Legendre and Related Functions - 14.9 Connection Formulas

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14.9.E1 π ⁢ sin ⁡ ( μ ⁢ π ) 2 ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ 𝖯 ν - μ ⁡ ( x ) = - 1 Γ ⁡ ( ν + μ + 1 ) ⁢ 𝖰 ν μ ⁡ ( x ) + cos ⁡ ( μ ⁢ π ) Γ ⁡ ( ν - μ + 1 ) ⁢ 𝖰 ν - μ ⁡ ( x ) 𝜋 𝜇 𝜋 2 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 1 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜇 𝜋 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\frac{\pi\sin\left(\mu\pi\right)}{2\Gamma\left(\nu% -\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)=-\frac{1}{\Gamma\left(\nu+% \mu+1\right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)+\frac{\cos\left(\mu\pi\right% )}{\Gamma\left(\nu-\mu+1\right)}\mathsf{Q}^{-\mu}_{\nu}\left(x\right)}}
\frac{\pi\sin@{\mu\pi}}{2\EulerGamma@{\nu-\mu+1}}\FerrersP[-\mu]{\nu}@{x} = -\frac{1}{\EulerGamma@{\nu+\mu+1}}\FerrersQ[\mu]{\nu}@{x}+\frac{\cos@{\mu\pi}}{\EulerGamma@{\nu-\mu+1}}\FerrersQ[-\mu]{\nu}@{x}		 ℜ ⁡ ( ν - μ + 1 ) > 0 , ℜ ⁡ ( ν + μ + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 , ℜ ⁡ ( ν + ( - μ ) + 1 ) > 0 , ℜ ⁡ ( ν - ( - μ ) + 1 ) > 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 1 2 1 2 𝑥 1 formulae-sequence 𝜈 𝜇 1 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\nu-\mu+1)>0,\Re(\nu+\mu+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1,\Re(\nu+(-\mu)+1)>0,\Re(\nu-(-\mu)+1)>0}} 
(Pi*sin(mu*Pi))/(2*GAMMA(nu - mu + 1))*LegendreP(nu, - mu, x) = -(1)/(GAMMA(nu + mu + 1))*LegendreQ(nu, mu, x)+(cos(mu*Pi))/(GAMMA(nu - mu + 1))*LegendreQ(nu, - mu, x)

Divide[Pi*Sin[\[Mu]*Pi],2*Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], x] == -Divide[1,Gamma[\[Nu]+ \[Mu]+ 1]]*LegendreQ[\[Nu], \[Mu], x]+Divide[Cos[\[Mu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreQ[\[Nu], - \[Mu], x]
Successful Successful - Successful [Tested: 135]
14.9.E2 2 ⁢ sin ⁡ ( μ ⁢ π ) π ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ 𝖰 ν - μ ⁡ ( x ) = 1 Γ ⁡ ( ν + μ + 1 ) ⁢ 𝖯 ν μ ⁡ ( x ) - cos ⁡ ( μ ⁢ π ) Γ ⁡ ( ν - μ + 1 ) ⁢ 𝖯 ν - μ ⁡ ( x ) 2 𝜇 𝜋 𝜋 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜇 𝜋 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\frac{2\sin\left(\mu\pi\right)}{\pi\Gamma\left(\nu% -\mu+1\right)}\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(\nu+% \mu+1\right)}\mathsf{P}^{\mu}_{\nu}\left(x\right)-\frac{\cos\left(\mu\pi\right% )}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)}}
\frac{2\sin@{\mu\pi}}{\pi\EulerGamma@{\nu-\mu+1}}\FerrersQ[-\mu]{\nu}@{x} = \frac{1}{\EulerGamma@{\nu+\mu+1}}\FerrersP[\mu]{\nu}@{x}-\frac{\cos@{\mu\pi}}{\EulerGamma@{\nu-\mu+1}}\FerrersP[-\mu]{\nu}@{x}		 ℜ ⁡ ( ν - μ + 1 ) > 0 , ℜ ⁡ ( ν + μ + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 , ℜ ⁡ ( ν + ( - μ ) + 1 ) > 0 , ℜ ⁡ ( ν - ( - μ ) + 1 ) > 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 1 2 1 2 𝑥 1 formulae-sequence 𝜈 𝜇 1 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\nu-\mu+1)>0,\Re(\nu+\mu+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1,\Re(\nu+(-\mu)+1)>0,\Re(\nu-(-\mu)+1)>0}} 
(2*sin(mu*Pi))/(Pi*GAMMA(nu - mu + 1))*LegendreQ(nu, - mu, x) = (1)/(GAMMA(nu + mu + 1))*LegendreP(nu, mu, x)-(cos(mu*Pi))/(GAMMA(nu - mu + 1))*LegendreP(nu, - mu, x)

