Discrete q-Hermite I

<< Stieltjes-Wigert

Discrete q-Hermite I

Discrete q-Hermite II >>

Basic hypergeometric representation

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

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}\left(qx,-qx;q\right)_{\infty}h_{m}\!\left(x;q\right)h_{n}\!\left(x;q\right)\,d_{q}x{}=(1-q)\left(q;q\right)_{n}\left(q,-1,-q;q\right)_{\infty}q^{\binomial{n}{2}}\,\delta_{m,n}}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle xh_{n}\!\left(x;q\right)=h_{n+1}\!\left(x;q\right)+q^{n-1}(1-q^{n})h_{n-1}\!\left(x;q\right)}}}$

Monic recurrence relation

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

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): ${\displaystyle{\displaystyle{\displaystyle y(x)=h_{n}\!\left(x;q\right)}}}$

Forward shift operator

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

Backward shift operator

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

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

Rodrigues-type formula

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

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

Generating function

${\displaystyle{\displaystyle{\displaystyle\frac{\left(t^{2};q^{2}\right)_{\infty}}{\left(xt;q\right)_{\infty}}=\sum_{n=0}^{\infty}\frac{h_{n}\!\left(x;q\right)}{\left(q;q\right)_{n}}t^{n}}}}$

Limit relations

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

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

Discrete q-Hermite I polynomial to Hermite polynomial

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

Remark

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

<< Stieltjes-Wigert

Discrete q-Hermite I

Discrete q-Hermite II >>