Discrete q-Hermite II

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Discrete q-Hermite II

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \discrqHermiteII{n}@{x}{q}=\iunit^{-n}\AlSalamCarlitzII{-1}{n}@{\iunit x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \discrqHermiteII{n}@{x}{q}=\iunit^{-n}q^{-\binomial{n}{2}}\,\qHyperrphis{2}{0}@@{q^{-n},\iunit x}{-}{q}{-q^n} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \discrqHermiteII{n}@{x}{q}=x^n \qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{0}{q^2}{-\frac{q^2}{x^2}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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^2q,-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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle c>0}}


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{\iunit x,-\iunit x}{q}{\infty}}=\frac{1}{\qPochhammer{-x^2}{q^2}{\infty}}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-\infty}^{\infty}\frac{\discrqHermiteII{m}@{x}{q}\discrqHermiteII{n}@{x}{q}}{\qPochhammer{-x^2}{q^2}{\infty}}\,d_qx =\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

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

q-Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \discrqHermiteII{n}@{x}{q}-(1+x^2)\discrqHermiteII{n}@{qx}{q}=-q^nx\discrqHermiteII{n+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)\discrqHermiteII{n}@{x}{q}\right] =-\frac{q^n}{1-q}w(x;q)\discrqHermiteII{n+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{\iunit x,-\iunit x}{q}{\infty}}=\frac{1}{\qPochhammer{-x^2}{q^2}{\infty}}}}


Rodrigues-type formula

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

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{\iunit x,-\iunit x}{q}{\infty}}=\frac{1}{\qPochhammer{-x^2}{q^2}{\infty}}}}


Generating functions

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

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

Discrete q-Hermite II polynomial to Hermite polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

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

Koornwinder Addendum: Discrete q-Hermite II

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \discrqHermiteII{n}@{x}{q}=x^n \qHyperrphis{2}{1}@@{q^{-n},q^{-n+1}}{0}{q^2}{-q^2 x^{-2}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqJacobi{n}@{0}{a}{b}{q}=1 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqJacobi{n}@{q^{-1}b^{-1}}{a}{b}{q}=(-qb)^{-n} q^{-\frac12 n(n-1)} \frac{\qPochhammer{qb}{q}{n}}{\qPochhammer{qa}{q}{n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqJacobi{n}@{1}{a}{b}{q}=(-a)^n q^{\frac12 n(n+1)} \frac{\qPochhammer{qb}{q}{n}}{\qPochhammer{qa}{q}{n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqJacobi{\alpha}{\beta}{n}@{-x }{ q}=(-1)^n q^{\frac12(\alpha-\beta)n} \ctsqJacobi{\beta}{\alpha}{n}@{x }{ q} }}

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