Divide[2*Sin[\[Mu]*Pi],Pi*Gamma[\[Nu]- \[Mu]+ 1]]*LegendreQ[\[Nu], - \[Mu], x] == Divide[1,Gamma[\[Nu]+ \[Mu]+ 1]]*LegendreP[\[Nu], \[Mu], x]-Divide[Cos[\[Mu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], x]
Successful Successful - Successful [Tested: 135]
14.9.E3 𝖯 ν - m ⁡ ( x ) = ( - 1 ) m ⁢ Γ ⁡ ( ν - m + 1 ) Γ ⁡ ( ν + m + 1 ) ⁢ 𝖯 ν m ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝑚 𝜈 𝑥 superscript 1 𝑚 Euler-Gamma 𝜈 𝑚 1 Euler-Gamma 𝜈 𝑚 1 Ferrers-Legendre-P-first-kind 𝑚 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{% \Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}\mathsf{P}^{m}_{\nu}% \left(x\right)}}
\FerrersP[-m]{\nu}@{x} = (-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\FerrersP[m]{\nu}@{x}		 ℜ ⁡ ( ν - m + 1 ) > 0 , ℜ ⁡ ( ν + m + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 formulae-sequence 𝜈 𝑚 1 0 formulae-sequence 𝜈 𝑚 1 0 1 2 1 2 𝑥 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))*LegendreP(nu, m, x)

LegendreP[\[Nu], - m, x] == (- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]*LegendreP[\[Nu], m, x]
Failure Failure Successful [Tested: 21] Successful [Tested: 21]
14.9.E4 𝖰 ν - m ⁡ ( x ) = ( - 1 ) m ⁢ Γ ⁡ ( ν - m + 1 ) Γ ⁡ ( ν + m + 1 ) ⁢ 𝖰 ν m ⁡ ( x ) Ferrers-Legendre-Q-first-kind 𝑚 𝜈 𝑥 superscript 1 𝑚 Euler-Gamma 𝜈 𝑚 1 Euler-Gamma 𝜈 𝑚 1 Ferrers-Legendre-Q-first-kind 𝑚 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{% \Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1\right)}\mathsf{Q}^{m}_{\nu}% \left(x\right)}}
\FerrersQ[-m]{\nu}@{x} = (-1)^{m}\frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\FerrersQ[m]{\nu}@{x}		 ν ≠ m - 1 , ℜ ⁡ ( ν - m + 1 ) > 0 , ℜ ⁡ ( ν + m + 1 ) > 0 , ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν + ( - m ) + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 , ℜ ⁡ ( ν - ( - m ) + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 formulae-sequence 𝜈 𝑚 1 formulae-sequence 𝜈 𝑚 1 0 formulae-sequence 𝜈 𝑚 1 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝑚 1 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝑚 1 0 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle\nu\neq m-1,\Re(\nu-m+1)>0,\Re(\nu+m+1)>0,\Re(\nu+% \mu+1)>0,\Re(\nu+(-m)+1)>0,\Re(\nu-\mu+1)>0,\Re(\nu-(-m)+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1}} 
LegendreQ(nu, - m, x) = (- 1)^(m)*(GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))*LegendreQ(nu, m, x)

LegendreQ[\[Nu], - m, x] == (- 1)^(m)*Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]*LegendreQ[\[Nu], m, x]
Failure Failure Error Successful [Tested: 21]
14.9#Ex1 𝖯 - ν - 1 μ ⁡ ( x ) = 𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{-\nu-1}\left(x\right)=\mathsf{P}% ^{\mu}_{\nu}\left(x\right)}}
\FerrersP[\mu]{-\nu-1}@{x} = \FerrersP[\mu]{\nu}@{x}		 | ( 1 2 - 1 2 ⁢ x ) | < 1 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(- nu - 1, mu, x) = LegendreP(nu, mu, x)

