Legendre and Related Functions - 14.3 Definitions and Hypergeometric Representations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.3.E1 𝖯 Ξ½ ΞΌ ⁑ ( x ) = ( 1 + x 1 - x ) ΞΌ / 2 ⁒ 𝐅 ⁑ ( Ξ½ + 1 , - Ξ½ ; 1 - ΞΌ ; 1 2 - 1 2 ⁒ x ) Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript 1 π‘₯ 1 π‘₯ πœ‡ 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 1 πœ‡ 1 2 1 2 π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=\left(\frac{1% +x}{1-x}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}% {2}x\right)}}
\FerrersP[\mu]{\nu}@{x} = \left(\frac{1+x}{1-x}\right)^{\mu/2}\hyperOlverF@{\nu+1}{-\nu}{1-\mu}{\tfrac{1}{2}-\tfrac{1}{2}x}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, mu, x) = ((1 + x)/(1 - x))^(mu/2)* hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)

LegendreP[\[Nu], \[Mu], x] == (Divide[1 + x,1 - x])^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]
Failure Failure
Failed [186 / 300]
Result: .299069150e-1-2.924977300*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: 1.647025838-2.840829287*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 2}

... skip entries to safe data
Failed [159 / 300]
Result: Complex[0.029906915825256147, -2.924977300264846]
Test Values: {Rule[x, 1.5], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.067091398010022, -0.8210135056644174]
Test Values: {Rule[x, 1.5], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.3.E2 𝖰 Ξ½ ΞΌ ⁑ ( x ) = Ο€ 2 ⁒ sin ⁑ ( ΞΌ ⁒ Ο€ ) ⁒ ( cos ⁑ ( ΞΌ ⁒ Ο€ ) ⁒ ( 1 + x 1 - x ) ΞΌ / 2 ⁒ 𝐅 ⁑ ( Ξ½ + 1 , - Ξ½ ; 1 - ΞΌ ; 1 2 - 1 2 ⁒ x ) - Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) Ξ“ ⁑ ( Ξ½ - ΞΌ + 1 ) ⁒ ( 1 - x 1 + x ) ΞΌ / 2 ⁒ 𝐅 ⁑ ( Ξ½ + 1 , - Ξ½ ; 1 + ΞΌ ; 1 2 - 1 2 ⁒ x ) ) Ferrers-Legendre-Q-first-kind πœ‡ 𝜈 π‘₯ πœ‹ 2 πœ‡ πœ‹ πœ‡ πœ‹ superscript 1 π‘₯ 1 π‘₯ πœ‡ 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 1 πœ‡ 1 2 1 2 π‘₯ Euler-Gamma 𝜈 πœ‡ 1 Euler-Gamma 𝜈 πœ‡ 1 superscript 1 π‘₯ 1 π‘₯ πœ‡ 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 1 πœ‡ 1 2 1 2 π‘₯ {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\frac{\pi}{2% \sin\left(\mu\pi\right)}\left(\cos\left(\mu\pi\right)\left(\frac{1+x}{1-x}% \right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x% \right)-\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)}\left% (\frac{1-x}{1+x}\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1+\mu;\tfrac{1}{2}-% \tfrac{1}{2}x\right)\right)}}
\FerrersQ[\mu]{\nu}@{x} = \frac{\pi}{2\sin@{\mu\pi}}\left(\cos@{\mu\pi}\left(\frac{1+x}{1-x}\right)^{\mu/2}\hyperOlverF@{\nu+1}{-\nu}{1-\mu}{\tfrac{1}{2}-\tfrac{1}{2}x}-\frac{\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+1}}\left(\frac{1-x}{1+x}\right)^{\mu/2}\hyperOlverF@{\nu+1}{-\nu}{1+\mu}{\tfrac{1}{2}-\tfrac{1}{2}x}\right)		 β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - ΞΌ + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝜈 πœ‡ 1 0 formulae-sequence 𝜈 πœ‡ 1 0 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1}} 
LegendreQ(nu, mu, x) = (Pi)/(2*sin(mu*Pi))*(cos(mu*Pi)*((1 + x)/(1 - x))^(mu/2)* hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)-(GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1))*((1 - x)/(1 + x))^(mu/2)* hypergeom([nu + 1, - nu], [1 + mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 + mu))

LegendreQ[\[Nu], \[Mu], x] == Divide[Pi,2*Sin[\[Mu]*Pi]]*(Cos[\[Mu]*Pi]*(Divide[1 + x,1 - x])^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]-Divide[Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]]*(Divide[1 - x,1 + x])^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 + \[Mu], Divide[1,2]-Divide[1,2]*x])
Failure Failure
Failed [52 / 120]
Result: -4.859700475+.2639835842*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


Result: -4.893385611-2.430027023*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 2}

... skip entries to safe data
Failed [54 / 135]
Result: Complex[-4.859700475422212, 0.2639835832089452]
Test Values: {Rule[x, 1.5], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.597069591108201, 8.997773008153189]
Test Values: {Rule[x, 1.5], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
14.3.E3 𝐅 ⁑ ( a , b ; c ; x ) = 1 Ξ“ ⁑ ( c ) ⁒ F ⁑ ( a , b ; c ; x ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 π‘₯ 1 Euler-Gamma 𝑐 Gauss-hypergeometric-F π‘Ž 𝑏 𝑐 π‘₯ {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;x\right)=\frac{1}{\Gamma% \left(c\right)}F\left(a,b;c;x\right)}}
\hyperOlverF@{a}{b}{c}{x} = \frac{1}{\EulerGamma@{c}}\hyperF@{a}{b}{c}{x}		 β„œ ⁑ c > 0 , | x | < 1 formulae-sequence 𝑐 0 π‘₯ 1 {\displaystyle{\displaystyle\Re c>0,|x|<1}} 
hypergeom([a, b], [c], x)/GAMMA(c) = (1)/(GAMMA(c))*hypergeom([a, b], [c], x)

Hypergeometric2F1Regularized[a, b, c, x] == Divide[1,Gamma[c]]*Hypergeometric2F1[a, b, c, x]
Successful Successful - Successful [Tested: 108]
14.3.E4 𝖯 Ξ½ m ⁑ ( x ) = ( - 1 ) m ⁒ Ξ“ ⁑ ( Ξ½ + m + 1 ) 2 m ⁒ Ξ“ ⁑ ( Ξ½ - m + 1 ) ⁒ ( 1 - x 2 ) m / 2 ⁒ 𝐅 ⁑ ( Ξ½ + m + 1 , m - Ξ½ ; m + 1 ; 1 2 - 1 2 ⁒ x ) Ferrers-Legendre-P-first-kind π‘š 𝜈 π‘₯ superscript 1 π‘š Euler-Gamma 𝜈 π‘š 1 superscript 2 π‘š Euler-Gamma 𝜈 π‘š 1 superscript 1 superscript π‘₯ 2 π‘š 2 scaled-hypergeometric-bold-F 𝜈 π‘š 1 π‘š 𝜈 π‘š 1 1 2 1 2 π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{% \Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(\nu-m+1\right)}\left(1-x^{2}% \right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x% \right)}}
\FerrersP[m]{\nu}@{x} = (-1)^{m}\frac{\EulerGamma@{\nu+m+1}}{2^{m}\EulerGamma@{\nu-m+1}}\left(1-x^{2}\right)^{m/2}\hyperOlverF@{\nu+m+1}{m-\nu}{m+1}{\tfrac{1}{2}-\tfrac{1}{2}x}		 β„œ ⁑ ( Ξ½ + m + 1 ) > 0 , β„œ ⁑ ( Ξ½ - m + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝜈 π‘š 1 0 formulae-sequence 𝜈 π‘š 1 0 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle\Re(\nu+m+1)>0,\Re(\nu-m+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1}} 
LegendreP(nu, m, x) = (- 1)^(m)*(GAMMA(nu + m + 1))/((2)^(m)* GAMMA(nu - m + 1))*(1 - (x)^(2))^(m/2)* hypergeom([nu + m + 1, m - nu], [m + 1], (1)/(2)-(1)/(2)*x)/GAMMA(m + 1)

