Q-Charlier
q-Charlier
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qCharlier{n}@{q^{-x}}{a}{q}=\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{0}{q}{-\frac{q^{n+1}}{a}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qCharlier{n}@{q^{-x}}{a}{q}=\qPochhammer{-a^{-1}q}{q}{n}\cdot\qHyperrphis{1}{1}@@{q^{-n}}{-a^{-1}q}{q}{-\frac{q^{n+1-x}}{a}} }}
Orthogonality relation(s)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{x=0}^{\infty}\frac{a^x}{\qPochhammer{q}{q}{x}}q^{\binomial{x}{2}}\qCharlier{m}@{q^{-x}}{a}{q}\qCharlier{n}@{q^{-x}}{a}{q} {}=q^{-n}\qPochhammer{-a}{q}{\infty}\qPochhammer{-a^{-1}q,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}(1-q^{-x})\qCharlier{n}@@{q^{-x}}{a}{q} {}=a\qCharlier{n+1}@@{q^{-x}}{a}{q}-\left[a+q(1-q^n)(a+q^n)\right]\qCharlier{n}@@{q^{-x}}{a}{q} {}+q(1-q^n)(a+q^n)\qCharlier{n-1}@@{q^{-x}}{a}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qCharlier{n}@@{q^{-x}}{a}{q}:=\qCharlier{n}@{q^{-x}}{a}{q} }}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicqCharlier{n}@@{x}{a}{q}=\monicqCharlier{n+1}@@{x}{a}{q}+\left[1+q^{-2n-1}\left\{a+q(1-q^n)(a+q^n)\right\}\right]\monicqCharlier{n}@@{x}{a}{q} {}+aq^{-4n+1}(1-q^n)(a+q^n)\monicqCharlier{n-1}@@{x}{a}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qCharlier{n}@{q^{-x}}{a}{q}=\frac{(-1)^nq^{n^2}}{a^n}\monicqCharlier{n}@@{q^{-x}}{a}{q} }}
q-Difference equation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q^ny(x)=aq^xy(x+1)-q^x(a-1)y(x)+(1-q^x)y(x-1) }}
Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qCharlier{n}@{q^{-x-1}}{a}{q}-\qCharlier{n}@{q^{-x}}{a}{q}=-a^{-1}q^{-x}(1-q^n)\qCharlier{n-1}@{q^{-x}}{aq^{-1}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Delta \qCharlier{n}@{q^{-x}}{a}{q}}{\Delta q^{-x}}=-\frac{q(1-q^n)}{a(1-q)}\qCharlier{n-1}@{q^{-x}}{aq^{-1}}{q} }}
Backward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qCharlier{n}@{q^{-x}}{a}{q}-a^{-1}q^{-x}(1-q^x)\qCharlier{n}@{q^{-x+1}}{a}{q}=\qCharlier{n+1}@{q^{-x}}{aq}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\nabla\left[w(x;a;q)\qCharlier{n}@{q^{-x}}{a}{q}\right]}{\nabla q^{-x}} =\frac{1}{1-q}w(x;aq;q)\qCharlier{n+1}@{q^{-x}}{aq}{q} }}
Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a;q)\qCharlier{n}@{q^{-x}}{a}{q}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;aq^{-n};q)\right] }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \nabla_q:=\frac{\nabla}{\nabla q^{-x}} }}
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{1}{1}@@{q^{-x}}{0}{q}{-a^{-1}qt} =\sum_{n=0}^{\infty}\frac{\qCharlier{n}@{q^{-x}}{a}{q}}{\qPochhammer{q}{q}{n}}t^n }}
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}@@{-}{-a^{-1}q}{q}{-a^{-1}q^{-x+1}t} =\sum_{n=0}^{\infty}\frac{\qCharlier{n}@{q^{-x}}{a}{q}}{\qPochhammer{-a^{-1}q,q}{q}{n}}t^n }}
Limit relations
q-Meixner polynomial to q-Charlier polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qMeixner{n}@{x}{0}{c}{q}=\qCharlier{n}@{x}{c}{q} }}
q-Krawtchouk polynomial to q-Charlier polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty}\qKrawtchouk{n}@{q^{-x}}{a^{-1}q^{-N}}{N}{q}=\qCharlier{n}@{q^{-x}}{a}{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} }}
q-Charlier polynomial to Charlier polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\qCharlier{n}@{q^{-x}}{a(1-q)}{q}=\Charlier{n}@{x}{a} }}
Remark
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qCharlier{n}@{-x}{-q^{-\alpha}}{q}}{\qPochhammer{q}{q}{n}}=\qLaguerre[\alpha]{n}@{x}{q} }}