# Formula:KLS:14.10:55

${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;\beta|q)C_{n}\!\left(x% ;\beta\,|\,q\right){}=\left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}\frac% {\left(\beta;q\right)_{n}}{\left(q,\beta^{2}q^{n};q\right)_{n}}\left(D_{q}% \right)^{n}\left[{\tilde{w}}(x;\beta q^{n}|q)\right]}}}$

## 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

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