LegendreP[- \[Nu]- 1, \[Mu], x] == LegendreP[\[Nu], \[Mu], x]
Successful Failure - Successful [Tested: 300]
14.9#Ex2 𝖯 - ν - 1 - μ ⁡ ( x ) = 𝖯 ν - μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{-\nu-1}\left(x\right)=\mathsf{P% }^{-\mu}_{\nu}\left(x\right)}}
\FerrersP[-\mu]{-\nu-1}@{x} = \FerrersP[-\mu]{\nu}@{x}		 | ( 1 2 - 1 2 ⁢ x ) | < 1 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
LegendreP(- nu - 1, - mu, x) = LegendreP(nu, - mu, x)

LegendreP[- \[Nu]- 1, - \[Mu], x] == LegendreP[\[Nu], - \[Mu], x]
Successful Failure - Successful [Tested: 300]
14.9.E6 π ⁢ cos ⁡ ( ν ⁢ π ) ⁢ cos ⁡ ( μ ⁢ π ) ⁢ 𝖯 ν μ ⁡ ( x ) = sin ⁡ ( ( ν + μ ) ⁢ π ) ⁢ 𝖰 ν μ ⁡ ( x ) - sin ⁡ ( ( ν - μ ) ⁢ π ) ⁢ 𝖰 - ν - 1 μ ⁡ ( x ) 𝜋 𝜈 𝜋 𝜇 𝜋 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜈 𝜇 𝜋 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜈 𝜇 𝜋 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 1 𝑥 {\displaystyle{\displaystyle\pi\cos\left(\nu\pi\right)\cos\left(\mu\pi\right)% \mathsf{P}^{\mu}_{\nu}\left(x\right)=\sin\left((\nu+\mu)\pi\right)\mathsf{Q}^{% \mu}_{\nu}\left(x\right)-\sin\left((\nu-\mu)\pi\right)\mathsf{Q}^{\mu}_{-\nu-1% }\left(x\right)}}
\pi\cos@{\nu\pi}\cos@{\mu\pi}\FerrersP[\mu]{\nu}@{x} = \sin@{(\nu+\mu)\pi}\FerrersQ[\mu]{\nu}@{x}-\sin@{(\nu-\mu)\pi}\FerrersQ[\mu]{-\nu-1}@{x}		 | ( 1 2 - 1 2 ⁢ x ) | < 1 , ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ( - ν - 1 ) + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 , ℜ ⁡ ( ( - ν - 1 ) - μ + 1 ) > 0 formulae-sequence 1 2 1 2 𝑥 1 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 1 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 𝜈 1 𝜇 1 0 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu+\mu+1)>0,% \Re((-\nu-1)+\mu+1)>0,\Re(\nu-\mu+1)>0,\Re((-\nu-1)-\mu+1)>0}} 
Pi*cos(nu*Pi)*cos(mu*Pi)*LegendreP(nu, mu, x) = sin((nu + mu)*Pi)*LegendreQ(nu, mu, x)- sin((nu - mu)*Pi)*LegendreQ(- nu - 1, mu, x)

Pi*Cos[\[Nu]*Pi]*Cos[\[Mu]*Pi]*LegendreP[\[Nu], \[Mu], x] == Sin[(\[Nu]+ \[Mu])*Pi]*LegendreQ[\[Nu], \[Mu], x]- Sin[(\[Nu]- \[Mu])*Pi]*LegendreQ[- \[Nu]- 1, \[Mu], x]
Successful Failure - Successful [Tested: 3]
14.9.E7 sin ⁡ ( ( ν - μ ) ⁢ π ) Γ ⁡ ( ν + μ + 1 ) ⁢ 𝖯 ν μ ⁡ ( x ) = sin ⁡ ( ν ⁢ π ) Γ ⁡ ( ν - μ + 1 ) ⁢ 𝖯 ν - μ ⁡ ( x ) - sin ⁡ ( μ ⁢ π ) Γ ⁡ ( ν - μ + 1 ) ⁢ 𝖯 ν - μ ⁡ ( - x ) 𝜈 𝜇 𝜋 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜈 𝜋 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜇 𝜋 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\frac{\sin\left((\nu-\mu)\pi\right)}{\Gamma\left(% \nu+\mu+1\right)}\mathsf{P}^{\mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi% \right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)-% \frac{\sin\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_% {\nu}\left(-x\right)}}
\frac{\sin@{(\nu-\mu)\pi}}{\EulerGamma@{\nu+\mu+1}}\FerrersP[\mu]{\nu}@{x} = \frac{\sin@{\nu\pi}}{\EulerGamma@{\nu-\mu+1}}\FerrersP[-\mu]{\nu}@{x}-\frac{\sin@{\mu\pi}}{\EulerGamma@{\nu-\mu+1}}\FerrersP[-\mu]{\nu}@{-x}		 ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 , | ( 1 2 - 1 2 ⁢ ( - x ) ) | < 1 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 1 2 1 2 𝑥 1 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1,|(\tfrac{1}{2}-\tfrac{1}{2}(-x))|<1}} 
(sin((nu - mu)*Pi))/(GAMMA(nu + mu + 1))*LegendreP(nu, mu, x) = (sin(nu*Pi))/(GAMMA(nu - mu + 1))*LegendreP(nu, - mu, x)-(sin(mu*Pi))/(GAMMA(nu - mu + 1))*LegendreP(nu, - mu, - x)

