Formula:KLS:09.08:01

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formula in Jacobi: Special cases

Jacobi: Special cases >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\lambda\neq 0}}}$

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
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}$ : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3

Bibliography

Equation in Section 9.8 of KLS.

URL links

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formula in Jacobi: Special cases

Jacobi: Special cases >>