LegendreP[\[Nu], m, x] == (- 1)^(m)*Divide[Gamma[\[Nu]+ m + 1],(2)^(m)* Gamma[\[Nu]- m + 1]]*(1 - (x)^(2))^(m/2)* Hypergeometric2F1Regularized[\[Nu]+ m + 1, m - \[Nu], m + 1, Divide[1,2]-Divide[1,2]*x]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
14.3.E5 𝖯 Ξ½ m ⁑ ( x ) = ( - 1 ) m ⁒ Ξ“ ⁑ ( Ξ½ + m + 1 ) Ξ“ ⁑ ( Ξ½ - m + 1 ) ⁒ ( 1 - x 1 + x ) m / 2 ⁒ 𝐅 ⁑ ( Ξ½ + 1 , - Ξ½ ; m + 1 ; 1 2 - 1 2 ⁒ x ) Ferrers-Legendre-P-first-kind π‘š 𝜈 π‘₯ superscript 1 π‘š Euler-Gamma 𝜈 π‘š 1 Euler-Gamma 𝜈 π‘š 1 superscript 1 π‘₯ 1 π‘₯ π‘š 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 π‘š 1 1 2 1 2 π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{% \Gamma\left(\nu+m+1\right)}{\Gamma\left(\nu-m+1\right)}\left(\frac{1-x}{1+x}% \right)^{m/2}\mathbf{F}\left(\nu+1,-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}
\FerrersP[m]{\nu}@{x} = (-1)^{m}\frac{\EulerGamma@{\nu+m+1}}{\EulerGamma@{\nu-m+1}}\left(\frac{1-x}{1+x}\right)^{m/2}\hyperOlverF@{\nu+1}{-\nu}{m+1}{\tfrac{1}{2}-\tfrac{1}{2}x}		 β„œ ⁑ ( Ξ½ + m + 1 ) > 0 , β„œ ⁑ ( Ξ½ - m + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝜈 π‘š 1 0 formulae-sequence 𝜈 π‘š 1 0 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle\Re(\nu+m+1)>0,\Re(\nu-m+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1}} 
LegendreP(nu, m, x) = (- 1)^(m)*(GAMMA(nu + m + 1))/(GAMMA(nu - m + 1))*((1 - x)/(1 + x))^(m/2)* hypergeom([nu + 1, - nu], [m + 1], (1)/(2)-(1)/(2)*x)/GAMMA(m + 1)

LegendreP[\[Nu], m, x] == (- 1)^(m)*Divide[Gamma[\[Nu]+ m + 1],Gamma[\[Nu]- m + 1]]*(Divide[1 - x,1 + x])^(m/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], m + 1, Divide[1,2]-Divide[1,2]*x]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
14.3.E6 P Ξ½ ΞΌ ⁑ ( x ) = ( x + 1 x - 1 ) ΞΌ / 2 ⁒ 𝐅 ⁑ ( Ξ½ + 1 , - Ξ½ ; 1 - ΞΌ ; 1 2 - 1 2 ⁒ x ) Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript π‘₯ 1 π‘₯ 1 πœ‡ 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 1 πœ‡ 1 2 1 2 π‘₯ {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\left(\frac{x+1}{x-1}% \right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x% \right)}}
\assLegendreP[\mu]{\nu}@{x} = \left(\frac{x+1}{x-1}\right)^{\mu/2}\hyperOlverF@{\nu+1}{-\nu}{1-\mu}{\tfrac{1}{2}-\tfrac{1}{2}x}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, mu, x) = ((x + 1)/(x - 1))^(mu/2)* hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)

LegendreP[\[Nu], \[Mu], 3, x] == (Divide[x + 1,x - 1])^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]
Failure Failure
Failed [106 / 300]
Result: -4.719014115+.3779003255*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}


Result: -1.667629478-3.026452547*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -1/2+1/2*I*3^(1/2), x = 1/2}

... skip entries to safe data
Failed [79 / 300]
Result: Complex[-4.719014112853729, 0.37790032166140924]
Test Values: {Rule[x, 0.5], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.667629477217065, -3.026452547389477]
Test Values: {Rule[x, 0.5], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Ξ½, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
14.3.E7 Q Ξ½ ΞΌ ⁑ ( x ) = e ΞΌ ⁒ Ο€ ⁒ i ⁒ Ο€ 1 / 2 ⁒ Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ ( x 2 - 1 ) ΞΌ / 2 2 Ξ½ + 1 ⁒ x Ξ½ + ΞΌ + 1 ⁒ 𝐅 ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 , 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ; Ξ½ + 3 2 ; 1 x 2 ) Legendre-Q-second-kind πœ‡ 𝜈 π‘₯ superscript 𝑒 πœ‡ πœ‹ 𝑖 superscript πœ‹ 1 2 Euler-Gamma 𝜈 πœ‡ 1 superscript superscript π‘₯ 2 1 πœ‡ 2 superscript 2 𝜈 1 superscript π‘₯ 𝜈 πœ‡ 1 scaled-hypergeometric-bold-F 1 2 𝜈 1 2 πœ‡ 1 1 2 𝜈 1 2 πœ‡ 1 2 𝜈 3 2 1 superscript π‘₯ 2 {\displaystyle{\displaystyle Q^{\mu}_{\nu}\left(x\right)=e^{\mu\pi i}\frac{\pi% ^{1/2}\Gamma\left(\nu+\mu+1\right)\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}x^{% \nu+\mu+1}}\mathbf{F}\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+% \tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right)}}
\assLegendreQ[\mu]{\nu}@{x} = e^{\mu\pi i}\frac{\pi^{1/2}\EulerGamma@{\nu+\mu+1}\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}x^{\nu+\mu+1}}\hyperOlverF@{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1}{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2}}{\nu+\tfrac{3}{2}}{\frac{1}{x^{2}}}		 β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 𝜈 πœ‡ 1 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0}} 
LegendreQ(nu, mu, x) = exp(mu*Pi*I)*((Pi)^(1/2)* GAMMA(nu + mu + 1)*((x)^(2)- 1)^(mu/2))/((2)^(nu + 1)* (x)^(nu + mu + 1))*hypergeom([(1)/(2)*nu +(1)/(2)*mu + 1, (1)/(2)*nu +(1)/(2)*mu +(1)/(2)], [nu +(3)/(2)], (1)/((x)^(2)))/GAMMA(nu +(3)/(2))

LegendreQ[\[Nu], \[Mu], 3, x] == Exp[\[Mu]*Pi*I]*Divide[(Pi)^(1/2)* Gamma[\[Nu]+ \[Mu]+ 1]*((x)^(2)- 1)^(\[Mu]/2),(2)^(\[Nu]+ 1)* (x)^(\[Nu]+ \[Mu]+ 1)]*Hypergeometric2F1Regularized[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1, Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2], \[Nu]+Divide[3,2], Divide[1,(x)^(2)]]
Failure Failure
Failed [28 / 200]
Result: Float(undefined)+Float(undefined)*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -3/2, x = 3/2, nu+mu = 1}


