Formula:KLS:14.10:37

<< Formula:KLS:14.10:36

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:38 >>

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

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
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 14.10 of KLS.

URL links

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<< Formula:KLS:14.10:36

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:38 >>