Hypergeometric Function - 15.9 Relations to Other Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
15.9.E1 P n ( α , β ) ⁡ ( x ) = ( α + 1 ) n n ! ⁢ F ⁡ ( - n , n + α + β + 1 α + 1 ; 1 - x 2 ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Pochhammer 𝛼 1 𝑛 𝑛 Gauss-hypergeometric-F 𝑛 𝑛 𝛼 𝛽 1 𝛼 1 1 𝑥 2 {\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=\frac{{\left% (\alpha+1\right)_{n}}}{n!}F\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}% {2}\right)}}
\JacobipolyP{\alpha}{\beta}{n}@{x} = \frac{\Pochhammersym{\alpha+1}{n}}{n!}\hyperF@@{-n}{n+\alpha+\beta+1}{\alpha+1}{\frac{1-x}{2}}		 

JacobiP(n, alpha, beta, x) = (pochhammer(alpha + 1, n))/(factorial(n))*hypergeom([- n, n + alpha + beta + 1], [alpha + 1], (1 - x)/(2))

JacobiP[n, \[Alpha], \[Beta], x] == Divide[Pochhammer[\[Alpha]+ 1, n],(n)!]*Hypergeometric2F1[- n, n + \[Alpha]+ \[Beta]+ 1, \[Alpha]+ 1, Divide[1 - x,2]]
Successful Successful - Successful [Tested: 81]
15.9.E2 C n ( λ ) ⁡ ( x ) = ( 2 ⁢ λ ) n n ! ⁢ F ⁡ ( - n , n + 2 ⁢ λ λ + 1 2 ; 1 - x 2 ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 Pochhammer 2 𝜆 𝑛 𝑛 Gauss-hypergeometric-F 𝑛 𝑛 2 𝜆 𝜆 1 2 1 𝑥 2 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2% \lambda\right)_{n}}}{n!}F\left({-n,n+2\lambda\atop\lambda+\frac{1}{2}};\frac{1% -x}{2}\right)}}
\ultrasphpoly{\lambda}{n}@{x} = \frac{\Pochhammersym{2\lambda}{n}}{n!}\hyperF@@{-n}{n+2\lambda}{\lambda+\frac{1}{2}}{\frac{1-x}{2}}		 

GegenbauerC(n, lambda, x) = (pochhammer(2*lambda, n))/(factorial(n))*hypergeom([- n, n + 2*lambda], [lambda +(1)/(2)], (1 - x)/(2))

GegenbauerC[n, \[Lambda], x] == Divide[Pochhammer[2*\[Lambda], n],(n)!]*Hypergeometric2F1[- n, n + 2*\[Lambda], \[Lambda]+Divide[1,2], Divide[1 - x,2]]
Successful Successful -
Failed [15 / 90]
Result: 0.375
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[λ, -1.5]}


Result: 0.4375
Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -1.5]}

... skip entries to safe data
15.9.E3 C n ( λ ) ⁡ ( x ) = ( 2 ⁢ x ) n ⁢ ( λ ) n n ! ⁢ F ⁡ ( - 1 2 ⁢ n , 1 2 ⁢ ( 1 - n ) 1 - λ - n ; 1 x 2 ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 2 𝑥 𝑛 Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F 1 2 𝑛 1 2 1 𝑛 1 𝜆 𝑛 1 superscript 𝑥 2 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=(2x)^{n}\frac{{% \left(\lambda\right)_{n}}}{n!}F\left({-\frac{1}{2}n,\frac{1}{2}(1-n)\atop 1-% \lambda-n};\frac{1}{x^{2}}\right)}}
\ultrasphpoly{\lambda}{n}@{x} = (2x)^{n}\frac{\Pochhammersym{\lambda}{n}}{n!}\hyperF@@{-\frac{1}{2}n}{\frac{1}{2}(1-n)}{1-\lambda-n}{\frac{1}{x^{2}}}		 

GegenbauerC(n, lambda, x) = (2*x)^(n)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([-(1)/(2)*n, (1)/(2)*(1 - n)], [1 - lambda - n], (1)/((x)^(2)))

GegenbauerC[n, \[Lambda], x] == (2*x)^(n)*Divide[Pochhammer[\[Lambda], n],(n)!]*Hypergeometric2F1[-Divide[1,2]*n, Divide[1,2]*(1 - n), 1 - \[Lambda]- n, Divide[1,(x)^(2)]]
Failure Failure
Failed [3 / 90]
Result: Float(undefined)+Float(undefined)*I
Test Values: {lambda = -2, x = 3/2, n = 3}


Result: Float(undefined)+Float(undefined)*I
Test Values: {lambda = -2, x = 1/2, n = 3}

... skip entries to safe data
Failed [3 / 90]
Result: Indeterminate
Test Values: {Rule[n, 3], Rule[x, 1.5], Rule[λ, -2]}


Result: Indeterminate
Test Values: {Rule[n, 3], Rule[x, 0.5], Rule[λ, -2]}

... skip entries to safe data
15.9.E4 C n ( λ ) ⁡ ( cos ⁡ θ ) = e n ⁢ i ⁢ θ ⁢ ( λ ) n n ! ⁢ F ⁡ ( - n , λ 1 - λ - n ; e - 2 ⁢ i ⁢ θ ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝜃 superscript 𝑒 𝑛 imaginary-unit 𝜃 Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F 𝑛 𝜆 1 𝜆 𝑛 superscript 𝑒 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(\cos\theta\right)=e^{n% \mathrm{i}\theta}\frac{{\left(\lambda\right)_{n}}}{n!}F\left({-n,\lambda\atop 1% -\lambda-n};e^{-2\mathrm{i}\theta}\right)}}
\ultrasphpoly{\lambda}{n}@{\cos@@{\theta}} = e^{n\iunit\theta}\frac{\Pochhammersym{\lambda}{n}}{n!}\hyperF@@{-n}{\lambda}{1-\lambda-n}{e^{-2\iunit\theta}}		 

