Stieltjes-Wigert

<< Continuous q-Hermite

Stieltjes-Wigert

Discrete q-Hermite I >>

Basic hypergeometric representation

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

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}\frac{S_{m}\!\left(x;q\right)S_{n}\!\left(x;q\right)}{\left(-x,-qx^{-1};q\right)_{\infty}}\,dx=-\frac{\ln q}{q^{n}}\frac{\left(q;q\right)_{\infty}}{\left(q;q\right)_{n}}\,\delta_{m,n}}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-q^{2n+1}xS_{n}\!\left(x;q\right){}=(1-q^{n+1})S_{n+1}\!\left(x;q\right)-[1+q-q^{n+1}]S_{n}\!\left(x;q\right)+qS_{n-1}\!\left(x;q\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{S}}_{n}\!\left(x\right)={\widehat{S}}_{n+1}\!\left(x\right)+q^{-2n-1}\left[1+q-q^{n+1}\right]{\widehat{S}}_{n}\!\left(x\right){}+q^{-4n+1}(1-q^{n}){\widehat{S}}_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle S_{n}\!\left(x;q\right)=\frac{(-1)^{n}q^{n^{2}}}{\left(q;q\right)_{n}}{\widehat{S}}_{n}\!\left(x\right)}}}$

q-Difference equation

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

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

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle S_{n}\!\left(x;q\right)-S_{n}\!\left(qx;q\right)=-qxS_{n-1}\!\left(q^{2}x;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}S_{n}\!\left(x;q\right)=-\frac{q}{1-q}S_{n-1}\!\left(q^{2}x;q\right)}}}$

Backward shift operator

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(-x,-qx^{-1};q\right)_{\infty}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x)=\frac{\gamma}{\sqrt{\pi}}\exp\left(-\gamma^{2}{\ln^{2}}x\right),\quad x>0,\quad\textrm{with}\quad\gamma^{2}=-\frac{1}{2\ln q}=\frac{\gamma}{\sqrt{\pi}}x^{-\frac{1}{2}}\exp\left(-\gamma^{2}{\ln^{2}}x\right),x>0{\rm with}\gamma^{2}}}}$

Rodrigues-type formula

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(-x,-qx^{-1};q\right)_{\infty}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x)=\frac{\gamma}{\sqrt{\pi}}\exp\left(-\gamma^{2}{\ln^{2}}x\right),\quad x>0,\quad\textrm{with}\quad\gamma^{2}=-\frac{1}{2\ln q}=\frac{\gamma}{\sqrt{\pi}}x^{-\frac{1}{2}}\exp\left(-\gamma^{2}{\ln^{2}}x\right),x>0{\rm with}\gamma^{2}}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(t;q\right)_{\infty}}\,\qHyperrphis{0}{1}@@{-}{0}{q}{-qxt}=\sum_{n=0}^{\infty}S_{n}\!\left(x;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot\qHyperrphis{0}{2}@@{-}{0,t}{q}{-qxt}=\sum_{n=0}^{\infty}(-1)^{n}q^{\binomial{n}{2}}S_{n}\!\left(x;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\left(\gamma t;q\right)_{\infty}}{\left(t;q\right)_{\infty}}\,\qHyperrphis{1}{2}@@{\gamma}{0,\gamma t}{q}{-qxt}=\sum_{n=0}^{\infty}\left(\gamma;q\right)_{n}S_{n}\!\left(x;q\right)t^{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

Limit relations

q-Laguerre polynomial to Stieltjes-Wigert polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}L^{(\alpha)}_{n}\!\left(xq^{-\alpha};q\right)=S_{n}\!\left(x;q\right)}}}$

q-Bessel polynomial to Stieltjes-Wigert polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow\infty}y_{n}\!\left(a^{-1}x;a;q\right)=\left(q;q\right)_{n}S_{n}\!\left(x;q\right)}}}$

q-Charlier polynomial to Stieltjes-Wigert polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow\infty}C_{n}\!\left(ax;a;q\right)=\left(q;q\right)_{n}S_{n}\!\left(x;q\right)}}}$

Stieltjes-Wigert polynomial to Hermite polynomial

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

Koornwinder Addendum: Stieltjes-Wigert

An alternative weight function

${\displaystyle{\displaystyle{\displaystyle w(x)=-\frac{1}{2\ln q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;q)=\frac{1}{\left(-x,-qx^{-1};q\right)_{\infty}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x)=\frac{\gamma}{\sqrt{\pi}}\exp\left(-\gamma^{2}{\ln^{2}}x\right),\quad x>0,\quad\textrm{with}\quad\gamma^{2}=-\frac{1}{2\ln q}=\frac{\gamma}{\sqrt{\pi}}x^{-\frac{1}{2}}\exp\left(-\gamma^{2}{\ln^{2}}x\right),x>0{\rm with}\gamma^{2}}}}$

<< Continuous q-Hermite

Stieltjes-Wigert

Discrete q-Hermite I >>