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

... skip entries to safe data
 Successful [Tested: 138]
14.3.E8 P Ξ½ m ⁑ ( x ) = Ξ“ ⁑ ( Ξ½ + m + 1 ) 2 m ⁒ Ξ“ ⁑ ( Ξ½ - m + 1 ) ⁒ ( x 2 - 1 ) m / 2 ⁒ 𝐅 ⁑ ( Ξ½ + m + 1 , m - Ξ½ ; m + 1 ; 1 2 - 1 2 ⁒ x ) Legendre-P-first-kind π‘š 𝜈 π‘₯ Euler-Gamma 𝜈 π‘š 1 superscript 2 π‘š Euler-Gamma 𝜈 π‘š 1 superscript superscript π‘₯ 2 1 π‘š 2 scaled-hypergeometric-bold-F 𝜈 π‘š 1 π‘š 𝜈 π‘š 1 1 2 1 2 π‘₯ {\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m% +1\right)}{2^{m}\Gamma\left(\nu-m+1\right)}\left(x^{2}-1\right)^{m/2}\mathbf{F% }\left(\nu+m+1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}
\assLegendreP[m]{\nu}@{x} = \frac{\EulerGamma@{\nu+m+1}}{2^{m}\EulerGamma@{\nu-m+1}}\left(x^{2}-1\right)^{m/2}\hyperOlverF@{\nu+m+1}{m-\nu}{m+1}{\tfrac{1}{2}-\tfrac{1}{2}x}		 β„œ ⁑ ( Ξ½ + m + 1 ) > 0 , β„œ ⁑ ( Ξ½ - m + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝜈 π‘š 1 0 formulae-sequence 𝜈 π‘š 1 0 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle\Re(\nu+m+1)>0,\Re(\nu-m+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1}} 
LegendreP(nu, m, x) = (GAMMA(nu + m + 1))/((2)^(m)* GAMMA(nu - m + 1))*((x)^(2)- 1)^(m/2)* hypergeom([nu + m + 1, m - nu], [m + 1], (1)/(2)-(1)/(2)*x)/GAMMA(m + 1)

LegendreP[\[Nu], m, 3, x] == Divide[Gamma[\[Nu]+ m + 1],(2)^(m)* Gamma[\[Nu]- m + 1]]*((x)^(2)- 1)^(m/2)* Hypergeometric2F1Regularized[\[Nu]+ m + 1, m - \[Nu], m + 1, Divide[1,2]-Divide[1,2]*x]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
14.3.E9 P Ξ½ - ΞΌ ⁑ ( x ) = ( x - 1 x + 1 ) ΞΌ / 2 ⁒ 𝐅 ⁑ ( Ξ½ + 1 , - Ξ½ ; ΞΌ + 1 ; 1 2 - 1 2 ⁒ x ) Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript π‘₯ 1 π‘₯ 1 πœ‡ 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 πœ‡ 1 1 2 1 2 π‘₯ {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\left(\frac{x-1}{x+1% }\right)^{\mu/2}\mathbf{F}\left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x% \right)}}
\assLegendreP[-\mu]{\nu}@{x} = \left(\frac{x-1}{x+1}\right)^{\mu/2}\hyperOlverF@{\nu+1}{-\nu}{\mu+1}{\tfrac{1}{2}-\tfrac{1}{2}x}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, - mu, x) = ((x - 1)/(x + 1))^(mu/2)* hypergeom([nu + 1, - nu], [mu + 1], (1)/(2)-(1)/(2)*x)/GAMMA(mu + 1)

LegendreP[\[Nu], - \[Mu], 3, x] == (Divide[x - 1,x + 1])^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], \[Mu]+ 1, Divide[1,2]-Divide[1,2]*x]
Failure Successful
Failed [27 / 300]
Result: Float(undefined)+Float(undefined)*I
Test Values: {mu = -2, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


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

... skip entries to safe data
 Successful [Tested: 300]
14.3.E10 𝑸 Ξ½ ΞΌ ⁑ ( x ) = e - ΞΌ ⁒ Ο€ ⁒ i ⁒ Q Ξ½ ΞΌ ⁑ ( x ) Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) associated-Legendre-black-Q πœ‡ 𝜈 π‘₯ superscript 𝑒 πœ‡ πœ‹ 𝑖 Legendre-Q-second-kind πœ‡ 𝜈 π‘₯ Euler-Gamma 𝜈 πœ‡ 1 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=e^{-\mu% \pi i}\frac{Q^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}}}
\assLegendreOlverQ[\mu]{\nu}@{x} = e^{-\mu\pi i}\frac{\assLegendreQ[\mu]{\nu}@{x}}{\EulerGamma@{\nu+\mu+1}}		 β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 𝜈 πœ‡ 1 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0}} 
exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = exp(- mu*Pi*I)*(LegendreQ(nu, mu, x))/(GAMMA(nu + mu + 1))

Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == Exp[- \[Mu]*Pi*I]*Divide[LegendreQ[\[Nu], \[Mu], 3, x],Gamma[\[Nu]+ \[Mu]+ 1]]
Successful Successful - Successful [Tested: 207]
14.3.E11 𝖯 Ξ½ ΞΌ ⁑ ( x ) = cos ⁑ ( 1 2 ⁒ ( Ξ½ + ΞΌ ) ⁒ Ο€ ) ⁒ w 1 ⁒ ( Ξ½ , ΞΌ , x ) + sin ⁑ ( 1 2 ⁒ ( Ξ½ + ΞΌ ) ⁒ Ο€ ) ⁒ w 2 ⁒ ( Ξ½ , ΞΌ , x ) Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ 1 2 𝜈 πœ‡ πœ‹ subscript 𝑀 1 𝜈 πœ‡ π‘₯ 1 2 𝜈 πœ‡ πœ‹ subscript 𝑀 2 𝜈 πœ‡ π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=\cos\left(% \tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(\nu,\mu,x)+\sin\left(\tfrac{1}{2}(\nu+\mu% )\pi\right)w_{2}(\nu,\mu,x)}}
\FerrersP[\mu]{\nu}@{x} = \cos@{\tfrac{1}{2}(\nu+\mu)\pi}w_{1}(\nu,\mu,x)+\sin@{\tfrac{1}{2}(\nu+\mu)\pi}w_{2}(\nu,\mu,x)		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, mu, x) = cos((1)/(2)*(nu + mu)*Pi)*w[1](nu , mu , x)+ sin((1)/(2)*(nu + mu)*Pi)*w[2](nu , mu , x)

LegendreP[\[Nu], \[Mu], x] == Cos[Divide[1,2]*(\[Nu]+ \[Mu])*Pi]*Subscript[w, 1][\[Nu], \[Mu], x]+ Sin[Divide[1,2]*(\[Nu]+ \[Mu])*Pi]*Subscript[w, 2][\[Nu], \[Mu], x]
Failure Failure
Failed [300 / 300]
Result: .1996315555-2.444256460*I+(-.424833882+3.265828322*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, 1.500000000)
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = 1/2*3^(1/2)+1/2*I, w[2] = 1/2*3^(1/2)+1/2*I}


