# Formula:KLS:14.19:21

$\displaystyle {\displaystyle \frac{1}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\,\qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta}}{q^{\alpha+1}}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\ctsqLaguerre{\alpha}{n}@{x}{q}t^n}{\qPochhammer{q^{\alpha+1}}{q}{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}} }$

## 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
$\displaystyle {\displaystyle {{}_{r}\phi_{s}}}$  : basic hypergeometric (or $\displaystyle {\displaystyle q}$ -hypergeometric) function : http://dlmf.nist.gov/17.4#E1
$\displaystyle {\displaystyle \Sigma}$  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
$\displaystyle {\displaystyle P^{(n)}_{\alpha}}$  : continuous $\displaystyle {\displaystyle q}$ -Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre
$\displaystyle {\displaystyle \mathrm{cos}}$  : cosine function : http://dlmf.nist.gov/4.14#E2