# Formula:KLS:14.10:57

$\displaystyle {\displaystyle \frac{1}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{\beta,\beta\expe^{2\iunit\theta}}{\beta^2}{q}{\expe^{-\iunit\theta}t} =\sum_{n=0}^{\infty}\frac{\ctsqUltra{n}@{x}{\beta}{q}}{\qPochhammer{\beta^2}{q}{n}}t^n }$

## Substitution(s)

$\displaystyle {\displaystyle x=\cos@@{\theta}}$

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

## Symbols List

$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -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
$\displaystyle {\displaystyle \Sigma}$  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
$\displaystyle {\displaystyle C_{n}}$  : 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