Formula:KLS:09.08:89

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x ⁢ P ^ n ⁡ ( x ) ⁢ x = P ^ n + 1 ⁡ ( x ) ⁢ x + n 2 ( 2 ⁢ n - 1 ) ⁢ ( 2 ⁢ n + 1 ) ⁢ P ^ n - 1 ⁡ ( x ) ⁢ x 𝑥 Legendre-spherical-polynomial-monic-p 𝑛 𝑥 𝑥 Legendre-spherical-polynomial-monic-p 𝑛 1 𝑥 𝑥 superscript 𝑛 2 2 𝑛 1 2 𝑛 1 Legendre-spherical-polynomial-monic-p 𝑛 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\left(x\right){x}% ={\widehat{P}}_{n+1}\left(x\right){x}+\frac{n^{2}}{(2n-1)(2n+1)}{\widehat{P}}_% {n-1}\left(x\right){x}}}}

Proof

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Symbols List

P ^ n subscript ^ 𝑃 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{P}}_{n}}}}  : monic Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:monicLegendrePoly

Bibliography

Equation in Section 9.8 of KLS.

URL links

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