Formula:KLS:09.08:29

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