Discrete q-Hermite I

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Discrete q-Hermite I

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Basic hypergeometric representation

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

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}\qPochhammer{qx,-qx}{q}{\infty}\discrqHermiteI{m}@{x}{q}\discrqHermiteI{n}@{x}{q}\,d_{q}x{}=(1-q)\qPochhammer{q}{q}{n}\qPochhammer{q,-1,-q}{q}{\infty}q^{\binomial{n}{2}}\,\Kronecker{m}{n}}}}$

Recurrence relation

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

Monic recurrence relation

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

q-Difference equation

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=\discrqHermiteI{n}@{x}{q}}}}$

Forward shift operator

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

Backward shift operator

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\qPochhammer{qx,-qx}{q}{\infty}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;q)\discrqHermiteI{n}@{x}{q}=(q-1)^{n}q^{\frac{1}{2}n(n-3)}\left(\mathcal{D}_{q^{-1}}\right)^{n}\left[w(x;q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\qPochhammer{qx,-qx}{q}{\infty}}}}$

Generating function

${\displaystyle{\displaystyle{\displaystyle\frac{\qPochhammer{t^{2}}{q^{2}}{\infty}}{\qPochhammer{xt}{q}{\infty}}=\sum_{n=0}^{\infty}\frac{\discrqHermiteI{n}@{x}{q}}{\qPochhammer{q}{q}{n}}t^{n}}}}$

Limit relations

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

${\displaystyle{\displaystyle{\displaystyle\AlSalamCarlitzI{-1}{n}@{x}{q}=\discrqHermiteI{n}@{x}{q}}}}$

Discrete q-Hermite I polynomial to Hermite polynomial

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

Remark

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

<< Stieltjes-Wigert

Discrete q-Hermite I

Discrete q-Hermite II >>