Discrete q-Hermite II

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Discrete q-Hermite II

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Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\discrqHermiteII{n}@{x}{q}=\iunit^{-n}\AlSalamCarlitzII{-1}{n}@{\iunit x}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\discrqHermiteII{n}@{x}{q}=\iunit^{-n}q^{-\binomial{n}{2}}\,{{}_{2}\phi_{0}}\!\left({q^{-n},\iunit x\atop-};q,-q^{n}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\discrqHermiteII{n}@{x}{q}=x^{n}{{}_{2}\phi_{1}}\!\left({q^{-n},q^{-n+1}\atop 0};q^{2},-\frac{q^{2}}{x^{2}}\right)}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{k=-\infty}^{\infty}\left[\discrqHermiteII{m}@{cq^{k}}{q}\discrqHermiteII{n}@{cq^{k}}{q}+\discrqHermiteII{m}@{-cq^{k}}{q}\discrqHermiteII{n}@{-cq^{k}}{q}\right]w(cq^{k};q)q^{k}{}=2\frac{\qPochhammer{q^{2},-c^{2}q,-c^{-2}q}{q^{2}}{\infty}}{\qPochhammer{q,-c^{2},-c^{-2}q^{2}}{q^{2}}{\infty}}\frac{\qPochhammer{q}{q}{n}}{q^{n^{2}}}\,\Kronecker{m}{n}}}}$

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\qPochhammer{\iunit x,-\iunit x}{q}{\infty}}=\frac{1}{\qPochhammer{-x^{2}}{q^{2}}{\infty}}}}}$

${\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}\frac{\discrqHermiteII{m}@{x}{q}\discrqHermiteII{n}@{x}{q}}{\qPochhammer{-x^{2}}{q^{2}}{\infty}}\,d_{q}x=\frac{\qPochhammer{q^{2},-q,-q}{q^{2}}{\infty}}{\qPochhammer{q^{3},-q^{2},-q^{2}}{q^{2}}{\infty}}\frac{\qPochhammer{q}{q}{n}}{q^{n^{2}}}\,\Kronecker{m}{n}}}}$

Recurrence relation

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

Monic recurrence relation

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

q-Difference equation

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

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\discrqHermiteII{n}@{x}{q}-\discrqHermiteII{n}@{qx}{q}=q^{-n+1}(1-q^{n})x\discrqHermiteII{n-1}@{qx}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\qderiv{q}\discrqHermiteII{n}@{x}{q}=\frac{q^{-n+1}(1-q^{n})}{1-q}\discrqHermiteII{n-1}@{qx}{q}}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle\discrqHermiteII{n}@{x}{q}-(1+x^{2})\discrqHermiteII{n}@{qx}{q}=-q^{n}x\discrqHermiteII{n+1}@{x}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\qderiv{q}\left[w(x;q)\discrqHermiteII{n}@{x}{q}\right]=-\frac{q^{n}}{1-q}w(x;q)\discrqHermiteII{n+1}@{x}{q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\qPochhammer{\iunit x,-\iunit x}{q}{\infty}}=\frac{1}{\qPochhammer{-x^{2}}{q^{2}}{\infty}}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;q)\discrqHermiteII{n}@{x}{q}=(q-1)^{n}q^{-\binomial{n}{2}}\left(\qderiv{q}\right)^{n}\left[w(x;q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\qPochhammer{\iunit x,-\iunit x}{q}{\infty}}=\frac{1}{\qPochhammer{-x^{2}}{q^{2}}{\infty}}}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\frac{\qPochhammer{-xt}{q}{\infty}}{\qPochhammer{-t^{2}}{q^{2}}{\infty}}=\sum_{n=0}^{\infty}\frac{q^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}\discrqHermiteII{n}@{x}{q}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\qPochhammer{-\iunit t}{q}{\infty}\cdot{{}_{1}\phi_{1}}\!\left({\iunit x\atop-\iunit t};q,\iunit t\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n(n-1)}}{\qPochhammer{q}{q}{n}}\discrqHermiteII{n}@{x}{q}t^{n}}}}$

Limit relations

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

${\displaystyle{\displaystyle{\displaystyle\iunit^{-n}\AlSalamCarlitzII{-1}{n}@{\iunit x}{q}=\discrqHermiteII{n}@{x}{q}}}}$

Discrete q-Hermite II polynomial to Hermite polynomial

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

Remark

${\displaystyle{\displaystyle{\displaystyle\discrqHermiteII{n}@{x}{q^{-1}}=\iunit^{-n}\discrqHermiteI{n}@{\iunit x}{q}}}}$

Koornwinder Addendum: Discrete q-Hermite II

Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\discrqHermiteII{n}@{x}{q}=x^{n}{{}_{2}\phi_{1}}\!\left({q^{-n},q^{-n+1}\atop 0};q^{2},-q^{2}x^{-2}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\littleqJacobi{n}@{0}{a}{b}{q}=1}}}$
${\displaystyle{\displaystyle{\displaystyle\littleqJacobi{n}@{q^{-1}b^{-1}}{a}{b}{q}=(-qb)^{-n}q^{-\frac{1}{2}n(n-1)}\frac{\qPochhammer{qb}{q}{n}}{\qPochhammer{qa}{q}{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle\littleqJacobi{n}@{1}{a}{b}{q}=(-a)^{n}q^{\frac{1}{2}n(n+1)}\frac{\qPochhammer{qb}{q}{n}}{\qPochhammer{qa}{q}{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle\ctsqJacobi{\alpha}{\beta}{n}@{-x}{q}=(-1)^{n}q^{\frac{1}{2}(\alpha-\beta)n}\ctsqJacobi{\beta}{\alpha}{n}@{x}{q}}}}$

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