GegenbauerC(n, lambda, cos(theta)) = exp(n*I*theta)*(pochhammer(lambda, n))/(factorial(n))*hypergeom([- n, lambda], [1 - lambda - n], exp(- 2*I*theta))

GegenbauerC[n, \[Lambda], Cos[\[Theta]]] == Exp[n*I*\[Theta]]*Divide[Pochhammer[\[Lambda], n],(n)!]*Hypergeometric2F1[- n, \[Lambda], 1 - \[Lambda]- n, Exp[- 2*I*\[Theta]]]
Failure Failure
Failed [10 / 300]
Result: Float(undefined)+Float(undefined)*I
Test Values: {lambda = -2, theta = 1/2*3^(1/2)+1/2*I, n = 3}


Result: Float(undefined)+Float(undefined)*I
Test Values: {lambda = -2, theta = -1/2+1/2*I*3^(1/2), n = 3}

... skip entries to safe data
Failed [10 / 300]
Result: Indeterminate
Test Values: {Rule[n, 3], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[λ, -2]}


Result: Indeterminate
Test Values: {Rule[n, 3], Rule[θ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]], Rule[λ, -2]}

... skip entries to safe data
15.9.E5 T n ⁡ ( x ) = F ⁡ ( - n , n 1 2 ; 1 - x 2 ) Chebyshev-polynomial-first-kind-T 𝑛 𝑥 Gauss-hypergeometric-F 𝑛 𝑛 1 2 1 𝑥 2 {\displaystyle{\displaystyle T_{n}\left(x\right)=F\left({-n,n\atop\frac{1}{2}}% ;\frac{1-x}{2}\right)}}
\ChebyshevpolyT{n}@{x} = \hyperF@@{-n}{n}{\frac{1}{2}}{\frac{1-x}{2}}		 

ChebyshevT(n, x) = hypergeom([- n, n], [(1)/(2)], (1 - x)/(2))

ChebyshevT[n, x] == Hypergeometric2F1[- n, n, Divide[1,2], Divide[1 - x,2]]
Successful Successful - Successful [Tested: 9]
15.9.E6 U n ⁡ ( x ) = ( n + 1 ) ⁢ F ⁡ ( - n , n + 2 3 2 ; 1 - x 2 ) Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 Gauss-hypergeometric-F 𝑛 𝑛 2 3 2 1 𝑥 2 {\displaystyle{\displaystyle U_{n}\left(x\right)=(n+1)F\left({-n,n+2\atop\frac% {3}{2}};\frac{1-x}{2}\right)}}
\ChebyshevpolyU{n}@{x} = (n+1)\hyperF@@{-n}{n+2}{\frac{3}{2}}{\frac{1-x}{2}}		 

ChebyshevU(n, x) = (n + 1)*hypergeom([- n, n + 2], [(3)/(2)], (1 - x)/(2))

ChebyshevU[n, x] == (n + 1)*Hypergeometric2F1[- n, n + 2, Divide[3,2], Divide[1 - x,2]]
Successful Failure - Successful [Tested: 9]
15.9.E7 P n ⁡ ( x ) = F ⁡ ( - n , n + 1 1 ; 1 - x 2 ) Legendre-spherical-polynomial 𝑛 𝑥 Gauss-hypergeometric-F 𝑛 𝑛 1 1 1 𝑥 2 {\displaystyle{\displaystyle P_{n}\left(x\right)=F\left({-n,n+1\atop 1};\frac{% 1-x}{2}\right)}}
\LegendrepolyP{n}@{x} = \hyperF@@{-n}{n+1}{1}{\frac{1-x}{2}}		 

LegendreP(n, x) = hypergeom([- n, n + 1], [1], (1 - x)/(2))

LegendreP[n, x] == Hypergeometric2F1[- n, n + 1, 1, Divide[1 - x,2]]
Successful Successful - Successful [Tested: 9]
15.9.E11 ϕ λ ( α , β ) ⁡ ( t ) = F ⁡ ( 1 2 ⁢ ( α + β + 1 - i ⁢ λ ) , 1 2 ⁢ ( α + β + 1 + i ⁢ λ ) α + 1 ; - sinh 2 ⁡ t ) Jacobi-hypergeometric-phi 𝛼 𝛽 𝜆 𝑡 Gauss-hypergeometric-F 1 2 𝛼 𝛽 1 imaginary-unit 𝜆 1 2 𝛼 𝛽 1 imaginary-unit 𝜆 𝛼 1 2 𝑡 {\displaystyle{\displaystyle\phi^{(\alpha,\beta)}_{\lambda}\left(t\right)=F% \left({\tfrac{1}{2}(\alpha+\beta+1-\mathrm{i}\lambda),\tfrac{1}{2}(\alpha+% \beta+1+\mathrm{i}\lambda)\atop\alpha+1};-{\sinh^{2}}t\right)}}
\Jacobiphi{\alpha}{\beta}{\lambda}@{t} = \hyperF@@{\tfrac{1}{2}(\alpha+\beta+1-\iunit\lambda)}{\tfrac{1}{2}(\alpha+\beta+1+\iunit\lambda)}{\alpha+1}{-\sinh^{2}@@{t}}		 

hypergeom([((alpha)+(beta)+1-I*(lambda))/2, ((alpha)+(beta)+1+I*(lambda))], [(alpha)+1], -sinh(t)^2) = hypergeom([(1)/(2)*(alpha + beta + 1 - I*lambda), (1)/(2)*(alpha + beta + 1 + I*lambda)], [alpha + 1], - (sinh(t))^(2))

