# Continuous q-Hermite

## Basic hypergeometric representation

$\displaystyle {\displaystyle \ctsqHermite{n}@{x}{q}=\expe^{\iunit n\theta}\,\qHyperrphis{2}{0}@@{q^{-n},0}{-}{q}{q^n\expe^{-2\iunit\theta}} }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@@{\theta}}$

## Orthogonality relation(s)

$\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x|q)}{\sqrt{1-x^2}}\ctsqHermite{m}@{x}{q}\ctsqHermite{n}@{x}{q}\,dx= \frac{\,\Kronecker{m}{n}}{\qPochhammer{q^{n+1}}{q}{\infty}} }$

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

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

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

## Recurrence relation

$\displaystyle {\displaystyle 2x\ctsqHermite{n}@{x}{q}=\ctsqHermite{n+1}@{x}{q}+(1-q^n)\ctsqHermite{n-1}@{x}{q} }$

## Monic recurrence relation

$\displaystyle {\displaystyle x\monicctsqHermite{n}@@{x}{q}=\monicctsqHermite{n+1}@@{x}{q}+\frac{1}{4}(1-q^n)\monicctsqHermite{n-1}@@{x}{q} }$
$\displaystyle {\displaystyle \ctsqHermite{n}@{x}{q}=2^n\monicctsqHermite{n}@@{x}{q} }$

## q-Difference equation

$\displaystyle {\displaystyle (1-q)^2D_q\left[{\tilde w}(x|q)D_qy(x)\right]+4q^{-n+1}(1-q^n){\tilde w}(x|q)y(x)=0 }$

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

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

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

## Forward shift operator

$\displaystyle {\displaystyle \delta_q\ctsqHermite{n}@{x}{q}=-q^{-\frac{1}{2}n}(1-q^n)(\expe^{\iunit\theta}-\expe^{-\iunit\theta})\ctsqHermite{n-1}@{x}{q} }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@@{\theta}}$

$\displaystyle {\displaystyle D_q\ctsqHermite{n}@{x}{q}=\frac{2q^{-\frac{1}{2}(n-1)}(1-q^n)}{1-q}\ctsqHermite{n-1}@{x}{q} }$

## Backward shift operator

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

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

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

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

$\displaystyle {\displaystyle D_q\left[{\tilde w}(x|q)\ctsqHermite{n}@{x}{q}\right]= -\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde w}(x|q)\ctsqHermite{n+1}@{x}{q} }$

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

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

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

## Rodrigues-type formula

$\displaystyle {\displaystyle {\tilde w}(x|q)\ctsqHermite{n}@{x}{q}=\left(\frac{q-1}{2}\right)^nq^{\frac{1}{4}n(n-1)} \left(D_q\right)^n\left[{\tilde w}(x|q)\right] }$

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

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

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

## Generating functions

$\displaystyle {\displaystyle \frac{1}{\left|\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}\right|^2}= \frac{1}{\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}}= \sum_{n=0}^{\infty}\frac{\ctsqHermite{n}@{x}{q}}{\qPochhammer{q}{q}{n}}t^n }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@@{\theta}}$

$\displaystyle {\displaystyle \qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}\cdot\qHyperrphis{1}{1}@@{0}{\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}\ctsqHermite{n}@{x}{q}t^n }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@@{\theta}}$

$\displaystyle {\displaystyle \frac{\qPochhammer{\gamma\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{\gamma,0}{\gamma\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{q}{q}{n}}\ctsqHermite{n}@{x}{q}t^n }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@@{\theta}}$ &
$\displaystyle {\displaystyle \gamma}$ arbitrary

## Limit relations

### Continuous big q-Hermite polynomial to Continuous q-Hermite polynomial

$\displaystyle {\displaystyle \ctsbigqHermite{n}@{x}{0}{q}=\ctsqHermite{n}@{x}{q} }$

### Continuous q-Laguerre polynomial to Continuous q-Hermite polynomial

$\displaystyle {\displaystyle \lim_{\alpha\rightarrow\infty} \frac{\ctsqLaguerre{\alpha}{n}@{x}{q}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}} =\frac{\ctsqHermite{n}@{x}{q}}{\qPochhammer{q}{q}{n}} }$

### Continuous q-Hermite polynomial to Hermite polynomial

$\displaystyle {\displaystyle \lim_{q\rightarrow 1} \frac{\ctsqHermite{n}@{x\sqrt{\frac{1}{2}(1-q)}}{q}} {\left(\frac{1-q}{2}\right)^{\frac{n}{2}}}=\Hermite{n}@{x} }$

## Remark

$\displaystyle {\displaystyle \ctsqHermite{n}@{x}{q}=\sum_{k=0}^n\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q}{q}{k}\qPochhammer{q}{q}{n-k}}\expe^{\iunit(n-2k)\theta} }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@@{\theta}}$