Divide[Sin[(\[Nu]- \[Mu])*Pi],Gamma[\[Nu]+ \[Mu]+ 1]]*LegendreP[\[Nu], \[Mu], x] == Divide[Sin[\[Nu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], x]-Divide[Sin[\[Mu]*Pi],Gamma[\[Nu]- \[Mu]+ 1]]*LegendreP[\[Nu], - \[Mu], - x]
Failure Failure Successful [Tested: 40] Successful [Tested: 45]
14.9.E8 1 2 ⁢ π ⁢ sin ⁡ ( ( ν - μ ) ⁢ π ) ⁢ 𝖯 ν - μ ⁡ ( x ) = - cos ⁡ ( ( ν - μ ) ⁢ π ) ⁢ 𝖰 ν - μ ⁡ ( x ) - 𝖰 ν - μ ⁡ ( - x ) 1 2 𝜋 𝜈 𝜇 𝜋 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜈 𝜇 𝜋 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\tfrac{1}{2}\pi\sin\left((\nu-\mu)\pi\right)% \mathsf{P}^{-\mu}_{\nu}\left(x\right)=-\cos\left((\nu-\mu)\pi\right)\mathsf{Q}% ^{-\mu}_{\nu}\left(x\right)-\mathsf{Q}^{-\mu}_{\nu}\left(-x\right)}}
\tfrac{1}{2}\pi\sin@{(\nu-\mu)\pi}\FerrersP[-\mu]{\nu}@{x} = -\cos@{(\nu-\mu)\pi}\FerrersQ[-\mu]{\nu}@{x}-\FerrersQ[-\mu]{\nu}@{-x}		 | ( 1 2 - 1 2 ⁢ x ) | < 1 , ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν + ( - μ ) + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 , ℜ ⁡ ( ν - ( - μ ) + 1 ) > 0 , | ( 1 2 - 1 2 ⁢ ( - x ) ) | < 1 formulae-sequence 1 2 1 2 𝑥 1 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 1 2 1 2 𝑥 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,\Re(\nu+\mu+1)>0,% \Re(\nu+(-\mu)+1)>0,\Re(\nu-\mu+1)>0,\Re(\nu-(-\mu)+1)>0,|(\tfrac{1}{2}-\tfrac% {1}{2}(-x))|<1}} 
(1)/(2)*Pi*sin((nu - mu)*Pi)*LegendreP(nu, - mu, x) = - cos((nu - mu)*Pi)*LegendreQ(nu, - mu, x)- LegendreQ(nu, - mu, - x)