Result: .1996315555-2.444256460*I+(.206784146+.21312792e-1*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, 1.500000000)
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = 1/2*3^(1/2)+1/2*I, w[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
14.3.E12 𝖰 Ξ½ ΞΌ ⁑ ( x ) = - 1 2 ⁒ Ο€ ⁒ sin ⁑ ( 1 2 ⁒ ( Ξ½ + ΞΌ ) ⁒ Ο€ ) ⁒ w 1 ⁒ ( Ξ½ , ΞΌ , x ) + 1 2 ⁒ Ο€ ⁒ cos ⁑ ( 1 2 ⁒ ( Ξ½ + ΞΌ ) ⁒ Ο€ ) ⁒ w 2 ⁒ ( Ξ½ , ΞΌ , x ) Ferrers-Legendre-Q-first-kind πœ‡ 𝜈 π‘₯ 1 2 πœ‹ 1 2 𝜈 πœ‡ πœ‹ subscript 𝑀 1 𝜈 πœ‡ π‘₯ 1 2 πœ‹ 1 2 𝜈 πœ‡ πœ‹ subscript 𝑀 2 𝜈 πœ‡ π‘₯ {\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\tfrac{1}{2}% \pi\sin\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(\nu,\mu,x)+\tfrac{1}{2}\pi% \cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x)}}
\FerrersQ[\mu]{\nu}@{x} = -\tfrac{1}{2}\pi\sin@{\tfrac{1}{2}(\nu+\mu)\pi}w_{1}(\nu,\mu,x)+\tfrac{1}{2}\pi\cos@{\tfrac{1}{2}(\nu+\mu)\pi}w_{2}(\nu,\mu,x)		 β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - ΞΌ + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝜈 πœ‡ 1 0 formulae-sequence 𝜈 πœ‡ 1 0 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1}} 
LegendreQ(nu, mu, x) = -(1)/(2)*Pi*sin((1)/(2)*(nu + mu)*Pi)*w[1](nu , mu , x)+(1)/(2)*Pi*cos((1)/(2)*(nu + mu)*Pi)*w[2](nu , mu , x)

LegendreQ[\[Nu], \[Mu], x] == -Divide[1,2]*Pi*Sin[Divide[1,2]*(\[Nu]+ \[Mu])*Pi]*Subscript[w, 1][\[Nu], \[Mu], x]+Divide[1,2]*Pi*Cos[Divide[1,2]*(\[Nu]+ \[Mu])*Pi]*Subscript[w, 2][\[Nu], \[Mu], x]
Failure Failure
Failed [300 / 300]
Result: -3.819326549-.1470472359*I+(5.421288855+1.025621334*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, 1.500000000)
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = 1/2*3^(1/2)+1/2*I, w[2] = 1/2*3^(1/2)+1/2*I}


Result: -3.819326549-.1470472359*I+(-.33478055e-1+.324815778*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, 1.500000000)
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2, w[1] = 1/2*3^(1/2)+1/2*I, w[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
14.3.E13 w 1 ⁒ ( Ξ½ , ΞΌ , x ) = 2 ΞΌ ⁒ Ξ“ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ) Ξ“ ⁑ ( 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 ) ⁒ ( 1 - x 2 ) - ΞΌ / 2 ⁒ 𝐅 ⁑ ( - 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ , 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 2 ; 1 2 ; x 2 ) subscript 𝑀 1 𝜈 πœ‡ π‘₯ superscript 2 πœ‡ Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 2 Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 superscript 1 superscript π‘₯ 2 πœ‡ 2 scaled-hypergeometric-bold-F 1 2 𝜈 1 2 πœ‡ 1 2 𝜈 1 2 πœ‡ 1 2 1 2 superscript π‘₯ 2 {\displaystyle{\displaystyle w_{1}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1% }{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}% {2}\mu+1\right)}\left(1-x^{2}\right)^{-\mu/2}\mathbf{F}\left(-\tfrac{1}{2}\nu-% \tfrac{1}{2}\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2};\tfrac{1}{2};x^{2% }\right)}}
w_{1}(\nu,\mu,x) = \frac{2^{\mu}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+1}}\left(1-x^{2}\right)^{-\mu/2}\hyperOlverF@{-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu}{\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2}}{\tfrac{1}{2}}{x^{2}}		 β„œ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 ) > 0 , | ( x 2 ) | < 1 formulae-sequence 1 2 𝜈 1 2 πœ‡ 1 2 0 formulae-sequence 1 2 𝜈 1 2 πœ‡ 1 0 superscript π‘₯ 2 1 {\displaystyle{\displaystyle\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})>0,% \Re(\frac{1}{2}\nu-\frac{1}{2}\mu+1)>0,|(x^{2})|<1}} 
w[1](nu , mu , x) = ((2)^(mu)* GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu + 1))*(1 - (x)^(2))^(- mu/2)* hypergeom([-(1)/(2)*nu -(1)/(2)*mu, (1)/(2)*nu -(1)/(2)*mu +(1)/(2)], [(1)/(2)], (x)^(2))/GAMMA((1)/(2))

Subscript[w, 1][\[Nu], \[Mu], x] == Divide[(2)^\[Mu]* Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1]]*(1 - (x)^(2))^(- \[Mu]/2)* Hypergeometric2F1Regularized[-Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu], Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2], Divide[1,2], (x)^(2)]
Failure Failure
Failed [300 / 300]
Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, .5000000000)-.6893070382-.1737378889*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2, w[1] = 1/2*3^(1/2)+1/2*I}


Result: (-.5000000000+.8660254040*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, .5000000000)-.6893070382-.1737378889*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2, w[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
14.3.E14 w 2 ⁒ ( Ξ½ , ΞΌ , x ) = 2 ΞΌ ⁒ Ξ“ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 ) Ξ“ ⁑ ( 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 2 ) ⁒ x ⁒ ( 1 - x 2 ) - ΞΌ / 2 ⁒ 𝐅 ⁑ ( 1 2 - 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ , 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 ; 3 2 ; x 2 ) subscript 𝑀 2 𝜈 πœ‡ π‘₯ superscript 2 πœ‡ Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 2 π‘₯ superscript 1 superscript π‘₯ 2 πœ‡ 2 scaled-hypergeometric-bold-F 1 2 1 2 𝜈 1 2 πœ‡ 1 2 𝜈 1 2 πœ‡ 1 3 2 superscript π‘₯ 2 {\displaystyle{\displaystyle w_{2}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1% }{2}\nu+\frac{1}{2}\mu+1\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+% \frac{1}{2}\right)}x\left(1-x^{2}\right)^{-\mu/2}\mathbf{F}\left(\tfrac{1}{2}-% \tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1;\tfrac{3}{2}% ;x^{2}\right)}}
w_{2}(\nu,\mu,x) = \frac{2^{\mu}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+1}}{\EulerGamma@{\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}}}x\left(1-x^{2}\right)^{-\mu/2}\hyperOlverF@{\tfrac{1}{2}-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu}{\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1}{\tfrac{3}{2}}{x^{2}}		 β„œ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 ) > 0 , β„œ ⁑ ( 1 2 ⁒ Ξ½ - 1 2 ⁒ ΞΌ + 1 2 ) > 0 , | ( x 2 ) | < 1 formulae-sequence 1 2 𝜈 1 2 πœ‡ 1 0 formulae-sequence 1 2 𝜈 1 2 πœ‡ 1 2 0 superscript π‘₯ 2 1 {\displaystyle{\displaystyle\Re(\frac{1}{2}\nu+\frac{1}{2}\mu+1)>0,\Re(\frac{1% }{2}\nu-\frac{1}{2}\mu+\frac{1}{2})>0,|(x^{2})|<1}} 
w[2](nu , mu , x) = ((2)^(mu)* GAMMA((1)/(2)*nu +(1)/(2)*mu + 1))/(GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)))*x*(1 - (x)^(2))^(- mu/2)* hypergeom([(1)/(2)-(1)/(2)*nu -(1)/(2)*mu, (1)/(2)*nu -(1)/(2)*mu + 1], [(3)/(2)], (x)^(2))/GAMMA((3)/(2))

