Q-Krawtchouk

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q-Krawtchouk

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},-pq^n}{q^{-N},0}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\frac{\qPochhammer{q^{x-N}}{q}{n}}{\qPochhammer{q^{-N}}{q}{n}q^{nx}}\ \qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{q^{N-x-n+1}}{q}{-pq^{n+N+1}} }}

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{q^{-N}}{q}{x}}{\qPochhammer{q}{q}{x}}(-p)^{-x}\qKrawtchouk{m}@{q^{-x}}{p}{N}{q}\qKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}=\frac{\qPochhammer{q,-pq^{N+1}}{q}{n}}{\qPochhammer{-p,q^{-N}}{q}{n}}\frac{(1+p)}{(1+pq^{2n})} {} \qPochhammer{-pq}{q}{N}p^{-N}q^{-\binomial{N+1}{2}} \left(-pq^{-N}\right)^nq^{n^2}\,\Kronecker{m}{n} }}

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


Recurrence relation

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=-pq^{2n-N-1}\frac{(1+pq^{n+N})(1-q^n)}{(1+pq^{2n-1})(1+pq^{2n})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-q^{n-N})(1+pq^n)}{(1+pq^{2n})(1+pq^{2n+1})}}}


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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=-pq^{2n-N-1}\frac{(1+pq^{n+N})(1-q^n)}{(1+pq^{2n-1})(1+pq^{2n})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-q^{n-N})(1+pq^n)}{(1+pq^{2n})(1+pq^{2n+1})}}}


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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\qKrawtchouk{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 \qKrawtchouk{n}@{q^{-x-1}}{p}{N}{q}-\qKrawtchouk{n}@{q^{-x}}{p}{N}{q} {}=\frac{q^{-n-x}(1-q^n)(1+pq^n)}{1-q^{-N}}\qKrawtchouk{n-1}@{q^{-x}}{pq^2}{N-1}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Delta \qKrawtchouk{n}@{q^{-x}}{p}{N}{q}}{\Delta q^{-x}}= \frac{q^{-n+1}(1-q^n)(1+pq^n)}{(1-q)(1-q^{-N})}\qKrawtchouk{n-1}@{q^{-x}}{pq^2}{N-1}{q} }}

Backward shift operator

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

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

Limit relations

q-Racah polynomial to q-Krawtchouk polynomial

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

q-Hahn polynomial to q-Krawtchouk polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\alpha\rightarrow 0} Q_n(q^{-x};\alpha,-\alpha^{-1}q^{-1}p,N|q)=\qKrawtchouk{n}@{q^{-x}}{p}{N}{q} }}

q-Krawtchouk polynomial to q-Bessel 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-N}}{p}{N}{q}=\qBesselPoly{n}@{q^x}{p}{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-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}\qKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\Krawtchouk{n}@{x}{(p+1)^{-1}}{N} }}

Remark

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

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


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

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


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