Divide[1,2]*Pi*Sin[(\[Nu]- \[Mu])*Pi]*LegendreP[\[Nu], - \[Mu], x] == - Cos[(\[Nu]- \[Mu])*Pi]*LegendreQ[\[Nu], - \[Mu], x]- LegendreQ[\[Nu], - \[Mu], - x]
Failure Failure Error Successful [Tested: 45]
14.9.E9 2 Γ ⁡ ( ν + μ + 1 ) ⁢ Γ ⁡ ( μ - ν ) ⁢ 𝖰 ν μ ⁡ ( x ) = - cos ⁡ ( ν ⁢ π ) ⁢ 𝖯 ν - μ ⁡ ( x ) + cos ⁡ ( μ ⁢ π ) ⁢ 𝖯 ν - μ ⁡ ( - x ) 2 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜇 𝜈 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜈 𝜋 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜇 𝜋 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\frac{2}{\Gamma\left(\nu+\mu+1\right)\Gamma\left(% \mu-\nu\right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\cos\left(\nu\pi\right)% \mathsf{P}^{-\mu}_{\nu}\left(x\right)+\cos\left(\mu\pi\right)\mathsf{P}^{-\mu}% _{\nu}\left(-x\right)}}
\frac{2}{\EulerGamma@{\nu+\mu+1}\EulerGamma@{\mu-\nu}}\FerrersQ[\mu]{\nu}@{x} = -\cos@{\nu\pi}\FerrersP[-\mu]{\nu}@{x}+\cos@{\mu\pi}\FerrersP[-\mu]{\nu}@{-x}		 ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( μ - ν ) > 0 , | ( 1 2 - 1 2 ⁢ x ) | < 1 , | ( 1 2 - 1 2 ⁢ ( - x ) ) | < 1 , ℜ ⁡ ( ν - μ + 1 ) > 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜇 𝜈 0 formulae-sequence 1 2 1 2 𝑥 1 formulae-sequence 1 2 1 2 𝑥 1 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\mu-\nu)>0,|(\tfrac{1}{2}-% \tfrac{1}{2}x)|<1,|(\tfrac{1}{2}-\tfrac{1}{2}(-x))|<1,\Re(\nu-\mu+1)>0}} 
(2)/(GAMMA(nu + mu + 1)*GAMMA(mu - nu))*LegendreQ(nu, mu, x) = - cos(nu*Pi)*LegendreP(nu, - mu, x)+ cos(mu*Pi)*LegendreP(nu, - mu, - x)

Divide[2,Gamma[\[Nu]+ \[Mu]+ 1]*Gamma[\[Mu]- \[Nu]]]*LegendreQ[\[Nu], \[Mu], x] == - Cos[\[Nu]*Pi]*LegendreP[\[Nu], - \[Mu], x]+ Cos[\[Mu]*Pi]*LegendreP[\[Nu], - \[Mu], - x]
Failure Failure Successful [Tested: 4] Successful [Tested: 8]
14.9.E10 ( 2 / π ) ⁢ sin ⁡ ( ( ν - μ ) ⁢ π ) ⁢ 𝖰 ν - μ ⁡ ( x ) = cos ⁡ ( ( ν - μ ) ⁢ π ) ⁢ 𝖯 ν - μ ⁡ ( x ) - 𝖯 ν - μ ⁡ ( - x ) 2 𝜋 𝜈 𝜇 𝜋 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜈 𝜇 𝜋 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle(2/\pi)\sin\left((\nu-\mu)\pi\right)\mathsf{Q}^{-% \mu}_{\nu}\left(x\right)=\cos\left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}% \left(x\right)-\mathsf{P}^{-\mu}_{\nu}\left(-x\right)}}
(2/\pi)\sin@{(\nu-\mu)\pi}\FerrersQ[-\mu]{\nu}@{x} = \cos@{(\nu-\mu)\pi}\FerrersP[-\mu]{\nu}@{x}-\FerrersP[-\mu]{\nu}@{-x}		 | ( 1 2 - 1 2 ⁢ x ) | < 1 , | ( 1 2 - 1 2 ⁢ ( - x ) ) | < 1 , ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν + ( - μ ) + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 , ℜ ⁡ ( ν - ( - μ ) + 1 ) > 0 formulae-sequence 1 2 1 2 𝑥 1 formulae-sequence 1 2 1 2 𝑥 1 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 formulae-sequence 𝜈 𝜇 1 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1,|(\tfrac{1}{2}-% \tfrac{1}{2}(-x))|<1,\Re(\nu+\mu+1)>0,\Re(\nu+(-\mu)+1)>0,\Re(\nu-\mu+1)>0,\Re% (\nu-(-\mu)+1)>0}} 
(2/Pi)*sin((nu - mu)*Pi)*LegendreQ(nu, - mu, x) = cos((nu - mu)*Pi)*LegendreP(nu, - mu, x)- LegendreP(nu, - mu, - x)

(2/Pi)*Sin[(\[Nu]- \[Mu])*Pi]*LegendreQ[\[Nu], - \[Mu], x] == Cos[(\[Nu]- \[Mu])*Pi]*LegendreP[\[Nu], - \[Mu], x]- LegendreP[\[Nu], - \[Mu], - x]
Failure Failure Error Successful [Tested: 45]
14.9#Ex3 P - ν - 1 - μ ⁡ ( x ) = P ν - μ ⁡ ( x ) Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle P^{-\mu}_{-\nu-1}\left(x\right)=P^{-\mu}_{\nu}% \left(x\right)}}
\assLegendreP[-\mu]{-\nu-1}@{x} = \assLegendreP[-\mu]{\nu}@{x}		 

