Affine q-Krawtchouk

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Affine q-Krawtchouk

Dual q-Krawtchouk >>

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},0,q^{-x}}{pq,q^{-N}}{q}{q} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n=0,1,2,\ldots,N}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\frac{(-pq)^nq^{\binomial{n}{2}}}{\qPochhammer{pq}{q}{n}}\ \qHyperrphis{2}{1}@@{q^{-n},q^{x-N}}{q^{-N}}{q}{\frac{q^{-x}}{p}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{x=0}^N\frac{\qPochhammer{pq}{q}{x}\qPochhammer{q}{q}{N}}{\qPochhammer{q}{q}{x}\qPochhammer{q}{q}{N-x}}(pq)^{-x}\AffqKrawtchouk{m}@{q^{-x}}{p}{N}{q}\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}=(pq)^{n-N}\frac{\qPochhammer{q}{q}{n}\qPochhammer{q}{q}{N-n}}{\qPochhammer{pq}{q}{n}\qPochhammer{q}{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<pq<1}}

Recurrence relation

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

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicAffqKrawtchouk{n}@@{x}{p}{N}{q}=\monicAffqKrawtchouk{n+1}@@{x}{p}{N}{q}+ \left[1-\left\{(1-q^{n-N})(1-pq^{n+1})-pq^{n-N}(1-q^n)\right\}\right]\monicAffqKrawtchouk{n}@@{x}{p}{N}{q} {}-pq^{n-N}(1-q^n)(1-pq^n)(1-q^{n-N-1})\monicAffqKrawtchouk{n-1}@@{x}{p}{N}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\frac{1}{\qPochhammer{pq,q^{-N}}{q}{n}}\monicAffqKrawtchouk{n}@@{q^{-x}}{p}{N}{q} }}

q-Difference equation

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

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

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

Forward shift operator

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

Backward shift operator

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;p,N;q)=\frac{\qPochhammer{pq}{q}{x}}{\qPochhammer{q}{q}{x}\qPochhammer{q}{q}{N-x}}p^{-x}}}

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;p,N;q)\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}=\frac{(-1)^nq^{-Nn+\binomial{n}{2}}(1-q)^n}{\qPochhammer{q^{-N}}{q}{n}}\left(\nabla_q\right)^n\left[w(x;pq^n,N-n;q)\right] }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;p,N;q)=\frac{\qPochhammer{pq}{q}{x}}{\qPochhammer{q}{q}{x}\qPochhammer{q}{q}{N-x}}p^{-x}}}

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 \qPochhammer{q^{-N}t}{q}{N-x}\cdot\qHyperrphis{1}{1}@@{q^{-x}}{pq}{q}{pqt}=\sum_{n=0}^N \frac{\qPochhammer{q^{-N}}{q}{n}}{\qPochhammer{q}{q}{n}}\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{-pq^{-N+1}t}{q}{x}\cdot\qHyperrphis{2}{0}@@{q^{x-N},pq^{x+1}}{-}{q}{-q^{-x}t} {}=\sum_{n=0}^N\frac{\qPochhammer{pq,q^{-N}}{q}{n}}{\qPochhammer{q}{q}{n}}q^{-\binomial{n}{2}}\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}t^n }}

Limit relations

q-Hahn polynomial to Affine q-Krawtchouk polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHahn{n}@{q^{-x}}{p}{0}{N}{q}=\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }}

Dual q-Hahn polynomial to Affine q-Krawtchouk polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \dualqHahn{n}@{\mu(x)}{p}{0}{N}{q}=\AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q} }}

Affine q-Krawtchouk polynomial to Little q-Laguerre / Wall polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty}\AffqKrawtchouk{n}@{q^{x-N}}{p}{N}{q}=\littleqLaguerre{n}@{q^x}{p}{q} }}

Affine q-Krawtchouk polynomial to Krawtchouk polynomial

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

Remarks

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

Koornwinder Addendum: Affine q-Krawtchouk

<< q-Krawtchouk

Affine q-Krawtchouk

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