Error
Failure Missing Macro Error
Failed [288 / 300]
Result: -.4877482336e-1+.1329197787e-1*I
Test Values: {alpha = 3/2, beta = 3/2, lambda = 1/2*3^(1/2)+1/2*I, t = -3/2}


Result: -.4877482336e-1+.1329197787e-1*I
Test Values: {alpha = 3/2, beta = 3/2, lambda = 1/2*3^(1/2)+1/2*I, t = 3/2}

... skip entries to safe data
 -
15.9.E15 C α ( λ ) ⁡ ( z ) = Γ ⁡ ( α + 2 ⁢ λ ) Γ ⁡ ( 2 ⁢ λ ) ⁢ Γ ⁡ ( α + 1 ) ⁢ F ⁡ ( - α , α + 2 ⁢ λ λ + 1 2 ; 1 - z 2 ) ultraspherical-Gegenbauer-polynomial 𝜆 𝛼 𝑧 Euler-Gamma 𝛼 2 𝜆 Euler-Gamma 2 𝜆 Euler-Gamma 𝛼 1 Gauss-hypergeometric-F 𝛼 𝛼 2 𝜆 𝜆 1 2 1 𝑧 2 {\displaystyle{\displaystyle C^{(\lambda)}_{\alpha}\left(z\right)=\frac{\Gamma% \left(\alpha+2\lambda\right)}{\Gamma\left(2\lambda\right)\Gamma\left(\alpha+1% \right)}F\left({-\alpha,\alpha+2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-z}{2% }\right)}}
\ultrasphpoly{\lambda}{\alpha}@{z} = \frac{\EulerGamma@{\alpha+2\lambda}}{\EulerGamma@{2\lambda}\EulerGamma@{\alpha+1}}\hyperF@@{-\alpha}{\alpha+2\lambda}{\lambda+\tfrac{1}{2}}{\frac{1-z}{2}}		 ℜ ⁡ ( α + 2 ⁢ λ ) > 0 , ℜ ⁡ ( 2 ⁢ λ ) > 0 , ℜ ⁡ ( α + 1 ) > 0 formulae-sequence 𝛼 2 𝜆 0 formulae-sequence 2 𝜆 0 𝛼 1 0 {\displaystyle{\displaystyle\Re(\alpha+2\lambda)>0,\Re(2\lambda)>0,\Re(\alpha+% 1)>0}} 
GegenbauerC(alpha, lambda, z) = (GAMMA(alpha + 2*lambda))/(GAMMA(2*lambda)*GAMMA(alpha + 1))*hypergeom([- alpha, alpha + 2*lambda], [lambda +(1)/(2)], (1 - z)/(2))

GegenbauerC[\[Alpha], \[Lambda], z] == Divide[Gamma[\[Alpha]+ 2*\[Lambda]],Gamma[2*\[Lambda]]*Gamma[\[Alpha]+ 1]]*Hypergeometric2F1[- \[Alpha], \[Alpha]+ 2*\[Lambda], \[Lambda]+Divide[1,2], Divide[1 - z,2]]
Successful Successful - Successful [Tested: 105]
15.9.E16 𝐅 ⁡ ( a , b 2 ⁢ b ; z ) = π Γ ⁡ ( b ) ⁢ z - b + ( 1 / 2 ) ⁢ ( 1 - z ) ( b - a - ( 1 / 2 ) ) / 2 ⁢ P a - b - ( 1 / 2 ) - b + ( 1 / 2 ) ⁡ ( 2 - z 2 ⁢ 1 - z ) scaled-hypergeometric-bold-F 𝑎 𝑏 2 𝑏 𝑧 𝜋 Euler-Gamma 𝑏 superscript 𝑧 𝑏 1 2 superscript 1 𝑧 𝑏 𝑎 1 2 2 Legendre-P-first-kind 𝑏 1 2 𝑎 𝑏 1 2 2 𝑧 2 1 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop 2b};z\right)=\frac{\sqrt% {\pi}}{\Gamma\left(b\right)}z^{-b+(\ifrac{1}{2})}(1-z)^{(b-a-(\ifrac{1}{2}))/2% }\*P^{-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\frac{2-z}{2\sqrt{1-z}}% \right)}}
\hyperOlverF@@{a}{b}{2b}{z} = \frac{\sqrt{\pi}}{\EulerGamma@{b}}z^{-b+(\ifrac{1}{2})}(1-z)^{(b-a-(\ifrac{1}{2}))/2}\*\assLegendreP[-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{\frac{2-z}{2\sqrt{1-z}}}		 | ph ⁡ ( 1 - z ) | < π , | 1 - z | < 1 , ℜ ⁡ b > 0 , | z | < 1 , ℜ ⁡ ( ( 2 ⁢ b ) + s ) > 0 formulae-sequence phase 1 𝑧 𝜋 formulae-sequence 1 𝑧 1 formulae-sequence 𝑏 0 formulae-sequence 𝑧 1 2 𝑏 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,|1-z|<1,% \Re b>0,|z|<1,\Re((2b)+s)>0}} 
hypergeom([a, b], [2*b], z)/GAMMA(2*b) = (sqrt(Pi))/(GAMMA(b))*(z)^(- b +((1)/(2)))*(1 - z)^((b - a -((1)/(2)))/2)* LegendreP(a - b -((1)/(2)), - b +((1)/(2)), (2 - z)/(2*sqrt(1 - z)))

