Formula:KLS:09.08:03

<< Formula:KLS:09.08:02

formula in Jacobi: Special cases

Formula:KLS:09.08:04 >>

${\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{\lambda}_{m}\left(x\right)C^{\lambda}_{n}\left(x\right)\,dx{}=\frac{\pi\Gamma\left(n+2\lambda\right)2^{1-2\lambda}}{\left\{\Gamma\left(\lambda\right)\right\}^{2}(n+\lambda)n!}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\lambda>-\frac{1}{2}\quad\lambda\neq 0}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}$ : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : 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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<< Formula:KLS:09.08:02

formula in Jacobi: Special cases

Formula:KLS:09.08:04 >>