Krawtchouk
Krawtchouk
Hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\HyperpFq{2}{1}@@{-n,-x}{-N}{\frac{1}{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\binomial{N}{x}p^x(1-p)^{N-x} \Krawtchouk{m}@{x}{p}{N}\Krawtchouk{n}@{x}{p}{N} {}=\frac{(-1)^nn!}{\pochhammer{-N}{n}}\left(\frac{1-p}{p}\right)^n\,\Kronecker{m}{n} }}
Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -x\Krawtchouk{n}@{x}{p}{N}=p(N-n)\Krawtchouk{n+1}@{x}{p}{N} {}-\left[p(N-n)+n(1-p)\right]\Krawtchouk{n}@{x}{p}{N} {}+n(1-p)\Krawtchouk{n-1}@{x}{p}{N} }}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicKrawtchouk{n}@@{x}{p}{N}=\monicKrawtchouk{n+1}@@{x}{p}{N}+\left[p(N-n)+n(1-p)\right]\monicKrawtchouk{n}@@{x}{p}{N} {}+np(1-p)(N+1-n)\monicKrawtchouk{n-1}@@{x}{p}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\frac{1}{\pochhammer{-N}{n}p^n}\monicKrawtchouk{n}@@{x}{p}{N} }}
Difference equation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -ny(x)=p(N-x)y(x+1) {}-\left[p(N-x)+x(1-p)\right]y(x)+x(1-p)y(x-1) }}
Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{n}@{x+1}{p}{N}-\Krawtchouk{n}@{x}{p}{N}=-\frac{n}{Np}\Krawtchouk{n-1}@{x}{p}{N-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Delta \Krawtchouk{n}@{x}{p}{N}=-\frac{n}{Np}\Krawtchouk{n-1}@{x}{p}{N-1} }}
Backward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (N+1-x)\Krawtchouk{n}@{x}{p}{N}-x\left(\frac{1-p}{p}\right)\Krawtchouk{n}@{x-1}{p}{N} {}=(N+1)\Krawtchouk{n+1}@{x}{p}{N+1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \nabla\left[\binomial{N}{x}\left(\frac{p}{1-p}\right)^x\Krawtchouk{n}@{x}{p}{N}\right]= \binomial{N+1}{x}\left(\frac{p}{1-p}\right)^x\Krawtchouk{n+1}@{x}{p}{N+1} }}
Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \binomial{N}{x}\left(\frac{p}{1-p}\right)^x\Krawtchouk{n}@{x}{p}{N}= \nabla^n\left[\binomial{N-n}{x}\left(\frac{p}{1-p}\right)^x\right] }}
Generating functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left(1-\frac{(1-p)}{p}t\right)^x(1+t)^{N-x}= \sum_{n=0}^N\binomial{N}{n}\Krawtchouk{n}@{x}{p}{N}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left[\expe^t\,\HyperpFq{1}{1}@@{-x}{-N}{-\frac{t}{p}}\right]_N= \sum_{n=0}^N\frac{\Krawtchouk{n}@{x}{p}{N}}{n!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left[(1-t)^{-\gamma}\,\HyperpFq{2}{1}@@{\gamma,-x}{-N}{\frac{t}{p(t-1)}}\right]_N {}=\sum_{n=0}^N\frac{\pochhammer{\gamma}{n}}{n!}\Krawtchouk{n}@{x}{p}{N}t^n }}
Limit relations
Hahn polynomial to Krawtchouk polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{t\rightarrow\infty}\Hahn{n}@{x}{pt}{(1-p)t}{N}=\Krawtchouk{n}@{x}{p}{N} }}
Dual Hahn polynomial to Krawtchouk polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{t\rightarrow\infty}\dualHahn{n}@{\lambda(x)}{pt}{(1-p)t}{N}=\Krawtchouk{n}@{x}{p}{N} }}
Krawtchouk polynomial to Charlier polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty}\Krawtchouk{n}@{x}{N^{-1}a}{N}=\Charlier{n}@{x}{a} }}
