Legendre and Related Functions - 14.10 Recurrence Relations and Derivatives

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
14.10.E1 𝖯 Ξ½ ΞΌ + 2 ⁑ ( x ) + 2 ⁒ ( ΞΌ + 1 ) ⁒ x ⁒ ( 1 - x 2 ) - 1 / 2 ⁒ 𝖯 Ξ½ ΞΌ + 1 ⁑ ( x ) + ( Ξ½ - ΞΌ ) ⁒ ( Ξ½ + ΞΌ + 1 ) ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) = 0 Ferrers-Legendre-P-first-kind πœ‡ 2 𝜈 π‘₯ 2 πœ‡ 1 π‘₯ superscript 1 superscript π‘₯ 2 1 2 Ferrers-Legendre-P-first-kind πœ‡ 1 𝜈 π‘₯ 𝜈 πœ‡ 𝜈 πœ‡ 1 Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ 0 {\displaystyle{\displaystyle{\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x% \left(1-x^{2}\right)^{-1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(% \nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0}}
{\FerrersP[\mu+2]{\nu}@{x}+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}\FerrersP[\mu+1]{\nu}@{x}}+(\nu-\mu)(\nu+\mu+1)\FerrersP[\mu]{\nu}@{x} = 0		 | ( 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 + 2, x)+ 2*(mu + 1)*x*(1 - (x)^(2))^(- 1/2)* LegendreP(nu, mu + 1, x)+(nu - mu)*(nu + mu + 1)*LegendreP(nu, mu, x) = 0

LegendreP[\[Nu], \[Mu]+ 2, x]+ 2*(\[Mu]+ 1)*x*(1 - (x)^(2))^(- 1/2)* LegendreP[\[Nu], \[Mu]+ 1, x]+(\[Nu]- \[Mu])*(\[Nu]+ \[Mu]+ 1)*LegendreP[\[Nu], \[Mu], x] == 0
Failure Successful Successful [Tested: 300] Successful [Tested: 300]
14.10.E2 ( 1 - x 2 ) 1 / 2 ⁒ 𝖯 Ξ½ ΞΌ + 1 ⁑ ( x ) - ( Ξ½ - ΞΌ + 1 ) ⁒ 𝖯 Ξ½ + 1 ΞΌ ⁑ ( x ) + ( Ξ½ + ΞΌ + 1 ) ⁒ x ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) = 0 superscript 1 superscript π‘₯ 2 1 2 Ferrers-Legendre-P-first-kind πœ‡ 1 𝜈 π‘₯ 𝜈 πœ‡ 1 Ferrers-Legendre-P-first-kind πœ‡ 𝜈 1 π‘₯ 𝜈 πœ‡ 1 π‘₯ Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ 0 {\displaystyle{\displaystyle{\left(1-x^{2}\right)^{1/2}\mathsf{P}^{\mu+1}_{\nu% }\left(x\right)-(\nu-\mu+1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)% x\mathsf{P}^{\mu}_{\nu}\left(x\right)=0}}
{\left(1-x^{2}\right)^{1/2}\FerrersP[\mu+1]{\nu}@{x}-(\nu-\mu+1)\FerrersP[\mu]{\nu+1}@{x}}+(\nu+\mu+1)x\FerrersP[\mu]{\nu}@{x} = 0		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
(1 - (x)^(2))^(1/2)* LegendreP(nu, mu + 1, x)-(nu - mu + 1)*LegendreP(nu + 1, mu, x)+(nu + mu + 1)*x*LegendreP(nu, mu, x) = 0

(1 - (x)^(2))^(1/2)* LegendreP[\[Nu], \[Mu]+ 1, x]-(\[Nu]- \[Mu]+ 1)*LegendreP[\[Nu]+ 1, \[Mu], x]+(\[Nu]+ \[Mu]+ 1)*x*LegendreP[\[Nu], \[Mu], x] == 0
Failure Successful Successful [Tested: 300] Successful [Tested: 300]
14.10.E3 ( Ξ½ - ΞΌ + 2 ) ⁒ 𝖯 Ξ½ + 2 ΞΌ ⁑ ( x ) - ( 2 ⁒ Ξ½ + 3 ) ⁒ x ⁒ 𝖯 Ξ½ + 1 ΞΌ ⁑ ( x ) + ( Ξ½ + ΞΌ + 1 ) ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) = 0 𝜈 πœ‡ 2 Ferrers-Legendre-P-first-kind πœ‡ 𝜈 2 π‘₯ 2 𝜈 3 π‘₯ Ferrers-Legendre-P-first-kind πœ‡ 𝜈 1 π‘₯ 𝜈 πœ‡ 1 Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ 0 {\displaystyle{\displaystyle{(\nu-\mu+2)\mathsf{P}^{\mu}_{\nu+2}\left(x\right)% -(2\nu+3)x\mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)\mathsf{P}^{\mu}_% {\nu}\left(x\right)=0}}
{(\nu-\mu+2)\FerrersP[\mu]{\nu+2}@{x}-(2\nu+3)x\FerrersP[\mu]{\nu+1}@{x}}+(\nu+\mu+1)\FerrersP[\mu]{\nu}@{x} = 0		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
(nu - mu + 2)*LegendreP(nu + 2, mu, x)-(2*nu + 3)*x*LegendreP(nu + 1, mu, x)+(nu + mu + 1)*LegendreP(nu, mu, x) = 0

