Formula:KLS:09.08:03

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∫ - 1 1 ( 1 - x 2 ) λ - 1 2 ⁢ C m λ ⁡ ( x ) ⁢ C n λ ⁡ ( x ) ⁢ 𝑑 x = π ⁢ Γ ⁡ ( n + 2 ⁢ λ ) ⁢ 2 1 - 2 ⁢ λ { Γ ⁡ ( λ ) } 2 ⁢ ( n + λ ) ⁢ n ! ⁢ δ m , n superscript subscript 1 1 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑚 𝑥 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 differential-d 𝑥 Euler-Gamma 𝑛 2 𝜆 superscript 2 1 2 𝜆 superscript Euler-Gamma 𝜆 2 𝑛 𝜆 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}(1-x^{2})^{\lambda-% \frac{1}{2}}C^{\lambda}_{m}\left(x\right)C^{\lambda}_{n}\left(x\right)\,dx{}=% \frac{\pi\Gamma\left(n+2\lambda\right)2^{1-2\lambda}}{\left\{\Gamma\left(% \lambda\right)\right\}^{2}(n+\lambda)n!}\,\delta_{m,n}}}}

Constraint(s)

λ > - 1 2   λ ≠ 0 formulae-sequence 𝜆 1 2 𝜆 0 {\displaystyle{\displaystyle{\displaystyle\lambda>-\frac{1}{2}\quad\lambda\neq 0% }}}


Proof

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Symbols List

∫ {\displaystyle{\displaystyle{\displaystyle\int}}}  : integral : http://dlmf.nist.gov/1.4#iv
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
π 𝜋 {\displaystyle{\displaystyle{\displaystyle\pi}}}  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
δ m , n subscript 𝛿 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.8 of KLS.

URL links

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