Krawtchouk polynomial to Hermite polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty} \sqrt{\binomial{N}{n}}\Krawtchouk{n}@{pN+x\sqrt{2p(1-p)N}}{p}{N} =\frac{\displaystyle (-1)^n\Hermite{n}@{x}}{\displaystyle\sqrt{2^nn!\left(\frac{p}{1-p}\right)^n}} }}
Remarks
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\Krawtchouk{x}@{n}{p}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{n=0}^N\binomial{N}{n}p^n(1-p)^{N-n} \Krawtchouk{n}@{x}{p}{N}\Krawtchouk{n}@{y}{p}{N}= \frac{\displaystyle\left(\frac{1-p}{p}\right)^x}{\dbinom{N}{x}}\Kronecker{x}{y} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\Meixner{n}@{x}{-N}{(p-1)^{-1}p} }}
Koornwinder Addendum: Krawtchouk
Krawtchouk: Special values
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{n}@{0}{p}{N}=1,\qquad \Krawtchouk{n}@{N}{p}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{n}@{0}{p}{N} =(1-p^{-1})^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\Krawtchouk{x}@{n}{p}{N}\qquad (n,x\in \{0,1,\ldots,N\}) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{N}@{x}{p}{N}=(1-p^{-1})^x\qquad(x\in\{0,1,\ldots,N\}) }}
Symmetry
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Krawtchouk{n}@{N-x}{p}{N}}{\Krawtchouk{n}@{N}{p}{N}}=\Krawtchouk{n}@{x}{1-p}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Krawtchouk{N-n}@{x}{p}{N}}{\Krawtchouk{N}@{x}{p}{N}}=\Krawtchouk{n}@{x}{1-p}{N} \qquad(n,x\in\{0,1,\ldots,N\}) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{N-n}@{N-x}{p}{N}=\left(\frac p{p-1}\right)^{n+x-N}\Krawtchouk{n}@{x}{p}{N} \qquad(n,x\in\{0,1,\ldots,N\}) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{n}@{N-x}{\frac12}{N}=(-1)^n \Krawtchouk{n}@{x}{\frac12}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{2m+1}@{N}{\frac12}{2N}=0 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{2m}@{N}{\frac12}{2N}=\frac{\pochhammer{\frac12}{m}}{\pochhammer{-N+\frac12}{m}} }}
Quadratic transformations
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{2m}@{x+N}{\frac12}{2N}=\frac{\pochhammer{\frac12}{m}}{\pochhammer{-N+\frac12}{m}} \dualHahn{m}@{x^2}{-\frac12}{-\frac12}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{2m+1}@{x+N}{\frac12}{2N}=- \frac{\pochhammer{\tfrac32}{m}}{N \pochhammer{-N+\frac12}{m}} x \dualHahn{m}@{x^2-1}{\frac12}{\frac12}{N-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{2m}@{x+N+1}{\frac12}{2N+1}=\frac{\pochhammer{\tfrac12}{m}}{\pochhammer{-N-\frac12}{m}} \dualHahn{m}@{x(x+1)}{-\frac12}{\frac12}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Krawtchouk{2m+1}@{x+N+1}{\frac12}{2N+1}=\frac{\pochhammer{\tfrac32}{m}}{\pochhammer{-N-\frac12}{m+1}} (x+\frac12) \dualHahn{m}@{x(x+1)}{\frac12}{-\frac12}{N} }}
Generating functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{x=0}^N\binom Nx \Krawtchouk{m}@{x}{p}{N}\Krawtchouk{n}@{x}{q}{N}z^x =\left(\frac{p-z+pz}p\right)^m \left(\frac{q-z+qz}q\right)^n (1+z)^{N-m-n} \Krawtchouk{m}@{n}{- \frac{(p-z+pz)(q-z+qz)}z}{N} }}