Formula:KLS:14.10:51

<< Formula:KLS:14.10:47

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:52 >>

${\displaystyle{\displaystyle{\displaystyle\delta_{q}C_{n}\!\left(x;\beta\,|\,q\right)=-q^{-\frac{1}{2}n}(1-\beta)({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}})C_{n-1}\!\left(x;\beta q\,|\,q\right)}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : 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:14.10:47

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:52 >>