# Formula:KLS:14.10:38

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}C_{m}\!\left(x;\beta\,|\,q\right)C_{n}\!\left(x;\beta\,|\,q% \right)\,dx{}=\frac{\left(\beta,\beta q;q\right)_{\infty}}{\left(\beta^{2},q;q% \right)_{\infty}}\frac{\left(\beta^{2};q\right)_{n}}{\left(q;q\right)_{n}}% \frac{(1-\beta)}{(1-\beta q^{n})}\,\delta_{m,n}}}}$

## Constraint(s)

${\displaystyle{\displaystyle{\displaystyle|\beta|<1}}}$

## Substitution(s)

$\displaystyle {\displaystyle w(x):=w(x;\beta|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{\beta^{\frac{1}{2}}\expe^{\iunit\theta},\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta} -\beta^{\frac{1}{2}}\expe^{\iunit\theta},-\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{\beta\expe^{2\iunit\theta}}{q}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,\beta^{\frac{1}{2}})h(x,\beta^{\frac{1}{2}}q^{\frac{1}{2}}) h(x,-\beta^{\frac{1}{2}})h(x,-\beta^{\frac{1}{2}}q^{\frac{1}{2}})}}$ &
${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

## Proof

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