Discrete q-Hermite I
Discrete q-Hermite I
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \discrqHermiteI{n}@{x}{q}=\AlSalamCarlitzI{-1}{n}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \discrqHermiteI{n}@{x}{q}=q^{\binomial{n}{2}}\,\qHyperrphis{2}{1}@@{q^{-n},x^{-1}}{0}{q}{-qx} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-1}^1\qPochhammer{qx,-qx}{q}{\infty}\discrqHermiteI{m}@{x}{q}\discrqHermiteI{n}@{x}{q}\,d_qx {}=(1-q)\qPochhammer{q}{q}{n}\qPochhammer{q,-1,-q}{q}{\infty}q^{\binomial{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\discrqHermiteI{n}@{x}{q}=\discrqHermiteI{n+1}@{x}{q}+q^{n-1}(1-q^n)\discrqHermiteI{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\monicdiscrqHermiteI{n}@@{x}{q}=\monicdiscrqHermiteI{n+1}@@{x}{q}+q^{n-1}(1-q^n)\monicdiscrqHermiteI{n-1}@@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \discrqHermiteI{n}@{x}{q}=\monicdiscrqHermiteI{n}@@{x}{q} }}
q-Difference equation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -q^{-n+1}x^2y(x)=y(qx)-(1+q)y(x)+q(1-x^2)y(q^{-1}x) }}
Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \discrqHermiteI{n}@{x}{q}-\discrqHermiteI{n}@{qx}{q}=(1-q^n)x\discrqHermiteI{n-1}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\discrqHermiteI{n}@{x}{q}=\frac{1-q^n}{1-q}\discrqHermiteI{n-1}@{x}{q} }}
Backward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \discrqHermiteI{n}@{x}{q}-(1-x^2)\discrqHermiteI{n}@{q^{-1}x}{q}=q^{-n}x\discrqHermiteI{n+1}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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} }}
Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;q)\discrqHermiteI{n}@{x}{q}=(q-1)^nq^{\frac{1}{2}n(n-3)} \left(\mathcal{D}_{q^{-1}}\right)^n\left[w(x;q)\right] }}
Generating function
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamCarlitzI{-1}{n}@{x}{q}=\discrqHermiteI{n}@{x}{q} }}
Discrete q-Hermite I 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{\discrqHermiteI{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 \discrqHermiteI{n}@{\iunit x}{q^{-1}}=\iunit^n\discrqHermiteII{n}@{x}{q} }}