-sequence π‘Ž 𝑑 𝑓 1 0 π‘Ž 𝑒 𝑓 1 0 {\displaystyle{\displaystyle\Re(a-d+1)>0,\Re(a-e+1)>0,\Re(a-f+1)>0,\Re(a-d-e-f% +1)>0,\Re(a+1)>0,\Re(a-d-e+1)>0,\Re(a-d-f+1)>0,\Re(a-e-f+1)>0}} 
hypergeom([a ,(1)/(2)*a + 1 , b , c , d , e , f], [(1)/(2)*a , a - b + 1 , a - c + 1 , a - d + 1 , a - e + 1 , a - f + 1], 1) = (GAMMA(a - d + 1)*GAMMA(a - e + 1)*GAMMA(a - f + 1)*GAMMA(a - d - e - f + 1))/(GAMMA(a + 1)*GAMMA(a - d - e + 1)*GAMMA(a - d - f + 1)*GAMMA(a - e - f + 1))*hypergeom([a - b - c + 1 , d , e , f], [a - b + 1 , a - c + 1 , d + e + f - a], 1)

HypergeometricPFQ[{a ,Divide[1,2]*a + 1 , b , c , d , e , f}, {Divide[1,2]*a , a - b + 1 , a - c + 1 , a - d + 1 , a - e + 1 , a - f + 1}, 1] == Divide[Gamma[a - d + 1]*Gamma[a - e + 1]*Gamma[a - f + 1]*Gamma[a - d - e - f + 1],Gamma[a + 1]*Gamma[a - d - e + 1]*Gamma[a - d - f + 1]*Gamma[a - e - f + 1]]*HypergeometricPFQ[{a - b - c + 1 , d , e , f}, {a - b + 1 , a - c + 1 , d + e + f - a}, 1]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
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