Formula:KLS:14.10:82

$\displaystyle {\displaystyle \ctsqUltra{n}@{\cos@@{\theta}}{\beta }{ q}=\sum_{k=0}^n \frac{\qPochhammer{\beta}{q}{k} \qPochhammer{\beta}{q}{n-k}}{\qPochhammer{q}{q}{k} \qPochhammer{q}{q}{n-k}} \expe^{\iunit(n-2k)\theta} }$

Proof

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Symbols List

$\displaystyle {\displaystyle C_{n}}$  : continuous $\displaystyle {\displaystyle q}$ -ultraspherical/Rogers polynomial : http://dlmf.nist.gov/18.28#E13
$\displaystyle {\displaystyle \mathrm{cos}}$  : cosine function : http://dlmf.nist.gov/4.14#E2
$\displaystyle {\displaystyle \Sigma}$  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
$\displaystyle {\displaystyle \mathrm{e}}$  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
$\displaystyle {\displaystyle \mathrm{i}}$  : imaginary unit : http://dlmf.nist.gov/1.9.i