Stieltjes-Wigert

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \StieltjesWigert{n}@{x}{q}=\frac{1}{\qPochhammer{q}{q}{n}}\,\qHyperrphis{1}{1}@@{q^{-n}}{0}{q}{-q^{n+1}x} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{0}^{\infty}\frac{\StieltjesWigert{m}@{x}{q}\StieltjesWigert{n}@{x}{q}}{\qPochhammer{-x,-qx^{-1}}{q}{\infty}}\,dx =-\frac{\ln@@{q}}{q^n}\frac{\qPochhammer{q}{q}{\infty}}{\qPochhammer{q}{q}{n}}\,\Kronecker{m}{n} }}

Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -q^{2n+1}x\StieltjesWigert{n}@{x}{q} {}=(1-q^{n+1})\StieltjesWigert{n+1}@{x}{q}-[1+q-q^{n+1}]\StieltjesWigert{n}@{x}{q}+q\StieltjesWigert{n-1}@{x}{q} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicStieltjesWigert{n}@@{x}{q}=\monicStieltjesWigert{n+1}@@{x}{q}+q^{-2n-1}\left[1+q-q^{n+1}\right]\monicStieltjesWigert{n}@@{x}{q} {}+q^{-4n+1}(1-q^n)\monicStieltjesWigert{n-1}@@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \StieltjesWigert{n}@{x}{q}=\frac{(-1)^nq^{n^2}}{\qPochhammer{q}{q}{n}}\monicStieltjesWigert{n}@@{x}{q} }}

q-Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -x(1-q^n)y(x)=xy(qx)-(x+1)y(x)+y(q^{-1}x) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\StieltjesWigert{n}@{x}{q}}}

Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \StieltjesWigert{n}@{x}{q}-\StieltjesWigert{n}@{qx}{q}=-qx\StieltjesWigert{n-1}@{q^2x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\StieltjesWigert{n}@{x}{q}=-\frac{q}{1-q}\StieltjesWigert{n-1}@{q^2x}{q} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \StieltjesWigert{n}@{x}{q}-x\StieltjesWigert{n}@{qx}{q}=(1-q^{n+1})\StieltjesWigert{n+1}@{q^{-1}x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\left[w(x;q)\StieltjesWigert{n}@{x}{q}\right]=\frac{1-q^{n+1}}{1-q}q^{-1}w(q^{-1}x;q)\StieltjesWigert{n+1}@{q^{-1}x}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;q)=\frac{1}{\qPochhammer{-x,-qx^{-1}}{q}{\infty}}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x)=\frac{\gamma}{\sqrt{\cpi}}\exp@{-\gamma^2\ln^2@@{x}},\quad x>0,\quad\textrm{with}\quad \gamma^2=-\frac{1}{2\ln@@{q}} =\frac\gamma{\sqrt\cpi} x^{-\frac12}\exp@{-\gamma^2\ln^2@@{x}}, x>0 {\rm with} \gamma^2}}

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;q)\StieltjesWigert{n}@{x}{q}=\frac{q^n(1-q)^n}{\qPochhammer{q}{q}{n}}\left(\left(\qderiv{q}\right)^n w\right)(q^nx;q) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;q)=\frac{1}{\qPochhammer{-x,-qx^{-1}}{q}{\infty}}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x)=\frac{\gamma}{\sqrt{\cpi}}\exp@{-\gamma^2\ln^2@@{x}},\quad x>0,\quad\textrm{with}\quad \gamma^2=-\frac{1}{2\ln@@{q}} =\frac\gamma{\sqrt\cpi} x^{-\frac12}\exp@{-\gamma^2\ln^2@@{x}}, x>0 {\rm with} \gamma^2}}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{0}{1}@@{-}{0}{q}{-qxt}= \sum_{n=0}^{\infty}\StieltjesWigert{n}@{x}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{t}{q}{\infty}\cdot\qHyperrphis{0}{2}@@{-}{0,t}{q}{-qxt}= \sum_{n=0}^{\infty}(-1)^nq^{\binomial{n}{2}}\StieltjesWigert{n}@{x}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{\gamma t}{q}{\infty}}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{1}{2}@@{\gamma}{0,\gamma t}{q}{-qxt} =\sum_{n=0}^{\infty}\qPochhammer{\gamma}{q}{n}\StieltjesWigert{n}@{x}{q}t^n }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary

Limit relations

q-Laguerre polynomial to Stieltjes-Wigert polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\alpha\rightarrow\infty}\qLaguerre[\alpha]{n}@{xq^{-\alpha}}{q}=\StieltjesWigert{n}@{x}{q} }}

q-Bessel polynomial to Stieltjes-Wigert polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{a\rightarrow\infty}\qBesselPoly{n}@{a^{-1}x}{a}{q}=\qPochhammer{q}{q}{n}\StieltjesWigert{n}@{x}{q} }}

q-Charlier polynomial to Stieltjes-Wigert polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{a\rightarrow\infty}\qCharlier{n}@{ax}{a}{q}=\qPochhammer{q}{q}{n}\StieltjesWigert{n}@{x}{q} }}