Formula:KLS:14.04:01

$\displaystyle {\displaystyle \frac{(a\expe^{\iunit\phi})^n\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab\expe^{2\iunit\phi},ac,ad}{q}{n}} {}=\qHyperrphis{4}{3}@@{q^{-n},abcdq^{n-1},a\expe^{\iunit(\theta+2\phi)},a\expe^{-\iunit\theta}}{ab\expe^{2\iunit\phi},ac,ad}{q}{q} }$

Substitution(s)

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

Proof

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Symbols List

$\displaystyle {\displaystyle \mathrm{e}}$  : 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
$\displaystyle {\displaystyle p_{n}}$  : 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
$\displaystyle {\displaystyle {{}_{r}\phi_{s}}}$  : basic hypergeometric (or $\displaystyle {\displaystyle q}$ -hypergeometric) function : http://dlmf.nist.gov/17.4#E1
$\displaystyle {\displaystyle \mathrm{cos}}$  : cosine function : http://dlmf.nist.gov/4.14#E2