Hypergeometric2F1Regularized[a, b, 2*b, z] == Divide[Sqrt[Pi],Gamma[b]]*(z)^(- b +(Divide[1,2]))*(1 - z)^((b - a -(Divide[1,2]))/2)* LegendreP[a - b -(Divide[1,2]), - b +(Divide[1,2]), 3, Divide[2 - z,2*Sqrt[1 - z]]]
Failure Failure Successful [Tested: 6] Successful [Tested: 18]
15.9.E17 𝐅 ⁡ ( a , a + 1 2 c ; z ) = 2 c - 1 ⁢ z ( 1 - c ) / 2 ⁢ ( 1 - z ) - a + ( ( c - 1 ) / 2 ) ⁢ P 2 ⁢ a - c 1 - c ⁡ ( 1 1 - z ) scaled-hypergeometric-bold-F 𝑎 𝑎 1 2 𝑐 𝑧 superscript 2 𝑐 1 superscript 𝑧 1 𝑐 2 superscript 1 𝑧 𝑎 𝑐 1 2 Legendre-P-first-kind 1 𝑐 2 𝑎 𝑐 1 1 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,a+\tfrac{1}{2}\atop c};z\right)% =2^{c-1}z^{\ifrac{(1-c)}{2}}(1-z)^{-a+(\ifrac{(c-1)}{2})}\*P^{1-c}_{2a-c}\left% (\frac{1}{\sqrt{1-z}}\right)}}
\hyperOlverF@@{a}{a+\tfrac{1}{2}}{c}{z} = 2^{c-1}z^{\ifrac{(1-c)}{2}}(1-z)^{-a+(\ifrac{(c-1)}{2})}\*\assLegendreP[1-c]{2a-c}@{\frac{1}{\sqrt{1-z}}}		 | ph ⁡ z | < π , | ph ⁡ ( 1 - z ) | < π , | z | < 1 , ℜ ⁡ ( c + s ) > 0 formulae-sequence phase 𝑧 𝜋 formulae-sequence phase 1 𝑧 𝜋 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1% -z\right)|<\pi,|z|<1,\Re(c+s)>0}} 
hypergeom([a, a +(1)/(2)], [c], z)/GAMMA(c) = (2)^(c - 1)* (z)^((1 - c)/(2))*(1 - z)^(- a +((c - 1)/(2)))* LegendreP(2*a - c, 1 - c, (1)/(sqrt(1 - z)))

Hypergeometric2F1Regularized[a, a +Divide[1,2], c, z] == (2)^(c - 1)* (z)^(Divide[1 - c,2])*(1 - z)^(- a +(Divide[c - 1,2]))* LegendreP[2*a - c, 1 - c, 3, Divide[1,Sqrt[1 - z]]]
Failure Failure Error Successful [Tested: 180]
15.9.E18 𝐅 ⁡ ( a , b a + b + 1 2 ; z ) = 2 a + b - ( 1 / 2 ) ⁢ ( - z ) ( - a - b + ( 1 / 2 ) ) / 2 ⁢ P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ⁡ ( 1 - z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑎 𝑏 1 2 𝑧 superscript 2 𝑎 𝑏 1 2 superscript 𝑧 𝑎 𝑏 1 2 2 Legendre-P-first-kind 𝑎 𝑏 1 2 𝑎 𝑏 1 2 1 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a+b+\tfrac{1}{2}};z% \right)=2^{a+b-(\ifrac{1}{2})}(-z)^{(-a-b+(\ifrac{1}{2}))/2}\*P^{-a-b+(\ifrac{% 1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{1-z}\right)}}
\hyperOlverF@@{a}{b}{a+b+\tfrac{1}{2}}{z} = 2^{a+b-(\ifrac{1}{2})}(-z)^{(-a-b+(\ifrac{1}{2}))/2}\*\assLegendreP[-a-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{\sqrt{1-z}}		 | ph ⁡ ( - z ) | < π , | z | < 1 , ℜ ⁡ ( ( a + b + 1 2 ) + s ) > 0 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝑧 1 𝑎 𝑏 1 2 𝑠 0 {\displaystyle{\displaystyle\left|\operatorname{ph}\left(-z\right)\right|<\pi,% |z|<1,\Re((a+b+\tfrac{1}{2})+s)>0}} 
hypergeom([a, b], [a + b +(1)/(2)], z)/GAMMA(a + b +(1)/(2)) = (2)^(a + b -((1)/(2)))*(- z)^((- a - b +((1)/(2)))/2)* LegendreP(a - b -((1)/(2)), - a - b +((1)/(2)), sqrt(1 - z))

Hypergeometric2F1Regularized[a, b, a + b +Divide[1,2], z] == (2)^(a + b -(Divide[1,2]))*(- z)^((- a - b +(Divide[1,2]))/2)* LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[1,2]), 3, Sqrt[1 - z]]
Failure Failure Error Successful [Tested: 144]
15.9.E19 𝐅 ⁡ ( a , b a - b + 1 ; z ) = z ( b - a ) / 2 ⁢ ( 1 - z ) - b ⁢ P - b b - a ⁡ ( 1 + z 1 - z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑎 𝑏 1 𝑧 superscript 𝑧 𝑏 𝑎 2 superscript 1 𝑧 𝑏 Legendre-P-first-kind 𝑏 𝑎 𝑏 1 𝑧 1 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a-b+1};z\right)=z^{% \ifrac{(b-a)}{2}}(1-z)^{-b}\*P^{b-a}_{-b}\left(\frac{1+z}{1-z}\right)}}
\hyperOlverF@@{a}{b}{a-b+1}{z} = z^{\ifrac{(b-a)}{2}}(1-z)^{-b}\*\assLegendreP[b-a]{-b}@{\frac{1+z}{1-z}}		 | ph ⁡ z | < π , | ph ⁡ ( 1 - z ) | < π , | z | < 1 , ℜ ⁡ ( ( a - b + 1 ) + s ) > 0 formulae-sequence phase 𝑧 𝜋 formulae-sequence phase 1 𝑧 𝜋 formulae-sequence 𝑧 1 𝑎 𝑏 1 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1% -z\right)|<\pi,|z|<1,\Re((a-b+1)+s)>0}} 
hypergeom([a, b], [a - b + 1], z)/GAMMA(a - b + 1) = (z)^((b - a)/(2))*(1 - z)^(- b)* LegendreP(- b, b - a, (1 + z)/(1 - z))

