# Formula:KLS:14.10:53

${\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;\beta|% q)C_{n}\!\left(x;\beta\,|\,q\right)\right]{}=q^{-\frac{1}{2}(n+1)}\frac{(1-q^{% n+1})(1-\beta^{2}q^{n-1})}{(1-\beta q^{-1})}({\mathrm{e}^{\mathrm{i}\theta}}-{% \mathrm{e}^{-\mathrm{i}\theta}}){}{\tilde{w}}(x;\beta q^{-1}|q)C_{n+1}\!\left(% x;\beta q^{-1}\,|\,q\right)}}}$

## Substitution(s)

${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;\beta|q):=\frac{w(x;% \beta|q)}{\sqrt{1-x^{2}}}}}}$ &

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

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

## Proof

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