Charlier

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Charlier

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Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;a\right)=\HyperpFq{2}{0}@@{-n,-x}{-}{-\frac{1}{a}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{a^{x}}{x!}C_{m}\!\left(x;a\right)C_{n}\!\left(x;a\right)=a^{-n}{\mathrm{e}^{a}}n!\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a>0}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-xC_{n}\!\left(x;a\right)=aC_{n+1}\!\left(x;a\right)-(n+a)C_{n}\!\left(x;a\right)+nC_{n-1}\!\left(x;a\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{C}}_{n}\!\left(x\right)={\widehat{C}}_{n+1}\!\left(x\right)+(n+a){\widehat{C}}_{n}\!\left(x\right)+na{\widehat{C}}_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;a\right)=\left(-\frac{1}{a}\right)^{n}{\widehat{C}}_{n}\!\left(x\right)}}}$

Difference equation

${\displaystyle{\displaystyle{\displaystyle-ny(x)=ay(x+1)-(x+a)y(x)+xy(x-1)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=C_{n}\!\left(x;a\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x+1;a\right)-C_{n}\!\left(x;a\right)=-\frac{n}{a}C_{n-1}\!\left(x;a\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\Delta C_{n}\!\left(x;a\right)=-\frac{n}{a}C_{n-1}\!\left(x;a\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;a\right)-\frac{x}{a}C_{n}\!\left(x-1;a\right)=C_{n+1}\!\left(x;a\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\nabla\left[\frac{a^{x}}{x!}C_{n}\!\left(x;a\right)\right]=\frac{a^{x}}{x!}C_{n+1}\!\left(x;a\right)}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle\frac{a^{x}}{x!}C_{n}\!\left(x;a\right)=\nabla^{n}\left[\frac{a^{x}}{x!}\right]}}}$

Generating function

${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\left(1-\frac{t}{a}\right)^{x}=\sum_{n=0}^{\infty}\frac{C_{n}\!\left(x;a\right)}{n!}t^{n}}}}$

Limit relations

Meixner polynomial to Charlier polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow\infty}M_{n}\!\left(x;\beta,(a+\beta)^{-1}a\right)=C_{n}\!\left(x;a\right)}}}$

Krawtchouk polynomial to Charlier polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K_{n}\!\left(x;N^{-1}a,N\right)=C_{n}\!\left(x;a\right)}}}$

Charlier polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow\infty}(2a)^{\frac{1}{2}n}C_{n}\!\left((2a)^{\frac{1}{2}}x+a;a\right)=(-1)^{n}H_{n}\left(x\right)}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle\frac{(-a)^{n}}{n!}C_{n}\!\left(x;a\right)=L^{x-n}_{n}\left(a\right)}}}$

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Charlier

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