# Formula:KLS:14.04:02

$\displaystyle {\displaystyle \frac{1}{4\cpi}\int_{-\cpi}^{\cpi}w(\cos@{\theta+\phi}) \ctsqHahn{m}@{\cos@{\theta+\phi}}{a}{b}{c}{d}{q}\ctsqHahn{n}@{\cos@{\theta+\phi}}{a}{b}{c}{d}{q}\,d\theta {}=\frac{\qPochhammer{abcdq^{n-1}}{q}{n}\qPochhammer{abcdq^{2n}}{q}{\infty}} {\qPochhammer{q^{n+1},abq^n\expe^{2\iunit\phi},acq^n,adq^n,bcq^n,bdq^n,cdq^n\expe^{-2\iunit\phi}}{q}{\infty}}\,\Kronecker{m}{n} }$

## Substitution(s)

$\displaystyle {\displaystyle w(x):=w(x;a,b,c,d;q) =\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit(\theta+2\phi)},b\expe^{\iunit(\theta+2\phi)} c\expe^{\iunit\theta},d\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a\expe^{\iunit\phi})h(x,b\expe^{\iunit\phi})h(x,c\expe^{-\iunit\phi})h(x,d\expe^{-\iunit\phi})}}$ &

$\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit(\theta+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{q}{\infty}}$ &

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

## 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

& : logical and
$\displaystyle {\displaystyle \int}$  : integral : http://dlmf.nist.gov/1.4#iv
$\displaystyle {\displaystyle \mathrm{cos}}$  : cosine function : http://dlmf.nist.gov/4.14#E2
$\displaystyle {\displaystyle p_{n}}$  : 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
$\displaystyle {\displaystyle \mathrm{e}}$  : 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
$\displaystyle {\displaystyle \delta_{m,n}}$  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4
$\displaystyle {\displaystyle \Pi}$  : product : http://drmf.wmflabs.org/wiki/Definition:prod