Hypergeometric2F1Regularized[a, b, a - b + 1, z] == (z)^(Divide[b - a,2])*(1 - z)^(- b)* LegendreP[- b, b - a, 3, Divide[1 + z,1 - z]]
Successful Failure - Successful [Tested: 180]
15.9.E20 𝐅 ⁡ ( a , b 1 2 ⁢ ( a + b + 1 ) ; z ) = ( - z ⁢ ( 1 - z ) ) ( 1 - a - b ) / 4 ⁢ P ( a - b - 1 ) / 2 ( 1 - a - b ) / 2 ⁡ ( 1 - 2 ⁢ z ) scaled-hypergeometric-bold-F 𝑎 𝑏 1 2 𝑎 𝑏 1 𝑧 superscript 𝑧 1 𝑧 1 𝑎 𝑏 4 Legendre-P-first-kind 1 𝑎 𝑏 2 𝑎 𝑏 1 2 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}(a+b+1)};z% \right)=\left(-z(1-z)\right)^{\ifrac{(1-a-b)}{4}}\*P^{\ifrac{(1-a-b)}{2}}_{% \ifrac{(a-b-1)}{2}}\left(1-2z\right)}}
\hyperOlverF@@{a}{b}{\tfrac{1}{2}(a+b+1)}{z} = \left(-z(1-z)\right)^{\ifrac{(1-a-b)}{4}}\*\assLegendreP[\ifrac{(1-a-b)}{2}]{\ifrac{(a-b-1)}{2}}@{1-2z}		 | ph ⁡ ( - z ) | < π , | z | < 1 , ℜ ⁡ ( ( 1 2 ⁢ ( a + b + 1 ) ) + s ) > 0 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝑧 1 1 2 𝑎 𝑏 1 𝑠 0 {\displaystyle{\displaystyle\left|\operatorname{ph}\left(-z\right)\right|<\pi,% |z|<1,\Re((\tfrac{1}{2}(a+b+1))+s)>0}} 
hypergeom([a, b], [(1)/(2)*(a + b + 1)], z)/GAMMA((1)/(2)*(a + b + 1)) = (- z*(1 - z))^((1 - a - b)/(4))* LegendreP((a - b - 1)/(2), (1 - a - b)/(2), 1 - 2*z)

Hypergeometric2F1Regularized[a, b, Divide[1,2]*(a + b + 1), z] == (- z*(1 - z))^(Divide[1 - a - b,4])* LegendreP[Divide[a - b - 1,2], Divide[1 - a - b,2], 3, 1 - 2*z]
Failure Failure Error Successful [Tested: 144]
15.9.E21 𝐅 ⁡ ( a , 1 - a c ; z ) = ( - z 1 - z ) ( 1 - c ) / 2 ⁢ P - a 1 - c ⁡ ( 1 - 2 ⁢ z ) scaled-hypergeometric-bold-F 𝑎 1 𝑎 𝑐 𝑧 superscript 𝑧 1 𝑧 1 𝑐 2 Legendre-P-first-kind 1 𝑐 𝑎 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,1-a\atop c};z\right)=\left(% \frac{-z}{1-z}\right)^{\ifrac{(1-c)}{2}}\*P^{1-c}_{-a}\left(1-2z\right)}}
\hyperOlverF@@{a}{1-a}{c}{z} = \left(\frac{-z}{1-z}\right)^{\ifrac{(1-c)}{2}}\*\assLegendreP[1-c]{-a}@{1-2z}		 | ph ⁡ ( - z ) | < π , | z | < 1 , ℜ ⁡ ( c + s ) > 0 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\displaystyle{\displaystyle\left|\operatorname{ph}\left(-z\right)\right|<\pi,% |z|<1,\Re(c+s)>0}} 
hypergeom([a, 1 - a], [c], z)/GAMMA(c) = ((- z)/(1 - z))^((1 - c)/(2))* LegendreP(- a, 1 - c, 1 - 2*z)

