Formula:KLS:14.10:71

<< Formula:KLS:14.10:70

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:72 >>

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;q\,|\,q\right)=\frac{\sin\left(n+1\right)\theta}{\sin\theta}=U_{n}\left(x\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle U_{n}}}}$ : Chebyshev polynomial of the second kind : http://dlmf.nist.gov/18.3#T1.t1.r11
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.10 of KLS.

URL links

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<< Formula:KLS:14.10:70

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:72 >>