Subscript[w, 2][\[Nu], \[Mu], x] == Divide[(2)^\[Mu]* Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]]*x*(1 - (x)^(2))^(- \[Mu]/2)* Hypergeometric2F1Regularized[Divide[1,2]-Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu], Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1, Divide[3,2], (x)^(2)]
Failure Failure
Failed [300 / 300]
Result: (.8660254040+.5000000000*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, .5000000000)-.4687612945-.2577588545*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2, w[2] = 1/2*3^(1/2)+1/2*I}


Result: (-.5000000000+.8660254040*I)*(.8660254040+.5000000000*I, .8660254040+.5000000000*I, .5000000000)-.4687612945-.2577588545*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2, w[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Error
14.3.E15 P Ξ½ - ΞΌ ⁑ ( x ) = 2 - ΞΌ ⁒ ( x 2 - 1 ) ΞΌ / 2 ⁒ 𝐅 ⁑ ( ΞΌ - Ξ½ , Ξ½ + ΞΌ + 1 ; ΞΌ + 1 ; 1 2 - 1 2 ⁒ x ) Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript 2 πœ‡ superscript superscript π‘₯ 2 1 πœ‡ 2 scaled-hypergeometric-bold-F πœ‡ 𝜈 𝜈 πœ‡ 1 πœ‡ 1 1 2 1 2 π‘₯ {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}\left(x^{2}-% 1\right)^{\mu/2}\mathbf{F}\left(\mu-\nu,\nu+\mu+1;\mu+1;\tfrac{1}{2}-\tfrac{1}% {2}x\right)}}
\assLegendreP[-\mu]{\nu}@{x} = 2^{-\mu}\left(x^{2}-1\right)^{\mu/2}\hyperOlverF@{\mu-\nu}{\nu+\mu+1}{\mu+1}{\tfrac{1}{2}-\tfrac{1}{2}x}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, - mu, x) = (2)^(- mu)*((x)^(2)- 1)^(mu/2)* hypergeom([mu - nu, nu + mu + 1], [mu + 1], (1)/(2)-(1)/(2)*x)/GAMMA(mu + 1)

LegendreP[\[Nu], - \[Mu], 3, x] == (2)^(- \[Mu])*((x)^(2)- 1)^(\[Mu]/2)* Hypergeometric2F1Regularized[\[Mu]- \[Nu], \[Nu]+ \[Mu]+ 1, \[Mu]+ 1, Divide[1,2]-Divide[1,2]*x]
Failure Failure
Failed [27 / 300]
Result: Float(undefined)+Float(undefined)*I
Test Values: {mu = -2, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


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

... skip entries to safe data
 Successful [Tested: 300]
14.3.E16 cos ⁑ ( Ξ½ ⁒ Ο€ ) ⁒ P Ξ½ - ΞΌ ⁑ ( x ) = 2 Ξ½ ⁒ Ο€ 1 / 2 ⁒ x Ξ½ - ΞΌ ⁒ ( x 2 - 1 ) ΞΌ / 2 Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ 𝐅 ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ , 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ + 1 2 ; 1 2 - Ξ½ ; 1 x 2 ) - Ο€ 1 / 2 ⁒ ( x 2 - 1 ) ΞΌ / 2 2 Ξ½ + 1 ⁒ Ξ“ ⁑ ( ΞΌ - Ξ½ ) ⁒ x Ξ½ + ΞΌ + 1 ⁒ 𝐅 ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 , 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ; Ξ½ + 3 2 ; 1 x 2 ) 𝜈 πœ‹ Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript 2 𝜈 superscript πœ‹ 1 2 superscript π‘₯ 𝜈 πœ‡ superscript superscript π‘₯ 2 1 πœ‡ 2 Euler-Gamma 𝜈 πœ‡ 1 scaled-hypergeometric-bold-F 1 2 πœ‡ 1 2 𝜈 1 2 πœ‡ 1 2 𝜈 1 2 1 2 𝜈 1 superscript π‘₯ 2 superscript πœ‹ 1 2 superscript superscript π‘₯ 2 1 πœ‡ 2 superscript 2 𝜈 1 Euler-Gamma πœ‡ 𝜈 superscript π‘₯ 𝜈 πœ‡ 1 scaled-hypergeometric-bold-F 1 2 𝜈 1 2 πœ‡ 1 1 2 𝜈 1 2 πœ‡ 1 2 𝜈 3 2 1 superscript π‘₯ 2 {\displaystyle{\displaystyle\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right% )=\frac{2^{\nu}\pi^{1/2}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}}{\Gamma\left(% \nu+\mu+1\right)}\mathbf{F}\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}% \mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\tfrac{1}{2}-\nu;\frac{1}{x^{2}}\right)-\frac% {\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}\Gamma\left(\mu-\nu\right)x^{% \nu+\mu+1}}\mathbf{F}\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+% \tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right)}}
\cos@{\nu\pi}\assLegendreP[-\mu]{\nu}@{x} = \frac{2^{\nu}\pi^{1/2}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}}{\EulerGamma@{\nu+\mu+1}}\hyperOlverF@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu}{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}}{\tfrac{1}{2}-\nu}{\frac{1}{x^{2}}}-\frac{\pi^{1/2}\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}\EulerGamma@{\mu-\nu}x^{\nu+\mu+1}}\hyperOlverF@{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1}{\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2}}{\nu+\tfrac{3}{2}}{\frac{1}{x^{2}}}		 β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 , β„œ ⁑ ( ΞΌ - Ξ½ ) > 0 formulae-sequence 𝜈 πœ‡ 1 0 πœ‡ 𝜈 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\mu-\nu)>0}} 
cos(nu*Pi)*LegendreP(nu, - mu, x) = ((2)^(nu)* (Pi)^(1/2)* (x)^(nu - mu)*((x)^(2)- 1)^(mu/2))/(GAMMA(nu + mu + 1))*hypergeom([(1)/(2)*mu -(1)/(2)*nu, (1)/(2)*mu -(1)/(2)*nu +(1)/(2)], [(1)/(2)- nu], (1)/((x)^(2)))/GAMMA((1)/(2)- nu)-((Pi)^(1/2)*((x)^(2)- 1)^(mu/2))/((2)^(nu + 1)* GAMMA(mu - nu)*(x)^(nu + mu + 1))*hypergeom([(1)/(2)*nu +(1)/(2)*mu + 1, (1)/(2)*nu +(1)/(2)*mu +(1)/(2)], [nu +(3)/(2)], (1)/((x)^(2)))/GAMMA(nu +(3)/(2))