(\[Nu]- \[Mu]+ 2)*LegendreP[\[Nu]+ 2, \[Mu], x]-(2*\[Nu]+ 3)*x*LegendreP[\[Nu]+ 1, \[Mu], x]+(\[Nu]+ \[Mu]+ 1)*LegendreP[\[Nu], \[Mu], x] == 0
Successful Successful - Successful [Tested: 300]
14.10.E4 ( 1 - x 2 ) ⁒ d 𝖯 Ξ½ ΞΌ ⁑ ( x ) d x = ( ΞΌ - Ξ½ - 1 ) ⁒ 𝖯 Ξ½ + 1 ΞΌ ⁑ ( x ) + ( Ξ½ + 1 ) ⁒ x ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) 1 superscript π‘₯ 2 derivative Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ π‘₯ πœ‡ 𝜈 1 Ferrers-Legendre-P-first-kind πœ‡ 𝜈 1 π‘₯ 𝜈 1 π‘₯ Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ {\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{% \mu}_{\nu}\left(x\right)}{\mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}% \left(x\right)+(\nu+1)x\mathsf{P}^{\mu}_{\nu}\left(x\right)}}}
\left(1-x^{2}\right)\deriv{\FerrersP[\mu]{\nu}@{x}}{x} = {(\mu-\nu-1)\FerrersP[\mu]{\nu+1}@{x}+(\nu+1)x\FerrersP[\mu]{\nu}@{x}}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
(1 - (x)^(2))*diff(LegendreP(nu, mu, x), x) = (mu - nu - 1)*LegendreP(nu + 1, mu, x)+(nu + 1)*x*LegendreP(nu, mu, x)

(1 - (x)^(2))*D[LegendreP[\[Nu], \[Mu], x], x] == (\[Mu]- \[Nu]- 1)*LegendreP[\[Nu]+ 1, \[Mu], x]+(\[Nu]+ 1)*x*LegendreP[\[Nu], \[Mu], x]
Successful Successful - Successful [Tested: 300]
14.10.E5 ( 1 - x 2 ) ⁒ d 𝖯 Ξ½ ΞΌ ⁑ ( x ) d x = ( Ξ½ + ΞΌ ) ⁒ 𝖯 Ξ½ - 1 ΞΌ ⁑ ( x ) - Ξ½ ⁒ x ⁒ 𝖯 Ξ½ ΞΌ ⁑ ( x ) 1 superscript π‘₯ 2 derivative Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ π‘₯ 𝜈 πœ‡ Ferrers-Legendre-P-first-kind πœ‡ 𝜈 1 π‘₯ 𝜈 π‘₯ Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ {\displaystyle{\displaystyle\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{% \mu}_{\nu}\left(x\right)}{\mathrm{d}x}=(\nu+\mu)\mathsf{P}^{\mu}_{\nu-1}\left(% x\right)-\nu x\mathsf{P}^{\mu}_{\nu}\left(x\right)}}
\left(1-x^{2}\right)\deriv{\FerrersP[\mu]{\nu}@{x}}{x} = (\nu+\mu)\FerrersP[\mu]{\nu-1}@{x}-\nu x\FerrersP[\mu]{\nu}@{x}		 | ( 1 2 - 1 2 ⁒ x ) | < 1 1 2 1 2 π‘₯ 1 {\displaystyle{\displaystyle|(\tfrac{1}{2}-\tfrac{1}{2}x)|<1}} 
(1 - (x)^(2))*diff(LegendreP(nu, mu, x), x) = (nu + mu)*LegendreP(nu - 1, mu, x)- nu*x*LegendreP(nu, mu, x)