Hypergeometric2F1Regularized[a, 1 - a, c, z] == (Divide[- z,1 - z])^(Divide[1 - c,2])* LegendreP[- a, 1 - c, 3, 1 - 2*z]
Failure Successful Error -
15.9.E22 𝐅 ⁡ ( a , b 1 2 ; z ) = 2 a + b - ( 3 / 2 ) π ⁢ Γ ⁡ ( a + 1 2 ) ⁢ Γ ⁡ ( b + 1 2 ) ⁢ ( z - 1 ) ( - a - b + ( 1 / 2 ) ) / 2 ⁢ ( e + π ⁢ i ⁢ ( a + b - ( 1 / 2 ) ) ⁢ P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ⁡ ( - z ) + P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ⁡ ( z ) ) scaled-hypergeometric-bold-F 𝑎 𝑏 1 2 𝑧 superscript 2 𝑎 𝑏 3 2 𝜋 Euler-Gamma 𝑎 1 2 Euler-Gamma 𝑏 1 2 superscript 𝑧 1 𝑎 𝑏 1 2 2 superscript 𝑒 𝜋 imaginary-unit 𝑎 𝑏 1 2 Legendre-P-first-kind 𝑎 𝑏 1 2 𝑎 𝑏 1 2 𝑧 Legendre-P-first-kind 𝑎 𝑏 1 2 𝑎 𝑏 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}};z\right)=% \frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left% (b+\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{+\pi\mathrm{% i}(a+b-(\ifrac{1}{2}))}P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(-% \sqrt{z}\right)+P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{z}% \right)\right)}}
\hyperOlverF@@{a}{b}{\tfrac{1}{2}}{z} = \frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\EulerGamma@{a+\tfrac{1}{2}}\EulerGamma@{b+\tfrac{1}{2}}\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{+\pi\iunit(a+b-(\ifrac{1}{2}))}\assLegendreP[-a-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{-\sqrt{z}}+\assLegendreP[-a-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{\sqrt{z}}\right)		 a ≠ - 1 2 , b ≠ - 1 2 , 0 < | ph ⁡ z | , | ph ⁡ z | < π , ℜ ⁡ ( a + 1 2 ) > 0 , ℜ ⁡ ( b + 1 2 ) > 0 , | z | < 1 , ℜ ⁡ ( ( 1 2 ) + s ) > 0 formulae-sequence 𝑎 1 2 formulae-sequence 𝑏 1 2 formulae-sequence 0 phase 𝑧 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝑎 1 2 0 formulae-sequence 𝑏 1 2 0 formulae-sequence 𝑧 1 1 2 𝑠 0 {\displaystyle{\displaystyle a\neq-\frac{1}{2},b\neq-\frac{1}{2},0<|% \operatorname{ph}z|,|\operatorname{ph}z|<\pi,\Re(a+\tfrac{1}{2})>0,\Re(b+% \tfrac{1}{2})>0,|z|<1,\Re((\tfrac{1}{2})+s)>0}} 
hypergeom([a, b], [(1)/(2)], z)/GAMMA((1)/(2)) = ((2)^(a + b -((3)/(2))))/(Pi)*GAMMA(a +(1)/(2))*GAMMA(b +(1)/(2))*(z - 1)^((- a - b +((1)/(2)))/2)*(exp(+ Pi*I*(a + b -((1)/(2))))*LegendreP(a - b -((1)/(2)), - a - b +((1)/(2)), -sqrt(z))+ LegendreP(a - b -((1)/(2)), - a - b +((1)/(2)), sqrt(z)))

Hypergeometric2F1Regularized[a, b, Divide[1,2], z] == Divide[(2)^(a + b -(Divide[3,2])),Pi]*Gamma[a +Divide[1,2]]*Gamma[b +Divide[1,2]]*(z - 1)^((- a - b +(Divide[1,2]))/2)*(Exp[+ Pi*I*(a + b -(Divide[1,2]))]*LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[1,2]), 3, -Sqrt[z]]+ LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[1,2]), 3, Sqrt[z]])
Failure Failure Error
Failed [10 / 36]
Result: Complex[-0.8582540688970105, -2.787267603366778]
Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]]}


Result: Complex[-0.09762832897349609, -0.474497895465574]
Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]]}

... skip entries to safe data
15.9.E22 𝐅 ⁡ ( a , b 1 2 ; z ) = 2 a + b - ( 3 / 2 ) π ⁢ Γ ⁡ ( a + 1 2 ) ⁢ Γ ⁡ ( b + 1 2 ) ⁢ ( z - 1 ) ( - a - b + ( 1 / 2 ) ) / 2 ⁢ ( e - π ⁢ i ⁢ ( a + b - ( 1 / 2 ) ) ⁢ P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ⁡ ( - z ) + P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ⁡ ( z ) ) scaled-hypergeometric-bold-F 𝑎 𝑏 1 2 𝑧 superscript 2 𝑎 𝑏 3 2 𝜋 Euler-Gamma 𝑎 1 2 Euler-Gamma 𝑏 1 2 superscript 𝑧 1 𝑎 𝑏 1 2 2 superscript 𝑒 𝜋 imaginary-unit 𝑎 𝑏 1 2 Legendre-P-first-kind 𝑎 𝑏 1 2 𝑎 𝑏 1 2 𝑧 Legendre-P-first-kind 𝑎 𝑏 1 2 𝑎 𝑏 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}};z\right)=% \frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left% (b+\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{-\pi\mathrm{% i}(a+b-(\ifrac{1}{2}))}P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(-% \sqrt{z}\right)+P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{z}% \right)\right)}}
\hyperOlverF@@{a}{b}{\tfrac{1}{2}}{z} = \frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\EulerGamma@{a+\tfrac{1}{2}}\EulerGamma@{b+\tfrac{1}{2}}\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{-\pi\iunit(a+b-(\ifrac{1}{2}))}\assLegendreP[-a-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{-\sqrt{z}}+\assLegendreP[-a-b+(\ifrac{1}{2})]{a-b-(\ifrac{1}{2})}@{\sqrt{z}}\right)		 a ≠ - 1 2 , b ≠ - 1 2 , 0 < | ph ⁡ z | , | ph ⁡ z | < π , ℜ ⁡ ( a + 1 2 ) > 0 , ℜ ⁡ ( b + 1 2 ) > 0 , | z | < 1 , ℜ ⁡ ( ( 1 2 ) + s ) > 0 formulae-sequence 𝑎 1 2 formulae-sequence 𝑏 1 2 formulae-sequence 0 phase 𝑧 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝑎 1 2 0 formulae-sequence 𝑏 1 2 0 formulae-sequence 𝑧 1 1 2 𝑠 0 {\displaystyle{\displaystyle a\neq-\frac{1}{2},b\neq-\frac{1}{2},0<|% \operatorname{ph}z|,|\operatorname{ph}z|<\pi,\Re(a+\tfrac{1}{2})>0,\Re(b+% \tfrac{1}{2})>0,|z|<1,\Re((\tfrac{1}{2})+s)>0}} 
hypergeom([a, b], [(1)/(2)], z)/GAMMA((1)/(2)) = ((2)^(a + b -((3)/(2))))/(Pi)*GAMMA(a +(1)/(2))*GAMMA(b +(1)/(2))*(z - 1)^((- a - b +((1)/(2)))/2)*(exp(- Pi*I*(a + b -((1)/(2))))*LegendreP(a - b -((1)/(2)), - a - b +((1)/(2)), -sqrt(z))+ LegendreP(a - b -((1)/(2)), - a - b +((1)/(2)), sqrt(z)))

