Q-Meixner

<< Little q-Jacobi: Special case

q-Meixner

Quantum q-Krawtchouk >>

Basic hypergeometric representation

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

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{\qPochhammer{bq}{q}{x}}{\qPochhammer{q,-bcq}{q}{x}}c^xq^{\binomial{x}{2}}\qMeixner{m}@{q^{-x}}{b}{c}{q}\qMeixner{n}@{q^{-x}}{b}{c}{q} {}=\frac{\qPochhammer{-c}{q}{\infty}}{\qPochhammer{-bcq}{q}{\infty}}\frac{\qPochhammer{q,-c^{-1}q}{q}{n}}{\qPochhammer{bq}{q}{n}}q^{-n}\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 0\leq bq<1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle c>0}}

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

Monic recurrence relation

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

q-Difference equation

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D(x)=(1-q^x)(1+bcq^x)}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=cq^x(1-bq^{x+1})}} &

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

Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qMeixner{n}@{q^{-x-1}}{b}{c}{q}-\qMeixner{n}@{q^{-x}}{b}{c}{q} {}=-\frac{q^{-x}(1-q^n)}{c(1-bq)}\qMeixner{n-1}@{q^{-x}}{bq}{cq^{-1}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Delta \qMeixner{n}@{q^{-x}}{b}{c}{q}}{\Delta q^{-x}} =-\frac{q(1-q^n)}{c(1-q)(1-bq)}\qMeixner{n-1}@{q^{-x}}{bq}{cq^{-1}}{q} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle cq^x(1-bq^x)\qMeixner{n}@{q^{-x}}{b}{c}{q}-(1-q^x)(1+bcq^x)\qMeixner{n}@{q^{-x+1}}{b}{c}{q} {}=cq^x(1-b)\qMeixner{n+1}@{q^{-x}}{bq^{-1}}{cq}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\nabla\left[w(x;b,c;q)\qMeixner{n}@{q^{-x}}{b}{c}{q}\right]}{\nabla q^{-x}} {}=\frac{1}{1-q}w(x;bq^{-1},cq;q)\qMeixner{n+1}@{q^{-x}}{bq^{-1}}{cq}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;b,c;q)=\frac{\qPochhammer{bq}{q}{x}}{\qPochhammer{q,-bcq}{q}{x}}c^xq^{\binomial{x+1}{2}}}}

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;b,c;q)\qMeixner{n}@{q^{-x}}{b}{c}{q}=(1-q)^n\left(\nabla_q\right)^n\left[w(x;bq^n,cq^{-n};q)\right] }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;b,c;q)=\frac{\qPochhammer{bq}{q}{x}}{\qPochhammer{q,-bcq}{q}{x}}c^xq^{\binomial{x+1}{2}}}}

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}}{bq}{q}{-c^{-1}qt} =\sum_{n=0}^{\infty}\frac{\qMeixner{n}@{q^{-x}}{b}{c}{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{1}{1}@@{-b^{-1}c^{-1}q^{-x}}{-c^{-1}q}{q}{bqt} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{bq}{q}{n}}{\qPochhammer{-c^{-1}q,q}{q}{n}}\qMeixner{n}@{q^{-x}}{b}{c}{q}t^n }}

Limit relations

Big q-Jacobi polynomial to q-Meixner polynomial

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

q-Hahn polynomial to q-Meixner polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty}Q_n(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=\qMeixner{n}@{q^{-x}}{b}{c}{q} }}

q-Meixner polynomial to q-Laguerre polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{c\rightarrow\infty}\qMeixner{n}@{cq^{\alpha}x}{q^{\alpha}}{c}{q}= \frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\qLaguerre[\alpha]{n}@{x}{q} }}

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-Meixner polynomial to Al-Salam-Carlitz II polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{c\rightarrow 0}\qMeixner{n}@{x}{-ac^{-1}}{c}{q}= \left(-\frac{1}{a}\right)^nq^{\binomial{n}{2}}\AlSalamCarlitzII{a}{n}@{x}{q} }}

q-Meixner polynomial to Meixner polynomial

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

Remarks

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qMeixner{n}@{q^{-x}}{b}{c}{q}=\littleqJacobi{n}@{-c^{-1}q^n}{b}{b^{-1}q^{-n-x-1}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\qMeixner{n}@{q^{-x}}{q^{-N-1}}{-p^{-1}}{q} }}

<< Little q-Jacobi: Special case

q-Meixner

Quantum q-Krawtchouk >>