Cos[\[Nu]*Pi]*LegendreP[\[Nu], - \[Mu], 3, x] == Divide[(2)^\[Nu]* (Pi)^(1/2)* (x)^(\[Nu]- \[Mu])*((x)^(2)- 1)^(\[Mu]/2),Gamma[\[Nu]+ \[Mu]+ 1]]*Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu], Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2], Divide[1,2]- \[Nu], Divide[1,(x)^(2)]]-Divide[(Pi)^(1/2)*((x)^(2)- 1)^(\[Mu]/2),(2)^(\[Nu]+ 1)* Gamma[\[Mu]- \[Nu]]*(x)^(\[Nu]+ \[Mu]+ 1)]*Hypergeometric2F1Regularized[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1, Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2], \[Nu]+Divide[3,2], Divide[1,(x)^(2)]]
Failure Failure
Failed [14 / 58]
Result: Float(undefined)+Float(undefined)*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -3/2, x = 3/2}


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

... skip entries to safe data
 Successful [Tested: 64]
14.3.E17 P Ξ½ - ΞΌ ⁑ ( x ) = Ο€ ⁒ ( x 2 - 1 ) ΞΌ / 2 2 ΞΌ ⁒ ( 𝐅 ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ , 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ; 1 2 ; x 2 ) Ξ“ ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ + 1 2 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 ) - x ⁒ 𝐅 ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ + 1 2 , 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 ; 3 2 ; x 2 ) Ξ“ ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ) ) Legendre-P-first-kind πœ‡ 𝜈 π‘₯ πœ‹ superscript superscript π‘₯ 2 1 πœ‡ 2 superscript 2 πœ‡ scaled-hypergeometric-bold-F 1 2 πœ‡ 1 2 𝜈 1 2 𝜈 1 2 πœ‡ 1 2 1 2 superscript π‘₯ 2 Euler-Gamma 1 2 πœ‡ 1 2 𝜈 1 2 Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 π‘₯ scaled-hypergeometric-bold-F 1 2 πœ‡ 1 2 𝜈 1 2 1 2 𝜈 1 2 πœ‡ 1 3 2 superscript π‘₯ 2 Euler-Gamma 1 2 πœ‡ 1 2 𝜈 Euler-Gamma 1 2 𝜈 1 2 πœ‡ 1 2 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{\pi\left(x^{2}% -1\right)^{\mu/2}}{2^{\mu}}\left(\frac{\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}% {2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{% \Gamma\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}\right)\Gamma\left(\frac{% 1}{2}\nu+\frac{1}{2}\mu+1\right)}-\frac{x\mathbf{F}\left(\frac{1}{2}\mu-\frac{% 1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}% {\Gamma\left(\frac{1}{2}\mu-\frac{1}{2}\nu\right)\Gamma\left(\frac{1}{2}\nu+% \frac{1}{2}\mu+\frac{1}{2}\right)}\right)}}
\assLegendreP[-\mu]{\nu}@{x} = \frac{\pi\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}}\left(\frac{\hyperOlverF@{\frac{1}{2}\mu-\frac{1}{2}\nu}{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}{\frac{1}{2}}{x^{2}}}{\EulerGamma@{\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+1}}-\frac{x\hyperOlverF@{\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}}{\frac{1}{2}\nu+\frac{1}{2}\mu+1}{\frac{3}{2}}{x^{2}}}{\EulerGamma@{\frac{1}{2}\mu-\frac{1}{2}\nu}\EulerGamma@{\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}}}\right)		 β„œ ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ + 1 2 ) > 0 , β„œ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 ) > 0 , β„œ ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ ) > 0 , β„œ ⁑ ( 1 2 ⁒ Ξ½ + 1 2 ⁒ ΞΌ + 1 2 ) > 0 , | ( x 2 ) | < 1 formulae-sequence 1 2 πœ‡ 1 2 𝜈 1 2 0 formulae-sequence 1 2 𝜈 1 2 πœ‡ 1 0 formulae-sequence 1 2 πœ‡ 1 2 𝜈 0 formulae-sequence 1 2 𝜈 1 2 πœ‡ 1 2 0 superscript π‘₯ 2 1 {\displaystyle{\displaystyle\Re(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2})>0,% \Re(\frac{1}{2}\nu+\frac{1}{2}\mu+1)>0,\Re(\frac{1}{2}\mu-\frac{1}{2}\nu)>0,% \Re(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2})>0,|(x^{2})|<1}} 
LegendreP(nu, - mu, x) = (Pi*((x)^(2)- 1)^(mu/2))/((2)^(mu))*((hypergeom([(1)/(2)*mu -(1)/(2)*nu, (1)/(2)*nu +(1)/(2)*mu +(1)/(2)], [(1)/(2)], (x)^(2))/GAMMA((1)/(2)))/(GAMMA((1)/(2)*mu -(1)/(2)*nu +(1)/(2))*GAMMA((1)/(2)*nu +(1)/(2)*mu + 1))-(x*hypergeom([(1)/(2)*mu -(1)/(2)*nu +(1)/(2), (1)/(2)*nu +(1)/(2)*mu + 1], [(3)/(2)], (x)^(2))/GAMMA((3)/(2)))/(GAMMA((1)/(2)*mu -(1)/(2)*nu)*GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2))))

LegendreP[\[Nu], - \[Mu], 3, x] == Divide[Pi*((x)^(2)- 1)^(\[Mu]/2),(2)^\[Mu]]*(Divide[Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu], Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2], Divide[1,2], (x)^(2)],Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]]-Divide[x*Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2], Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1, Divide[3,2], (x)^(2)],Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]]*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]]])
Failure Failure Successful [Tested: 29] Successful [Tested: 32]
14.3.E18 P Ξ½ - ΞΌ ⁑ ( x ) = 2 - ΞΌ ⁒ x Ξ½ - ΞΌ ⁒ ( x 2 - 1 ) ΞΌ / 2 ⁒ 𝐅 ⁑ ( 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ , 1 2 ⁒ ΞΌ - 1 2 ⁒ Ξ½ + 1 2 ; ΞΌ + 1 ; 1 - 1 x 2 ) Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript 2 πœ‡ superscript π‘₯ 𝜈 πœ‡ superscript superscript π‘₯ 2 1 πœ‡ 2 scaled-hypergeometric-bold-F 1 2 πœ‡ 1 2 𝜈 1 2 πœ‡ 1 2 𝜈 1 2 πœ‡ 1 1 1 superscript π‘₯ 2 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}x^{\nu-\mu}% \left(x^{2}-1\right)^{\mu/2}\mathbf{F}\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,% \tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\mu+1;1-\frac{1}{x^{2}}\right)}}
\assLegendreP[-\mu]{\nu}@{x} = 2^{-\mu}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}\hyperOlverF@{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu}{\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}}{\mu+1}{1-\frac{1}{x^{2}}}		 

LegendreP(nu, - mu, x) = (2)^(- mu)* (x)^(nu - mu)*((x)^(2)- 1)^(mu/2)* hypergeom([(1)/(2)*mu -(1)/(2)*nu, (1)/(2)*mu -(1)/(2)*nu +(1)/(2)], [mu + 1], 1 -(1)/((x)^(2)))/GAMMA(mu + 1)

LegendreP[\[Nu], - \[Mu], 3, x] == (2)^(- \[Mu])* (x)^(\[Nu]- \[Mu])*((x)^(2)- 1)^(\[Mu]/2)* Hypergeometric2F1Regularized[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu], Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2], \[Mu]+ 1, 1 -Divide[1,(x)^(2)]]
Failure Failure
Failed [18 / 200]
Result: Float(undefined)+Float(undefined)*I
Test Values: {mu = -2, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}


