# Formula:KLS:14.08:07

$\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\AlSalamChihara{m}@{x}{a}{b}{q}\AlSalamChihara{n}@{x}{a}{b}{q}\,dx {}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1

## Substitution(s)

$\displaystyle {\displaystyle w_k=\frac{\qPochhammer{a^{-2}}{q}{\infty}}{\qPochhammer{q,ab,a^{-1}b}{q}{\infty}} \frac{(1-a^2q^{2k})\qPochhammer{a^2,ab}{q}{k}}{(1-a^2)\qPochhammer{q,ab^{-1}q}{q}{k}} q^{-k^2}\left(\frac{1}{a^3b}\right)^k}$ &

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

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

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.