Hypergeometric2F1Regularized[a, b, Divide[1,2], z] == Divide[(2)^(a + b -(Divide[3,2])),Pi]*Gamma[a +Divide[1,2]]*Gamma[b +Divide[1,2]]*(z - 1)^((- a - b +(Divide[1,2]))/2)*(Exp[- Pi*I*(a + b -(Divide[1,2]))]*LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[1,2]), 3, -Sqrt[z]]+ LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[1,2]), 3, Sqrt[z]])
Failure Failure Error
Failed [10 / 36]
Result: Complex[1.7877768256534143, 6.989426464541403]
Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.26682868759795453, 0.7163138167399228]
Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
15.9.E23 𝐅 ⁡ ( a , b 3 2 ; z ) = 2 a + b - ( 5 / 2 ) π ⁢ z ⁢ Γ ⁡ ( a - 1 2 ) ⁢ Γ ⁡ ( b - 1 2 ) ⁢ ( z - 1 ) ( - a - b + ( 3 / 2 ) ) / 2 ⁢ ( e + π ⁢ i ⁢ ( a + b - ( 3 / 2 ) ) ⁢ P a - b - ( 1 / 2 ) - a - b + ( 3 / 2 ) ⁡ ( - z ) - P a - b - ( 1 / 2 ) - a - b + ( 3 / 2 ) ⁡ ( z ) ) scaled-hypergeometric-bold-F 𝑎 𝑏 3 2 𝑧 superscript 2 𝑎 𝑏 5 2 𝜋 𝑧 Euler-Gamma 𝑎 1 2 Euler-Gamma 𝑏 1 2 superscript 𝑧 1 𝑎 𝑏 3 2 2 superscript 𝑒 𝜋 imaginary-unit 𝑎 𝑏 3 2 Legendre-P-first-kind 𝑎 𝑏 3 2 𝑎 𝑏 1 2 𝑧 Legendre-P-first-kind 𝑎 𝑏 3 2 𝑎 𝑏 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{3}{2}};z\right)=% \frac{2^{a+b-(\ifrac{5}{2})}}{\pi\sqrt{z}}\Gamma\left(a-\tfrac{1}{2}\right)% \Gamma\left(b-\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{3}{2}))/2}\*\left(e^{+% \pi\mathrm{i}(a+b-(\ifrac{3}{2}))}P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}% \left(-\sqrt{z}\right)-P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt% {z}\right)\right)}}
\hyperOlverF@@{a}{b}{\tfrac{3}{2}}{z} = \frac{2^{a+b-(\ifrac{5}{2})}}{\pi\sqrt{z}}\EulerGamma@{a-\tfrac{1}{2}}\EulerGamma@{b-\tfrac{1}{2}}\*(z-1)^{(-a-b+(\ifrac{3}{2}))/2}\*\left(e^{+\pi\iunit(a+b-(\ifrac{3}{2}))}\assLegendreP[-a-b+(\ifrac{3}{2})]{a-b-(\ifrac{1}{2})}@{-\sqrt{z}}-\assLegendreP[-a-b+(\ifrac{3}{2})]{a-b-(\ifrac{1}{2})}@{\sqrt{z}}\right)		 a ≠ 1 2 , b ≠ 1 2 , 0 < | ph ⁡ z | , | ph ⁡ z | < π , ℜ ⁡ ( a - 1 2 ) > 0 , ℜ ⁡ ( b - 1 2 ) > 0 , | z | < 1 , ℜ ⁡ ( ( 3 2 ) + s ) > 0 formulae-sequence 𝑎 1 2 formulae-sequence 𝑏 1 2 formulae-sequence 0 phase 𝑧 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝑎 1 2 0 formulae-sequence 𝑏 1 2 0 formulae-sequence 𝑧 1 3 2 𝑠 0 {\displaystyle{\displaystyle a\neq\frac{1}{2},b\neq\frac{1}{2},0<|% \operatorname{ph}z|,|\operatorname{ph}z|<\pi,\Re(a-\tfrac{1}{2})>0,\Re(b-% \tfrac{1}{2})>0,|z|<1,\Re((\tfrac{3}{2})+s)>0}} 
hypergeom([a, b], [(3)/(2)], z)/GAMMA((3)/(2)) = ((2)^(a + b -((5)/(2))))/(Pi*sqrt(z))*GAMMA(a -(1)/(2))*GAMMA(b -(1)/(2))*(z - 1)^((- a - b +((3)/(2)))/2)*(exp(+ Pi*I*(a + b -((3)/(2))))*LegendreP(a - b -((1)/(2)), - a - b +((3)/(2)), -sqrt(z))- LegendreP(a - b -((1)/(2)), - a - b +((3)/(2)), sqrt(z)))

