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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}
{\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		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}
{\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		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {(\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}
{(\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		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}
\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}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(1-x^{2}\right)\deriv{\FerrersP[\mu]{\nu}@{x}}{x} = (\nu+\mu)\FerrersP[\mu]{\nu-1}@{x}-\nu x\FerrersP[\mu]{\nu}@{x}}
\left(1-x^{2}\right)\deriv{\FerrersP[\mu]{\nu}@{x}}{x} = (\nu+\mu)\FerrersP[\mu]{\nu-1}@{x}-\nu x\FerrersP[\mu]{\nu}@{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}
{\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		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}
{\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		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
((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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