(1 - (x)^(2))*D[LegendreP[\[Nu], \[Mu], x], x] == (\[Nu]+ \[Mu])*LegendreP[\[Nu]- 1, \[Mu], x]- \[Nu]*x*LegendreP[\[Nu], \[Mu], x]
Successful Successful - Successful [Tested: 300]
14.10.E6 P Ξ½ ΞΌ + 2 ⁑ ( x ) + 2 ⁒ ( ΞΌ + 1 ) ⁒ x ⁒ ( x 2 - 1 ) - 1 / 2 ⁒ P Ξ½ ΞΌ + 1 ⁑ ( x ) - ( Ξ½ - ΞΌ ) ⁒ ( Ξ½ + ΞΌ + 1 ) ⁒ P Ξ½ ΞΌ ⁑ ( x ) = 0 Legendre-P-first-kind πœ‡ 2 𝜈 π‘₯ 2 πœ‡ 1 π‘₯ superscript superscript π‘₯ 2 1 1 2 Legendre-P-first-kind πœ‡ 1 𝜈 π‘₯ 𝜈 πœ‡ 𝜈 πœ‡ 1 Legendre-P-first-kind πœ‡ 𝜈 π‘₯ 0 {\displaystyle{\displaystyle{P^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(x^{2% }-1\right)^{-1/2}P^{\mu+1}_{\nu}\left(x\right)}-(\nu-\mu)(\nu+\mu+1)P^{\mu}_{% \nu}\left(x\right)=0}}
{\assLegendreP[\mu+2]{\nu}@{x}+2(\mu+1)x\left(x^{2}-1\right)^{-1/2}\assLegendreP[\mu+1]{\nu}@{x}}-(\nu-\mu)(\nu+\mu+1)\assLegendreP[\mu]{\nu}@{x} = 0		 

LegendreP(nu, mu + 2, x)+ 2*(mu + 1)*x*((x)^(2)- 1)^(- 1/2)* LegendreP(nu, mu + 1, x)-(nu - mu)*(nu + mu + 1)*LegendreP(nu, mu, x) = 0

LegendreP[\[Nu], \[Mu]+ 2, 3, x]+ 2*(\[Mu]+ 1)*x*((x)^(2)- 1)^(- 1/2)* LegendreP[\[Nu], \[Mu]+ 1, 3, x]-(\[Nu]- \[Mu])*(\[Nu]+ \[Mu]+ 1)*LegendreP[\[Nu], \[Mu], 3, x] == 0
Failure Failure Successful [Tested: 300] Successful [Tested: 300]
14.10.E7 ( x 2 - 1 ) 1 / 2 ⁒ P Ξ½ ΞΌ + 1 ⁑ ( x ) - ( Ξ½ - ΞΌ + 1 ) ⁒ P Ξ½ + 1 ΞΌ ⁑ ( x ) + ( Ξ½ + ΞΌ + 1 ) ⁒ x ⁒ P Ξ½ ΞΌ ⁑ ( x ) = 0 superscript superscript π‘₯ 2 1 1 2 Legendre-P-first-kind πœ‡ 1 𝜈 π‘₯ 𝜈 πœ‡ 1 Legendre-P-first-kind πœ‡ 𝜈 1 π‘₯ 𝜈 πœ‡ 1 π‘₯ Legendre-P-first-kind πœ‡ 𝜈 π‘₯ 0 {\displaystyle{\displaystyle{\left(x^{2}-1\right)^{1/2}P^{\mu+1}_{\nu}\left(x% \right)-(\nu-\mu+1)P^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)xP^{\mu}_{\nu}% \left(x\right)=0}}
{\left(x^{2}-1\right)^{1/2}\assLegendreP[\mu+1]{\nu}@{x}-(\nu-\mu+1)\assLegendreP[\mu]{\nu+1}@{x}}+(\nu+\mu+1)x\assLegendreP[\mu]{\nu}@{x} = 0		 

((x)^(2)- 1)^(1/2)* LegendreP(nu, mu + 1, x)-(nu - mu + 1)*LegendreP(nu + 1, mu, x)+(nu + mu + 1)*x*LegendreP(nu, mu, x) = 0

((x)^(2)- 1)^(1/2)* LegendreP[\[Nu], \[Mu]+ 1, 3, x]-(\[Nu]- \[Mu]+ 1)*LegendreP[\[Nu]+ 1, \[Mu], 3, x]+(\[Nu]+ \[Mu]+ 1)*x*LegendreP[\[Nu], \[Mu], 3, x] == 0
Failure Failure Successful [Tested: 300] Successful [Tested: 300]
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