LegendreP(- nu - 1, - mu, x) = LegendreP(nu, - mu, x)

LegendreP[- \[Nu]- 1, - \[Mu], 3, x] == LegendreP[\[Nu], - \[Mu], 3, x]
Successful Successful - Successful [Tested: 300]
14.9#Ex4 P - ν - 1 μ ⁡ ( x ) = P ν μ ⁡ ( x ) Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle P^{\mu}_{-\nu-1}\left(x\right)=P^{\mu}_{\nu}\left% (x\right)}}
\assLegendreP[\mu]{-\nu-1}@{x} = \assLegendreP[\mu]{\nu}@{x}		 

LegendreP(- nu - 1, mu, x) = LegendreP(nu, mu, x)

LegendreP[- \[Nu]- 1, \[Mu], 3, x] == LegendreP[\[Nu], \[Mu], 3, x]
Successful Successful - Successful [Tested: 300]
14.9.E12 cos ⁡ ( ν ⁢ π ) ⁢ P ν - μ ⁡ ( x ) = - 𝑸 ν μ ⁡ ( x ) Γ ⁡ ( μ - ν ) + 𝑸 - ν - 1 μ ⁡ ( x ) Γ ⁡ ( ν + μ + 1 ) 𝜈 𝜋 Legendre-P-first-kind 𝜇 𝜈 𝑥 associated-Legendre-black-Q 𝜇 𝜈 𝑥 Euler-Gamma 𝜇 𝜈 associated-Legendre-black-Q 𝜇 𝜈 1 𝑥 Euler-Gamma 𝜈 𝜇 1 {\displaystyle{\displaystyle\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right% )=-\frac{\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\mu-\nu\right)}% +\frac{\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)}{\Gamma\left(\nu+\mu+1% \right)}}}
\cos@{\nu\pi}\assLegendreP[-\mu]{\nu}@{x} = -\frac{\assLegendreOlverQ[\mu]{\nu}@{x}}{\EulerGamma@{\mu-\nu}}+\frac{\assLegendreOlverQ[\mu]{-\nu-1}@{x}}{\EulerGamma@{\nu+\mu+1}}		 ℜ ⁡ ( μ - ν ) > 0 , ℜ ⁡ ( ν + μ + 1 ) > 0 formulae-sequence 𝜇 𝜈 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\mu-\nu)>0,\Re(\nu+\mu+1)>0}} 
cos(nu*Pi)*LegendreP(nu, - mu, x) = -(exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1))/(GAMMA(mu - nu))+(exp(-(mu)*Pi*I)*LegendreQ(- nu - 1,mu,x)/GAMMA(- nu - 1+mu+1))/(GAMMA(nu + mu + 1))

Cos[\[Nu]*Pi]*LegendreP[\[Nu], - \[Mu], 3, x] == -Divide[Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1],Gamma[\[Mu]- \[Nu]]]+Divide[Exp[-(\[Mu]) Pi I] LegendreQ[- \[Nu]- 1, \[Mu], 3, x]/Gamma[- \[Nu]- 1 + \[Mu] + 1],Gamma[\[Nu]+ \[Mu]+ 1]]
Failure Failure
Failed [36 / 87]
Result: -9.22033570+3.98641277*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, nu = -1/2+1/2*I*3^(1/2), x = 3/2}


Result: 4.85982369+35.02749311*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
 Successful [Tested: 96]
14.9.E13 P ν - m ⁡ ( x ) = Γ ⁡ ( ν - m + 1 ) Γ ⁡ ( ν + m + 1 ) ⁢ P ν m ⁡ ( x ) Legendre-P-first-kind 𝑚 𝜈 𝑥 Euler-Gamma 𝜈 𝑚 1 Euler-Gamma 𝜈 𝑚 1 Legendre-P-first-kind 𝑚 𝜈 𝑥 {\displaystyle{\displaystyle P^{-m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu-% m+1\right)}{\Gamma\left(\nu+m+1\right)}P^{m}_{\nu}\left(x\right)}}
\assLegendreP[-m]{\nu}@{x} = \frac{\EulerGamma@{\nu-m+1}}{\EulerGamma@{\nu+m+1}}\assLegendreP[m]{\nu}@{x}		 ν ≠ m - 1 , ℜ ⁡ ( ν - m + 1 ) > 0 , ℜ ⁡ ( ν + m + 1 ) > 0 formulae-sequence 𝜈 𝑚 1 formulae-sequence 𝜈 𝑚 1 0 𝜈 𝑚 1 0 {\displaystyle{\displaystyle\nu\neq m-1,\Re(\nu-m+1)>0,\Re(\nu+m+1)>0}} 
LegendreP(nu, - m, x) = (GAMMA(nu - m + 1))/(GAMMA(nu + m + 1))*LegendreP(nu, m, x)