Hypergeometric2F1Regularized[a, b, Divide[3,2], z] == Divide[(2)^(a + b -(Divide[5,2])),Pi*Sqrt[z]]*Gamma[a -Divide[1,2]]*Gamma[b -Divide[1,2]]*(z - 1)^((- a - b +(Divide[3,2]))/2)*(Exp[+ Pi*I*(a + b -(Divide[3,2]))]*LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[3,2]), 3, -Sqrt[z]]- LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[3,2]), 3, Sqrt[z]])
Failure Failure Error
Failed [4 / 16]
Result: Complex[2.2779820596001903, -1.628954540775632]
Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]]}


Result: Complex[0.907830443893564, 0.19750251034857133]
Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]]}

... skip entries to safe data
15.9.E23 𝐅 ⁡ ( a , b 3 2 ; z ) = 2 a + b - ( 5 / 2 ) π ⁢ z ⁢ Γ ⁡ ( a - 1 2 ) ⁢ Γ ⁡ ( b - 1 2 ) ⁢ ( z - 1 ) ( - a - b + ( 3 / 2 ) ) / 2 ⁢ ( e - π ⁢ i ⁢ ( a + b - ( 3 / 2 ) ) ⁢ P a - b - ( 1 / 2 ) - a - b + ( 3 / 2 ) ⁡ ( - z ) - P a - b - ( 1 / 2 ) - a - b + ( 3 / 2 ) ⁡ ( z ) ) scaled-hypergeometric-bold-F 𝑎 𝑏 3 2 𝑧 superscript 2 𝑎 𝑏 5 2 𝜋 𝑧 Euler-Gamma 𝑎 1 2 Euler-Gamma 𝑏 1 2 superscript 𝑧 1 𝑎 𝑏 3 2 2 superscript 𝑒 𝜋 imaginary-unit 𝑎 𝑏 3 2 Legendre-P-first-kind 𝑎 𝑏 3 2 𝑎 𝑏 1 2 𝑧 Legendre-P-first-kind 𝑎 𝑏 3 2 𝑎 𝑏 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{3}{2}};z\right)=% \frac{2^{a+b-(\ifrac{5}{2})}}{\pi\sqrt{z}}\Gamma\left(a-\tfrac{1}{2}\right)% \Gamma\left(b-\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{3}{2}))/2}\*\left(e^{-% \pi\mathrm{i}(a+b-(\ifrac{3}{2}))}P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}% \left(-\sqrt{z}\right)-P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt% {z}\right)\right)}}
\hyperOlverF@@{a}{b}{\tfrac{3}{2}}{z} = \frac{2^{a+b-(\ifrac{5}{2})}}{\pi\sqrt{z}}\EulerGamma@{a-\tfrac{1}{2}}\EulerGamma@{b-\tfrac{1}{2}}\*(z-1)^{(-a-b+(\ifrac{3}{2}))/2}\*\left(e^{-\pi\iunit(a+b-(\ifrac{3}{2}))}\assLegendreP[-a-b+(\ifrac{3}{2})]{a-b-(\ifrac{1}{2})}@{-\sqrt{z}}-\assLegendreP[-a-b+(\ifrac{3}{2})]{a-b-(\ifrac{1}{2})}@{\sqrt{z}}\right)		 a ≠ 1 2 , b ≠ 1 2 , 0 < | ph ⁡ z | , | ph ⁡ z | < π , ℜ ⁡ ( a - 1 2 ) > 0 , ℜ ⁡ ( b - 1 2 ) > 0 , | z | < 1 , ℜ ⁡ ( ( 3 2 ) + s ) > 0 formulae-sequence 𝑎 1 2 formulae-sequence 𝑏 1 2 formulae-sequence 0 phase 𝑧 formulae-sequence phase 𝑧 𝜋 formulae-sequence 𝑎 1 2 0 formulae-sequence 𝑏 1 2 0 formulae-sequence 𝑧 1 3 2 𝑠 0 {\displaystyle{\displaystyle a\neq\frac{1}{2},b\neq\frac{1}{2},0<|% \operatorname{ph}z|,|\operatorname{ph}z|<\pi,\Re(a-\tfrac{1}{2})>0,\Re(b-% \tfrac{1}{2})>0,|z|<1,\Re((\tfrac{3}{2})+s)>0}} 
hypergeom([a, b], [(3)/(2)], z)/GAMMA((3)/(2)) = ((2)^(a + b -((5)/(2))))/(Pi*sqrt(z))*GAMMA(a -(1)/(2))*GAMMA(b -(1)/(2))*(z - 1)^((- a - b +((3)/(2)))/2)*(exp(- Pi*I*(a + b -((3)/(2))))*LegendreP(a - b -((1)/(2)), - a - b +((3)/(2)), -sqrt(z))- LegendreP(a - b -((1)/(2)), - a - b +((3)/(2)), sqrt(z)))

Hypergeometric2F1Regularized[a, b, Divide[3,2], z] == Divide[(2)^(a + b -(Divide[5,2])),Pi*Sqrt[z]]*Gamma[a -Divide[1,2]]*Gamma[b -Divide[1,2]]*(z - 1)^((- a - b +(Divide[3,2]))/2)*(Exp[- Pi*I*(a + b -(Divide[3,2]))]*LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[3,2]), 3, -Sqrt[z]]- LegendreP[a - b -(Divide[1,2]), - a - b +(Divide[3,2]), 3, Sqrt[z]])
Failure Failure Error
Failed [4 / 16]
Result: Complex[4.158519870861856, 2.5132294016879406]
Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.2196744558627868, 0.17160454696174166]
Test Values: {Rule[a, 1.5], Rule[b, 1.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
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