Formula:KLS:09.08:09

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( 1 - x 2 ) ⁢ d d ⁢ x ⁢ C n λ ⁡ ( x ) + ( 1 - 2 ⁢ λ ) ⁢ x ⁢ C n λ ⁡ ( x ) = - ( n + 1 ) ⁢ ( 2 ⁢ λ + n - 1 ) 2 ⁢ ( λ - 1 ) ⁢ C n + 1 λ - 1 ⁡ ( x ) 1 superscript 𝑥 2 𝑑 𝑑 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 1 2 𝜆 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 𝑛 1 2 𝜆 𝑛 1 2 𝜆 1 ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})\frac{d}{dx}C^{\lambda}_{n}% \left(x\right)+(1-2\lambda)xC^{\lambda}_{n}\left(x\right)=-\frac{(n+1)(2% \lambda+n-1)}{2(\lambda-1)}C^{\lambda-1}_{n+1}\left(x\right)}}}

Proof

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Symbols List

C n μ subscript superscript 𝐶 𝜇 𝑛 {\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}  : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5

Bibliography

Equation in Section 9.8 of KLS.

URL links

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