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

... skip entries to safe data
 Successful [Tested: 200]
14.3.E19 𝑸 Ξ½ ΞΌ ⁑ ( x ) = 2 Ξ½ ⁒ Ξ“ ⁑ ( Ξ½ + 1 ) ⁒ ( x + 1 ) ΞΌ / 2 ( x - 1 ) ( ΞΌ / 2 ) + Ξ½ + 1 ⁒ 𝐅 ⁑ ( Ξ½ + 1 , Ξ½ + ΞΌ + 1 ; 2 ⁒ Ξ½ + 2 ; 2 1 - x ) associated-Legendre-black-Q πœ‡ 𝜈 π‘₯ superscript 2 𝜈 Euler-Gamma 𝜈 1 superscript π‘₯ 1 πœ‡ 2 superscript π‘₯ 1 πœ‡ 2 𝜈 1 scaled-hypergeometric-bold-F 𝜈 1 𝜈 πœ‡ 1 2 𝜈 2 2 1 π‘₯ {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{2^{% \nu}\Gamma\left(\nu+1\right)(x+1)^{\mu/2}}{(x-1)^{(\mu/2)+\nu+1}}\mathbf{F}% \left(\nu+1,\nu+\mu+1;2\nu+2;\frac{2}{1-x}\right)}}
\assLegendreOlverQ[\mu]{\nu}@{x} = \frac{2^{\nu}\EulerGamma@{\nu+1}(x+1)^{\mu/2}}{(x-1)^{(\mu/2)+\nu+1}}\hyperOlverF@{\nu+1}{\nu+\mu+1}{2\nu+2}{\frac{2}{1-x}}		 β„œ ⁑ ( Ξ½ + 1 ) > 0 𝜈 1 0 {\displaystyle{\displaystyle\Re(\nu+1)>0}} 
exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = ((2)^(nu)* GAMMA(nu + 1)*(x + 1)^(mu/2))/((x - 1)^((mu/2)+ nu + 1))*hypergeom([nu + 1, nu + mu + 1], [2*nu + 2], (2)/(1 - x))/GAMMA(2*nu + 2)

Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == Divide[(2)^\[Nu]* Gamma[\[Nu]+ 1]*(x + 1)^(\[Mu]/2),(x - 1)^((\[Mu]/2)+ \[Nu]+ 1)]*Hypergeometric2F1Regularized[\[Nu]+ 1, \[Nu]+ \[Mu]+ 1, 2*\[Nu]+ 2, Divide[2,1 - x]]
Failure Failure Error Skip - No test values generated
14.3.E20 2 ⁒ sin ⁑ ( ΞΌ ⁒ Ο€ ) Ο€ ⁒ 𝑸 Ξ½ ΞΌ ⁑ ( x ) = ( x + 1 ) ΞΌ / 2 Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) ⁒ ( x - 1 ) ΞΌ / 2 ⁒ 𝐅 ⁑ ( Ξ½ + 1 , - Ξ½ ; 1 - ΞΌ ; 1 2 - 1 2 ⁒ x ) - ( x - 1 ) ΞΌ / 2 Ξ“ ⁑ ( Ξ½ - ΞΌ + 1 ) ⁒ ( x + 1 ) ΞΌ / 2 ⁒ 𝐅 ⁑ ( Ξ½ + 1 , - Ξ½ ; ΞΌ + 1 ; 1 2 - 1 2 ⁒ x ) 2 πœ‡ πœ‹ πœ‹ associated-Legendre-black-Q πœ‡ 𝜈 π‘₯ superscript π‘₯ 1 πœ‡ 2 Euler-Gamma 𝜈 πœ‡ 1 superscript π‘₯ 1 πœ‡ 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 1 πœ‡ 1 2 1 2 π‘₯ superscript π‘₯ 1 πœ‡ 2 Euler-Gamma 𝜈 πœ‡ 1 superscript π‘₯ 1 πœ‡ 2 scaled-hypergeometric-bold-F 𝜈 1 𝜈 πœ‡ 1 1 2 1 2 π‘₯ {\displaystyle{\displaystyle\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}% ^{\mu}_{\nu}\left(x\right)=\frac{(x+1)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)(x% -1)^{\mu/2}}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)% -\frac{(x-1)^{\mu/2}}{\Gamma\left(\nu-\mu+1\right)(x+1)^{\mu/2}}\mathbf{F}% \left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right)}}
\frac{2\sin@{\mu\pi}}{\pi}\assLegendreOlverQ[\mu]{\nu}@{x} = \frac{(x+1)^{\mu/2}}{\EulerGamma@{\nu+\mu+1}(x-1)^{\mu/2}}\hyperOlverF@{\nu+1}{-\nu}{1-\mu}{\tfrac{1}{2}-\tfrac{1}{2}x}-\frac{(x-1)^{\mu/2}}{\EulerGamma@{\nu-\mu+1}(x+1)^{\mu/2}}\hyperOlverF@{\nu+1}{-\nu}{\mu+1}{\tfrac{1}{2}-\tfrac{1}{2}x}		 β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - ΞΌ + 1 ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 𝜈 πœ‡ 1 0 formulae-sequence 𝜈 πœ‡ 1 0 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1}} 
(2*sin(mu*Pi))/(Pi)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = ((x + 1)^(mu/2))/(GAMMA(nu + mu + 1)*(x - 1)^(mu/2))*hypergeom([nu + 1, - nu], [1 - mu], (1)/(2)-(1)/(2)*x)/GAMMA(1 - mu)-((x - 1)^(mu/2))/(GAMMA(nu - mu + 1)*(x + 1)^(mu/2))*hypergeom([nu + 1, - nu], [mu + 1], (1)/(2)-(1)/(2)*x)/GAMMA(mu + 1)

Divide[2*Sin[\[Mu]*Pi],Pi]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == Divide[(x + 1)^(\[Mu]/2),Gamma[\[Nu]+ \[Mu]+ 1]*(x - 1)^(\[Mu]/2)]*Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], 1 - \[Mu], Divide[1,2]-Divide[1,2]*x]-Divide[(x - 1)^(\[Mu]/2),Gamma[\[Nu]- \[Mu]+ 1]*(x + 1)^(\[Mu]/2)]*Hypergeometric2F1Regularized[\[Nu]+ 1, - \[Nu], \[Mu]+ 1, Divide[1,2]-Divide[1,2]*x]
Failure Successful
Failed [12 / 120]
Result: Float(undefined)+Float(undefined)*I
Test Values: {mu = -2, nu = 3/2, x = 3/2}


