Formula:KLS:09.08:05

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x ⁢ C ^ n λ ⁡ ( x ) ⁢ x = C ^ n + 1 λ ⁡ ( x ) ⁢ x + n ⁢ ( n + 2 ⁢ λ - 1 ) 4 ⁢ ( n + λ - 1 ) ⁢ ( n + λ ) ⁢ C ^ n - 1 λ ⁡ ( x ) ⁢ x 𝑥 ultraspherical-Gegenbauer-polynomial-monic-p 𝜆 𝑛 𝑥 𝑥 ultraspherical-Gegenbauer-polynomial-monic-p 𝜆 𝑛 1 𝑥 𝑥 𝑛 𝑛 2 𝜆 1 4 𝑛 𝜆 1 𝑛 𝜆 ultraspherical-Gegenbauer-polynomial-monic-p 𝜆 𝑛 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle x{\widehat{C}}^{\lambda}_{n}\left(x% \right){x}={\widehat{C}}^{\lambda}_{n+1}\left(x\right){x}+\frac{n(n+2\lambda-1% )}{4(n+\lambda-1)(n+\lambda)}{\widehat{C}}^{\lambda}_{n-1}\left(x\right){x}}}}

Proof

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Symbols List

C ^ n μ subscript superscript ^ 𝐶 𝜇 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{C}}^{\mu}_{n}}}}  : monic ultraspherical/Gegenbauer polynomial : http://drmf.wmflabs.org/wiki/Definition:monicUltra

Bibliography

Equation in Section 9.8 of KLS.

URL links

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