# Formula:KLS:14.10:61

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

## 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

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