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

... skip entries to safe data
 Successful [Tested: 135]
14.3.E21 𝖯 Ξ½ ΞΌ ⁑ ( x ) = 2 ΞΌ ⁒ Ξ“ ⁑ ( 1 - 2 ⁒ ΞΌ ) ⁒ Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) Ξ“ ⁑ ( Ξ½ - ΞΌ + 1 ) ⁒ Ξ“ ⁑ ( 1 - ΞΌ ) ⁒ ( 1 - x 2 ) ΞΌ / 2 ⁒ C Ξ½ + ΞΌ ( 1 2 - ΞΌ ) ⁑ ( x ) Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript 2 πœ‡ Euler-Gamma 1 2 πœ‡ Euler-Gamma 𝜈 πœ‡ 1 Euler-Gamma 𝜈 πœ‡ 1 Euler-Gamma 1 πœ‡ superscript 1 superscript π‘₯ 2 πœ‡ 2 ultraspherical-Gegenbauer-polynomial 1 2 πœ‡ 𝜈 πœ‡ π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)=\frac{2^{\mu}% \Gamma\left(1-2\mu\right)\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1% \right)\Gamma\left(1-\mu\right)\left(1-x^{2}\right)^{\mu/2}}C^{(\frac{1}{2}-% \mu)}_{\nu+\mu}\left(x\right)}}
\FerrersP[\mu]{\nu}@{x} = \frac{2^{\mu}\EulerGamma@{1-2\mu}\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+1}\EulerGamma@{1-\mu}\left(1-x^{2}\right)^{\mu/2}}\ultrasphpoly{\frac{1}{2}-\mu}{\nu+\mu}@{x}		 β„œ ⁑ ( 1 - 2 ⁒ ΞΌ ) > 0 , β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - ΞΌ + 1 ) > 0 , β„œ ⁑ ( 1 - ΞΌ ) > 0 , | ( 1 2 - 1 2 ⁒ x ) | < 1 formulae-sequence 1 2 πœ‡ 0 formulae-sequence 𝜈 πœ‡ 1 0 formulae-sequence 𝜈 πœ‡ 1 0 formulae-sequence 1 πœ‡ 0 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle\Re(1-2\mu)>0,\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,% \Re(1-\mu)>0,|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(nu, mu, x) = ((2)^(mu)* GAMMA(1 - 2*mu)*GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1)*GAMMA(1 - mu)*(1 - (x)^(2))^(mu/2))*GegenbauerC(nu + mu, (1)/(2)- mu, x)

LegendreP[\[Nu], \[Mu], x] == Divide[(2)^\[Mu]* Gamma[1 - 2*\[Mu]]*Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]*Gamma[1 - \[Mu]]*(1 - (x)^(2))^(\[Mu]/2)]*GegenbauerC[\[Nu]+ \[Mu], Divide[1,2]- \[Mu], x]
Failure Failure Successful [Tested: 60] Successful [Tested: 69]
14.3.E22 P Ξ½ ΞΌ ⁑ ( x ) = 2 ΞΌ ⁒ Ξ“ ⁑ ( 1 - 2 ⁒ ΞΌ ) ⁒ Ξ“ ⁑ ( Ξ½ + ΞΌ + 1 ) Ξ“ ⁑ ( Ξ½ - ΞΌ + 1 ) ⁒ Ξ“ ⁑ ( 1 - ΞΌ ) ⁒ ( x 2 - 1 ) ΞΌ / 2 ⁒ C Ξ½ + ΞΌ ( 1 2 - ΞΌ ) ⁑ ( x ) Legendre-P-first-kind πœ‡ 𝜈 π‘₯ superscript 2 πœ‡ Euler-Gamma 1 2 πœ‡ Euler-Gamma 𝜈 πœ‡ 1 Euler-Gamma 𝜈 πœ‡ 1 Euler-Gamma 1 πœ‡ superscript superscript π‘₯ 2 1 πœ‡ 2 ultraspherical-Gegenbauer-polynomial 1 2 πœ‡ 𝜈 πœ‡ π‘₯ {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{2^{\mu}\Gamma% \left(1-2\mu\right)\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)% \Gamma\left(1-\mu\right)\left(x^{2}-1\right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{% \nu+\mu}\left(x\right)}}
\assLegendreP[\mu]{\nu}@{x} = \frac{2^{\mu}\EulerGamma@{1-2\mu}\EulerGamma@{\nu+\mu+1}}{\EulerGamma@{\nu-\mu+1}\EulerGamma@{1-\mu}\left(x^{2}-1\right)^{\mu/2}}\ultrasphpoly{\frac{1}{2}-\mu}{\nu+\mu}@{x}		 β„œ ⁑ ( 1 - 2 ⁒ ΞΌ ) > 0 , β„œ ⁑ ( Ξ½ + ΞΌ + 1 ) > 0 , β„œ ⁑ ( Ξ½ - ΞΌ + 1 ) > 0 , β„œ ⁑ ( 1 - ΞΌ ) > 0 formulae-sequence 1 2 πœ‡ 0 formulae-sequence 𝜈 πœ‡ 1 0 formulae-sequence 𝜈 πœ‡ 1 0 1 πœ‡ 0 {\displaystyle{\displaystyle\Re(1-2\mu)>0,\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,% \Re(1-\mu)>0}} 
LegendreP(nu, mu, x) = ((2)^(mu)* GAMMA(1 - 2*mu)*GAMMA(nu + mu + 1))/(GAMMA(nu - mu + 1)*GAMMA(1 - mu)*((x)^(2)- 1)^(mu/2))*GegenbauerC(nu + mu, (1)/(2)- mu, x)

LegendreP[\[Nu], \[Mu], 3, x] == Divide[(2)^\[Mu]* Gamma[1 - 2*\[Mu]]*Gamma[\[Nu]+ \[Mu]+ 1],Gamma[\[Nu]- \[Mu]+ 1]*Gamma[1 - \[Mu]]*((x)^(2)- 1)^(\[Mu]/2)]*GegenbauerC[\[Nu]+ \[Mu], Divide[1,2]- \[Mu], x]
Failure Failure Successful [Tested: 60] Successful [Tested: 69]
14.3.E23 P Ξ½ ΞΌ ⁑ ( x ) = 1 Ξ“ ⁑ ( 1 - ΞΌ ) ⁒ ( x + 1 x - 1 ) ΞΌ / 2 ⁒ Ο• - i ⁒ ( 2 ⁒ Ξ½ + 1 ) ( - ΞΌ , ΞΌ ) ⁑ ( arcsinh ⁑ ( ( 1 2 ⁒ x - 1 2 ) 1 / 2 ) ) Legendre-P-first-kind πœ‡ 𝜈 π‘₯ 1 Euler-Gamma 1 πœ‡ superscript π‘₯ 1 π‘₯ 1 πœ‡ 2 Jacobi-hypergeometric-phi πœ‡ πœ‡ imaginary-unit 2 𝜈 1 hyperbolic-inverse-sine superscript 1 2 π‘₯ 1 2 1 2 {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(% 1-\mu\right)}\left(\frac{x+1}{x-1}\right)^{\mu/2}\phi^{(-\mu,\mu)}_{-\mathrm{i% }(2\nu+1)}\left(\operatorname{arcsinh}\left((\tfrac{1}{2}x-\tfrac{1}{2})^{% \ifrac{1}{2}}\right)\right)}}
\assLegendreP[\mu]{\nu}@{x} = \frac{1}{\EulerGamma@{1-\mu}}\left(\frac{x+1}{x-1}\right)^{\mu/2}\Jacobiphi{-\mu}{\mu}{-\iunit(2\nu+1)}@{\asinh@{(\tfrac{1}{2}x-\tfrac{1}{2})^{\ifrac{1}{2}}}}		 β„œ ⁑ ( 1 - ΞΌ ) > 0 1 πœ‡ 0 {\displaystyle{\displaystyle\Re(1-\mu)>0}} 
LegendreP(nu, mu, x) = (1)/(GAMMA(1 - mu))*((x + 1)/(x - 1))^(mu/2)* hypergeom([((- mu)+(mu)+1-I*(- I*(2*nu + 1)))/2, ((- mu)+(mu)+1+I*(- I*(2*nu + 1)))], [(- mu)+1], -sinh(arcsinh(((1)/(2)*x -(1)/(2))^((1)/(2))))^2)

Error
 Failure Missing Macro Error
Failed [240 / 240]
Result: -.318116688-.9248307299*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 3/2}

Result: -5.010614457+.9472052439*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
 -
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