LegendreP[\[Nu], - m, 3, x] == Divide[Gamma[\[Nu]- m + 1],Gamma[\[Nu]+ m + 1]]*LegendreP[\[Nu], m, 3, x]
Failure Failure
Failed [15 / 21]
Result: -.1566814731+1.035406980*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2, m = 1}


Result: .9394863529-.1899097116*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2, m = 1}

... skip entries to safe data
 Successful [Tested: 21]
14.9.E14 𝑸 ν - μ ⁡ ( x ) = 𝑸 ν μ ⁡ ( x ) associated-Legendre-black-Q 𝜇 𝜈 𝑥 associated-Legendre-black-Q 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\boldsymbol{Q}^{-\mu}_{\nu}\left(x\right)=% \boldsymbol{Q}^{\mu}_{\nu}\left(x\right)}}
\assLegendreOlverQ[-\mu]{\nu}@{x} = \assLegendreOlverQ[\mu]{\nu}@{x}		 

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] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1]
Error Successful -
Failed [36 / 300]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[μ, -1.5], Rule[ν, -1.5]}


Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[μ, -1.5], Rule[ν, -0.5]}

... skip entries to safe data
14.9.E15 2 ⁢ sin ⁡ ( μ ⁢ π ) π ⁢ 𝑸 ν μ ⁡ ( x ) = P ν μ ⁡ ( x ) Γ ⁡ ( ν + μ + 1 ) - P ν - μ ⁡ ( x ) Γ ⁡ ( ν - μ + 1 ) 2 𝜇 𝜋 𝜋 associated-Legendre-black-Q 𝜇 𝜈 𝑥 Legendre-P-first-kind 𝜇 𝜈 𝑥 Euler-Gamma 𝜈 𝜇 1 Legendre-P-first-kind 𝜇 𝜈 𝑥 Euler-Gamma 𝜈 𝜇 1 {\displaystyle{\displaystyle\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}% ^{\mu}_{\nu}\left(x\right)=\frac{P^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+% \mu+1\right)}-\frac{P^{-\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu-\mu+1\right)% }}}
\frac{2\sin@{\mu\pi}}{\pi}\assLegendreOlverQ[\mu]{\nu}@{x} = \frac{\assLegendreP[\mu]{\nu}@{x}}{\EulerGamma@{\nu+\mu+1}}-\frac{\assLegendreP[-\mu]{\nu}@{x}}{\EulerGamma@{\nu-\mu+1}}		 ℜ ⁡ ( ν + μ + 1 ) > 0 , ℜ ⁡ ( ν - μ + 1 ) > 0 formulae-sequence 𝜈 𝜇 1 0 𝜈 𝜇 1 0 {\displaystyle{\displaystyle\Re(\nu+\mu+1)>0,\Re(\nu-\mu+1)>0}} 
(2*sin(mu*Pi))/(Pi)*exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = (LegendreP(nu, mu, x))/(GAMMA(nu + mu + 1))-(LegendreP(nu, - mu, x))/(GAMMA(nu - mu + 1))

