Formula:KLS:14.04:23

$\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{a\expe^{\iunit(\theta+2\phi)},d\expe^{\iunit\theta}}{ad}{q}{\expe^{-\iunit(\theta+\phi)}t}\ \qHyperrphis{2}{1}@@{b\expe^{-\iunit\theta},c\expe^{-\iunit(\theta+2\phi)}}{bc}{q}{\expe^{\iunit(\theta+\phi)}t} {}=\sum_{n=0}^{\infty}\frac{\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ad,bc,q}{q}{n}}t^n }$

Substitution(s)

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

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
$\displaystyle {\displaystyle p_{n}}$  : continuous $\displaystyle {\displaystyle q}$ -Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqHahn
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
$\displaystyle {\displaystyle \mathrm{cos}}$  : cosine function : http://dlmf.nist.gov/4.14#E2