Formula:KLS:14.10:72

<< Formula:KLS:14.10:71

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:73 >>

${\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow 1}\frac{1-q^{n}}{2(1-\beta)}C_{n}\!\left(x;\beta\,|\,q\right)=\cos n\theta=T_{n}\left(x\right),\quad x=\cos\theta}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\ldots}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle C_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-ultraspherical/Rogers polynomial : http://dlmf.nist.gov/18.28#E13
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle T_{n}}}}$ : Chebyshev polynomial of the first kind : http://dlmf.nist.gov/18.3#T1.t1.r8

Bibliography

Equation in Section 14.10 of KLS.

URL links

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<< Formula:KLS:14.10:71

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:73 >>