Divide[2*Sin[\[Mu]*Pi],Pi]*Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == Divide[LegendreP[\[Nu], \[Mu], 3, x],Gamma[\[Nu]+ \[Mu]+ 1]]-Divide[LegendreP[\[Nu], - \[Mu], 3, x],Gamma[\[Nu]- \[Mu]+ 1]]
Failure Successful
Failed [108 / 120]
Result: 3.058402749-19.69019192*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: .1602155595-16.40144782*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
 Successful [Tested: 135]
14.9.E16 𝑸 ν μ ⁡ ( x ) = ( 1 2 ⁢ π ) 1 / 2 ⁢ ( x 2 - 1 ) - 1 / 4 ⁢ P - μ - ( 1 / 2 ) - ν - ( 1 / 2 ) ⁡ ( x ⁢ ( x 2 - 1 ) - 1 / 2 ) associated-Legendre-black-Q 𝜇 𝜈 𝑥 superscript 1 2 𝜋 1 2 superscript superscript 𝑥 2 1 1 4 Legendre-P-first-kind 𝜈 1 2 𝜇 1 2 𝑥 superscript superscript 𝑥 2 1 1 2 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\left(% \tfrac{1}{2}\pi\right)^{1/2}\left(x^{2}-1\right)^{-1/4}\*P^{-\nu-(1/2)}_{-\mu-% (1/2)}\left(x\left(x^{2}-1\right)^{-1/2}\right)}}
\assLegendreOlverQ[\mu]{\nu}@{x} = \left(\tfrac{1}{2}\pi\right)^{1/2}\left(x^{2}-1\right)^{-1/4}\*\assLegendreP[-\nu-(1/2)]{-\mu-(1/2)}@{x\left(x^{2}-1\right)^{-1/2}}		 

exp(-(mu)*Pi*I)*LegendreQ(nu,mu,x)/GAMMA(nu+mu+1) = ((1)/(2)*Pi)^(1/2)*((x)^(2)- 1)^(- 1/4)* LegendreP(- mu -(1/2), - nu -(1/2), x*((x)^(2)- 1)^(- 1/2))

Exp[-(\[Mu]) Pi I] LegendreQ[\[Nu], \[Mu], 3, x]/Gamma[\[Nu] + \[Mu] + 1] == (Divide[1,2]*Pi)^(1/2)*((x)^(2)- 1)^(- 1/4)* LegendreP[- \[Mu]-(1/2), - \[Nu]-(1/2), 3, x*((x)^(2)- 1)^(- 1/2)]
Failure Failure
Failed [292 / 300]
Result: 13.31105553-5.485346831*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: 8.925040493-5.300266523*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
Failed [21 / 300]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[μ, -1.5], Rule[ν, -1.5]}


Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[μ, -1.5], Rule[ν, -0.5]}

... skip entries to safe data
14.9.E17 P ν μ ⁡ ( x ) = ( 2 / π ) 1 / 2 ⁢ ( x 2 - 1 ) - 1 / 4 ⁢ 𝑸 - μ - ( 1 / 2 ) ν + ( 1 / 2 ) ⁡ ( x ⁢ ( x 2 - 1 ) - 1 / 2 ) Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript 2 𝜋 1 2 superscript superscript 𝑥 2 1 1 4 associated-Legendre-black-Q 𝜈 1 2 𝜇 1 2 𝑥 superscript superscript 𝑥 2 1 1 2 {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=(2/\pi)^{1/2}\left(x^% {2}-1\right)^{-1/4}\*\boldsymbol{Q}^{\nu+(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}% -1\right)^{-1/2}\right)}}
\assLegendreP[\mu]{\nu}@{x} = (2/\pi)^{1/2}\left(x^{2}-1\right)^{-1/4}\*\assLegendreOlverQ[\nu+(1/2)]{-\mu-(1/2)}@{x\left(x^{2}-1\right)^{-1/2}}		 

LegendreP(nu, mu, x) = (2/Pi)^(1/2)*((x)^(2)- 1)^(- 1/4)* exp(-(nu +(1/2))*Pi*I)*LegendreQ(- mu -(1/2),nu +(1/2),x*((x)^(2)- 1)^(- 1/2))/GAMMA(- mu -(1/2)+nu +(1/2)+1)

LegendreP[\[Nu], \[Mu], 3, x] == (2/Pi)^(1/2)*((x)^(2)- 1)^(- 1/4)* Exp[-(\[Nu]+(1/2)) Pi I] LegendreQ[- \[Mu]-(1/2), \[Nu]+(1/2), 3, x*((x)^(2)- 1)^(- 1/2)]/Gamma[- \[Mu]-(1/2) + \[Nu]+(1/2) + 1]
Failure Failure
Failed [297 / 300]
Result: 15.05963282-19.56004465*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: 2.964591568-6.756538622*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
Failed [21 / 300]
Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[μ, 1.5], Rule[ν, -1.5]}


Result: Indeterminate
Test Values: {Rule[x, 1.5], Rule[μ, 1.5], Rule[ν, -0.5]}

... skip entries to safe data
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