Continuous big q-Hermite

Basic hypergeometric representation

$\displaystyle {\displaystyle \ctsbigqHermite{n}@{x}{a}{q}=a^{-n}\,\qHyperrphis{3}{2}@@{q^{-n},a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{0,0}{q}{q} }$

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

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

Orthogonality relation(s)

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

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

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

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

$\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\ctsbigqHermite{m}@{x}{a}{q}\ctsbigqHermite{n}@{x}{a}{q}\,dx {}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1

Substitution(s): $\displaystyle {\displaystyle w_k=\frac{\qPochhammer{a^{-2}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} \frac{(1-a^2q^{2k})\qPochhammer{a^2}{q}{k}}{(1-a^2)\qPochhammer{q}{q}{k}} q^{-\frac{3}{2}k^2-\frac{1}{2}k}\left(-\frac{1}{a^4}\right)^k}$ &

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

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

Recurrence relation

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

Monic recurrence relation

$\displaystyle {\displaystyle x\monicctsbigqHermite{n}@@{x}{a}{q}=\monicctsbigqHermite{n+1}@@{x}{a}{q}+\frac{1}{2}aq^n\monicctsbigqHermite{n}@@{x}{a}{q}+\frac{1}{4}(1-q^n)\monicctsbigqHermite{n-1}@@{x}{a}{q} }$
$\displaystyle {\displaystyle \ctsbigqHermite{n}@{x}{a}{q}=2^n\monicctsbigqHermite{n}@@{x}{a}{q} }$

q-Difference equations

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

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

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

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

$\displaystyle {\displaystyle \LegendrePoly{n}@{z}:=a^{-n}\,\qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{0,0}{q}{q} }$
$\displaystyle {\displaystyle q^{-n}(1-q^n)\LegendrePoly{n}@{z}=A(z)\LegendrePoly{n}@{qz}-\left[A(z)+A(z^{-1})\right]\LegendrePoly{n}@{z} {}+A(z^{-1})\LegendrePoly{n}@{q^{-1}z} }$

Substitution(s): $\displaystyle {\displaystyle A(z)=\frac{(1-az)}{(1-z^2)(1-qz^2)}}$

Forward shift operator

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

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

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

Backward shift operator

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

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

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

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

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

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

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

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

Rodrigues-type formula

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

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

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

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

Generating functions

$\displaystyle {\displaystyle \frac{\qPochhammer{at}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}} =\sum_{n=0}^{\infty}\frac{\ctsbigqHermite{n}@{x}{a}{q}}{\qPochhammer{q}{q}{n}}t^n }$

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

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

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

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

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

Limit relations

Al-Salam-Chihara polynomial to Continuous big q-Hermite polynomial

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

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

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

Continuous big q-Hermite polynomial to Hermite polynomial

$\displaystyle {\displaystyle \lim_{q\rightarrow 1} \frac{\ctsbigqHermite{n}@{x\sqrt{\frac{1}{2}(1-q)}}{0}{q}} {\left(\frac{1-q}{2}\right)^{\frac{n}{2}}}=\Hermite{n}@{x} }$
$\displaystyle {\displaystyle \lim_{q\rightarrow 1} \frac{\ctsbigqHermite{n}@{x\sqrt{\frac{1}{2}(1-q)}}{a\sqrt{2(1-q)}}{q}} {\left(\frac{1-q}{2}\right)^{\frac{n}{2}}}=\Hermite{n}@{x-a} }$