Formula:KLS:14.08:28

$\displaystyle {\displaystyle \frac{\qPochhammer{\gamma \expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{3}{2}@@{\gamma,a\expe^{\iunit\theta},b\expe^{\iunit\theta}}{ab,\gamma \expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{ab,q}{q}{n}}\AlSalamChihara{n}@{x}{a}{b}{q}t^n }$

Substitution(s)

$\displaystyle {\displaystyle x=\cos@@{\theta}}$ &
$\displaystyle {\displaystyle \gamma}$ arbitrary

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

& : logical and
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -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
$\displaystyle {\displaystyle Q_{n}}$  : Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara
$\displaystyle {\displaystyle \mathrm{cos}}$  : cosine function : http://dlmf.nist.gov/4.14#E2