Charlier

<< Bessel

Charlier

Hermite >>

Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Charlier{n}@{x}{a}=\HyperpFq{2}{0}@@{-n,-x}{-}{-\frac{1}{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}{x!}\Charlier{m}@{x}{a}\Charlier{n}@{x}{a}= a^{-n}\expe^an!\,\Kronecker{m}{n} }}

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

Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -x\Charlier{n}@{x}{a}=a\Charlier{n+1}@{x}{a}-(n+a)\Charlier{n}@{x}{a}+n\Charlier{n-1}@{x}{a} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicCharlier{n}@@{x}{a}=\monicCharlier{n+1}@@{x}{a}+(n+a)\monicCharlier{n}@@{x}{a}+na\monicCharlier{n-1}@@{x}{a} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Charlier{n}@{x}{a}=\left(-\frac{1}{a}\right)^n\monicCharlier{n}@@{x}{a} }}

Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\Charlier{n}@{x}{a}}}

Forward shift operator

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

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Charlier{n}@{x}{a}-\frac{x}{a}\Charlier{n}@{x-1}{a}=\Charlier{n+1}@{x}{a} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \nabla\left[\frac{a^x}{x!}\Charlier{n}@{x}{a}\right]=\frac{a^x}{x!}\Charlier{n+1}@{x}{a} }}

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{a^x}{x!}\Charlier{n}@{x}{a}=\nabla^n\left[\frac{a^x}{x!}\right] }}

Generating function

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^t\left(1-\frac{t}{a}\right)^x=\sum_{n=0}^{\infty}\frac{\Charlier{n}@{x}{a}}{n!}t^n }}

Limit relations

Meixner polynomial to Charlier polynomial

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

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} }}

Charlier polynomial to Hermite polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{a\rightarrow\infty} (2a)^{\frac{1}{2}n}\Charlier{n}@{(2a)^{\frac{1}{2}}x+a}{a}=(-1)^n\Hermite{n}@{x} }}

Remark

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{(-a)^n}{n!}\Charlier{n}@{x}{a}=\Laguerre[x-n]{n}@{a} }}

<< Bessel

Charlier

Hermite >>