Formula:KLS:14.10:82

<< Formula:KLS:14.10:81

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:83 >>

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(\cos\theta;\beta\,|\,q\right)=\sum_{k=0}^{n}\frac{\left(\beta;q\right)_{k}\left(\beta;q\right)_{n-k}}{\left(q;q\right)_{k}\left(q;q\right)_{n-k}}{\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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

Bibliography

Equation in Section 14.10 of KLS.

URL links

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<< Formula:KLS:14.10:81

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:83 >>