# Formula:KLS:14.18:20

$\displaystyle {\displaystyle \delta_q\left[{\tilde w}(x;a|q)\ctsbigqHermite{n}@{x}{a}{q}\right] {}=q^{-\frac{1}{2}(n+1)}(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {} {\tilde w}(x;aq^{-\frac{1}{2}}|q)\ctsbigqHermite{n+1}@{x}{aq^{-\frac{1}{2}}}{q} }$

## Substitution(s)

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

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

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

## 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
$\displaystyle {\displaystyle H_{n}}$  : continuous big $\displaystyle {\displaystyle q}$ -Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsbigqHermite
$\displaystyle {\displaystyle \mathrm{e}}$  : 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
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
$\displaystyle {\displaystyle \Pi}$  : product : http://drmf.wmflabs.org/wiki/Definition:prod
$\displaystyle {\displaystyle \mathrm{cos}}$  : cosine function : http://dlmf.nist.gov/4.14#E2