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formula in Jacobi: Special cases

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${\displaystyle{\displaystyle{\displaystyle\frac{C^{\alpha+\frac{1}{2}}_{2n+1}\left(x\right)}{C^{\alpha+\frac{1}{2}}_{2n+1}\left(1\right)}=\frac{xP^{(\alpha,\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(\alpha,\frac{1}{2})}_{n}\left(1\right)}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}$ : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}$ : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3

Bibliography

Equation in Section of KLS.

URL links

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formula in Jacobi: Special cases

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