Formula:KLS:14.10:35

<< Continuous q-Jacobi

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:36 >>

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;\beta\,|\,q\right)=\frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}}\beta^{-\frac{1}{2}n}\,\qHyperrphis{4}{3}@@{q^{-n},\beta^{2}q^{n}\beta^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}\theta}},\beta^{\frac{1}{2}}{\mathrm{e}^{-\mathrm{i}\theta}}}{\beta q^{\frac{1}{2}}-\beta,-\beta q^{\frac{1}{2}}}{q}{q}}}}$

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

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:36 >>