Discrete q-Hermite II

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Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)={\mathrm{i}^{-n}}V^{(-1)}_{n}\!\left(\mathrm{i}x;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)={\mathrm{i}^{-n}}q^{-\binomial{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},\mathrm{i}x}{-}{q}{-q^{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)=x^{n}\qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{0}{q^{2}}{-\frac{q^{2}}{x^{2}}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{k=-\infty}^{\infty}\left[\tilde{h}_{m}\!\left(cq^{k};q\right)\tilde{h}_{n}\!\left(cq^{k};q\right)+\tilde{h}_{m}\!\left(-cq^{k};q\right)\tilde{h}_{n}\!\left(-cq^{k};q\right)\right]w(cq^{k};q)q^{k}{}=2\frac{\left(q^{2},-c^{2}q,-c^{-2}q;q^{2}\right)_{\infty}}{\left(q,-c^{2},-c^{-2}q^{2};q^{2}\right)_{\infty}}\frac{\left(q;q\right)_{n}}{q^{n^{2}}}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle c>0}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(\mathrm{i}x,-\mathrm{i}x;q\right)_{\infty}}=\frac{1}{\left(-x^{2};q^{2}\right)_{\infty}}}}}$

${\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}\frac{\tilde{h}_{m}\!\left(x;q\right)\tilde{h}_{n}\!\left(x;q\right)}{\left(-x^{2};q^{2}\right)_{\infty}}\,d_{q}x=\frac{\left(q^{2},-q,-q;q^{2}\right)_{\infty}}{\left(q^{3},-q^{2},-q^{2};q^{2}\right)_{\infty}}\frac{\left(q;q\right)_{n}}{q^{n^{2}}}\,\delta_{m,n}}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle x\tilde{h}_{n}\!\left(x;q\right)=\tilde{h}_{n+1}\!\left(x;q\right)+q^{-2n+1}(1-q^{n})\tilde{h}_{n-1}\!\left(x;q\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{\tilde{h}}_{n}}\!\left(x\right)={\widehat{\tilde{h}}_{n+1}}\!\left(x\right)+q^{-2n+1}(1-q^{n}){\widehat{\tilde{h}}_{n-1}}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)={\widehat{\tilde{h}}_{n}}\!\left(x\right)}}}$

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle-(1-q^{n})x^{2}\tilde{h}_{n}\!\left(x;q\right){}=(1+x^{2})\tilde{h}_{n}\!\left(qx;q\right)-(1+x^{2}+q)\tilde{h}_{n}\!\left(x;q\right)+q\tilde{h}_{n}\!\left(q^{-1}x;q\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)-\tilde{h}_{n}\!\left(qx;q\right)=q^{-n+1}(1-q^{n})x\tilde{h}_{n-1}\!\left(qx;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}\tilde{h}_{n}\!\left(x;q\right)=\frac{q^{-n+1}(1-q^{n})}{1-q}\tilde{h}_{n-1}\!\left(qx;q\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)-(1+x^{2})\tilde{h}_{n}\!\left(qx;q\right)=-q^{n}x\tilde{h}_{n+1}\!\left(x;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}\left[w(x;q)\tilde{h}_{n}\!\left(x;q\right)\right]=-\frac{q^{n}}{1-q}w(x;q)\tilde{h}_{n+1}\!\left(x;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(\mathrm{i}x,-\mathrm{i}x;q\right)_{\infty}}=\frac{1}{\left(-x^{2};q^{2}\right)_{\infty}}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;q)\tilde{h}_{n}\!\left(x;q\right)=(q-1)^{n}q^{-\binomial{n}{2}}\left(\mathcal{D}_{q}\right)^{n}\left[w(x;q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(\mathrm{i}x,-\mathrm{i}x;q\right)_{\infty}}=\frac{1}{\left(-x^{2};q^{2}\right)_{\infty}}}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\frac{\left(-xt;q\right)_{\infty}}{\left(-t^{2};q^{2}\right)_{\infty}}=\sum_{n=0}^{\infty}\frac{q^{\binomial{n}{2}}}{\left(q;q\right)_{n}}\tilde{h}_{n}\!\left(x;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(-\mathrm{i}t;q\right)_{\infty}\cdot\qHyperrphis{1}{1}@@{\mathrm{i}x}{-\mathrm{i}t}{q}{\mathrm{i}t}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n(n-1)}}{\left(q;q\right)_{n}}\tilde{h}_{n}\!\left(x;q\right)t^{n}}}}$

Limit relations

Al-Salam-Carlitz II polynomial to Discrete q-Hermite II polynomial

${\displaystyle{\displaystyle{\displaystyle{\mathrm{i}^{-n}}V^{(-1)}_{n}\!\left(\mathrm{i}x;q\right)=\tilde{h}_{n}\!\left(x;q\right)}}}$

Discrete q-Hermite II polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{\tilde{h}_{n}\!\left(x\sqrt{1-q^{2}};q\right)}{(1-q^{2})^{\frac{n}{2}}}=\frac{H_{n}\left(x\right)}{2^{n}}}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q^{-1}\right)={\mathrm{i}^{-n}}h_{n}\!\left(\mathrm{i}x;q\right)}}}$

Koornwinder Addendum: Discrete q-Hermite II

Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\tilde{h}_{n}\!\left(x;q\right)=x^{n}\qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{0}{q^{2}}{-q^{2}x^{-2}}}}}$
${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(0;a,b;q\right)=1}}}$
${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(q^{-1}b^{-1};a,b;q\right)=(-qb)^{-n}q^{-\frac{1}{2}n(n-1)}\frac{\left(qb;q\right)_{n}}{\left(qa;q\right)_{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(1;a,b;q\right)=(-a)^{n}q^{\frac{1}{2}n(n+1)}\frac{\left(qb;q\right)_{n}}{\left(qa;q\right)_{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(-x|q\right)=(-1)^{n}q^{\frac{1}{2}(\alpha-\beta)n}P^{(\beta,\alpha)}_{n}\!\left(x|q\right)}}}$

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