View source for 14.10

{{DISPLAYTITLE:Legendre and Related Functions - 14.10 Recurrence Relations and Derivatives}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/14.10.E1 14.10.E1] || <math qid="Q4810">{\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</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\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</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || Failure || Successful || Successful [Tested: 300] || Successful [Tested: 300]
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| [https://dlmf.nist.gov/14.10.E2 14.10.E2] || <math qid="Q4811">{\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</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\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</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>(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</syntaxhighlight> || <syntaxhighlight lang=mathematica>(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</syntaxhighlight> || Failure || Successful || Successful [Tested: 300] || Successful [Tested: 300]
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| [https://dlmf.nist.gov/14.10.E3 14.10.E3] || <math qid="Q4812">{(\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</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{(\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</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>(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</syntaxhighlight> || <syntaxhighlight lang=mathematica>(\[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</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 300]
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| [https://dlmf.nist.gov/14.10.E4 14.10.E4] || <math qid="Q4813">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>(1 - (x)^(2))*diff(LegendreP(nu, mu, x), x) = (mu - nu - 1)*LegendreP(nu + 1, mu, x)+(nu + 1)*x*LegendreP(nu, mu, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(1 - (x)^(2))*D[LegendreP[\[Nu], \[Mu], x], x] == (\[Mu]- \[Nu]- 1)*LegendreP[\[Nu]+ 1, \[Mu], x]+(\[Nu]+ 1)*x*LegendreP[\[Nu], \[Mu], x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 300]
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| [https://dlmf.nist.gov/14.10.E5 14.10.E5] || <math qid="Q4814">\left(1-x^{2}\right)\deriv{\FerrersP[\mu]{\nu}@{x}}{x} = (\nu+\mu)\FerrersP[\mu]{\nu-1}@{x}-\nu x\FerrersP[\mu]{\nu}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(1-x^{2}\right)\deriv{\FerrersP[\mu]{\nu}@{x}}{x} = (\nu+\mu)\FerrersP[\mu]{\nu-1}@{x}-\nu x\FerrersP[\mu]{\nu}@{x}</syntaxhighlight> || <math>|(\tfrac{1}{2}-\tfrac{1}{2}x)| < 1</math> || <syntaxhighlight lang=mathematica>(1 - (x)^(2))*diff(LegendreP(nu, mu, x), x) = (nu + mu)*LegendreP(nu - 1, mu, x)- nu*x*LegendreP(nu, mu, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(1 - (x)^(2))*D[LegendreP[\[Nu], \[Mu], x], x] == (\[Nu]+ \[Mu])*LegendreP[\[Nu]- 1, \[Mu], x]- \[Nu]*x*LegendreP[\[Nu], \[Mu], x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 300]
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| [https://dlmf.nist.gov/14.10.E6 14.10.E6] || <math qid="Q4815">{\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</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\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</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || <syntaxhighlight lang=mathematica>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</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || Successful [Tested: 300]
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| [https://dlmf.nist.gov/14.10.E7 14.10.E7] || <math qid="Q4816">{\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</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>{\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</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>((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</syntaxhighlight> || <syntaxhighlight lang=mathematica>((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</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || Successful [Tested: 300]
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