# Formula:KLS:14.01:02

$\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\AskeyWilson{m}@{x}{a}{b}{c}{d}{q}\AskeyWilson{n}@{x}{a}{b}{c}{d}{q}\,dx=h_n\,\Kronecker{m}{n} }$

## Substitution(s)

$\displaystyle {\displaystyle h_n=\frac{\qPochhammer{abcdq^{n-1}}{q}{n}\qPochhammer{abcdq^{2n}}{q}{\infty}}{\qPochhammer{q^{n+1},abq^n,acq^n,adq^n,bcq^n,bdq^n,cdq^n}{q}{\infty}}}$ &

$\displaystyle {\displaystyle w(x):=w(x;a,b,c,d|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta} c\expe^{\iunit\theta},d\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a)h(x,b)h(x,c)h(x,d)}}$ &
$\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}$ &

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

## 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
$\displaystyle {\displaystyle p_{n}}$  : Askey-Wilson polynomial : http://dlmf.nist.gov/18.28#E1
$\displaystyle {\displaystyle \delta_{m,n}}$  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -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
$\displaystyle {\displaystyle \Pi}$  : product : http://drmf.wmflabs.org/wiki/Definition:prod
$\displaystyle {\displaystyle \mathrm{cos}}$  : cosine function : http://dlmf.nist.gov/4.14#E2