Formula:KLS:09.08:46

<< Formula:KLS:09.08:45

formula in Jacobi: Special cases

Formula:KLS:09.08:47 >>

${\displaystyle{\displaystyle{\displaystyle U_{n}\left(x\right)=(n+1)\frac{P^{(\frac{1}{2},\frac{1}{2})}_{n}\left(x\right)}{P^{(\frac{1}{2},\frac{1}{2})}_{n}\left(1\right)}=(n+1)\,\HyperpFq{2}{1}@@{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle U_{n}}}}$ : Chebyshev polynomial of the second kind : http://dlmf.nist.gov/18.3#T1.t1.r11
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}$ : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3
${\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}$ : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1

Bibliography

Equation in Section 9.8 of KLS.

URL links

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<< Formula:KLS:09.08:45

formula in Jacobi: Special cases

Formula:KLS:09.08:47 >>