# Formula:KLS:14.10:37

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

## Proof

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## Symbols List

$\displaystyle {\displaystyle C_{n}}$  : continuous $\displaystyle {\displaystyle q}$ -ultraspherical/Rogers polynomial : http://dlmf.nist.gov/18.28#E13
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -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