Meixner

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Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(x;\beta,c\right)=\HyperpFq{2}{1}@@{-n,-x}{\beta}{1-\frac{1}{c}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{{\left(\beta\right)_{x}}}{x!}c^{x}M_{m}\!\left(x;\beta,c\right)M_{n}\!\left(x;\beta,c\right){}=\frac{c^{-n}n!}{{\left(\beta\right)_{n}}(1-c)^{\beta}}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\beta>0\quad 0<c<1}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle(c-1)xM_{n}\!\left(x;\beta,c\right)=c(n+\beta)M_{n+1}\!\left(x;\beta,c\right){}-\left[n+(n+\beta)c\right]M_{n}\!\left(x;\beta,c\right)+nM_{n-1}\!\left(x;\beta,c\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{M}}_{n}\!\left(x;\beta,c\right)={\widehat{M}}_{n+1}\!\left(x;\beta,c\right)+\frac{n+(n+\beta)c}{1-c}{\widehat{M}}_{n}\!\left(x;\beta,c\right)+\frac{n(n+\beta-1)c}{(1-c)^{2}}{\widehat{M}}_{n-1}\!\left(x;\beta,c\right)}}}$
${\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(x;\beta,c\right)=\frac{1}{{\left(\beta\right)_{n}}}\left(\frac{c-1}{c}\right)^{n}{\widehat{M}}_{n}\!\left(x;\beta,c\right)}}}$

Difference equation

${\displaystyle{\displaystyle{\displaystyle n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=M_{n}\!\left(x;\beta,c\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(x+1;\beta,c\right)-M_{n}\!\left(x;\beta,c\right)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}\!\left(x;\beta+1,c\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\Delta M_{n}\!\left(x;\beta,c\right)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}\!\left(x;\beta+1,c\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle c(\beta+x-1)M_{n}\!\left(x;\beta,c\right)-xM_{n}\!\left(x-1;\beta,c\right)=c(\beta-1)M_{n+1}\!\left(x;\beta-1,c\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\nabla\left[\frac{{\left(\beta\right)_{x}}c^{x}}{x!}M_{n}\!\left(x;\beta,c\right)\right]=\frac{{\left(\beta-1\right)_{x}}c^{x}}{x!}M_{n+1}\!\left(x;\beta-1,c\right)}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle\frac{{\left(\beta\right)_{x}}c^{x}}{x!}M_{n}\!\left(x;\beta,c\right)=\nabla^{n}\left[\frac{{\left(\beta+n\right)_{x}}c^{x}}{x!}\right]}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\left(1-\frac{t}{c}\right)^{x}(1-t)^{-x-\beta}=\sum_{n=0}^{\infty}\frac{{\left(\beta\right)_{n}}}{n!}M_{n}\!\left(x;\beta,c\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\,\HyperpFq{1}{1}@@{-x}{\beta}{\left(\frac{1-c}{c}\right)t}=\sum_{n=0}^{\infty}\frac{M_{n}\!\left(x;\beta,c\right)}{n!}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle(1-t)^{-\gamma}\,\HyperpFq{2}{1}@@{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}}{n!}M_{n}\!\left(x;\beta,c\right)t^{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

Limit relations

Hahn polynomial to Meixner polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}\Hahn{n}@{x}{b-1}{N(1-c)c^{-1}}{N}=\Meixner{n}@{x}{b}{c}}}}$

Dual Hahn polynomial to Meixner polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}\dualHahn{n}@{\lambda(x)}{\beta-1}{N(1-c)c^{-1}}{N}=\Meixner{n}@{x}{\beta}{c}}}}$

Meixner polynomial to Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow 1}M_{n}\!\left((1-c)^{-1}x;\alpha+1,c\right)=\frac{L^{\alpha}_{n}\left(x\right)}{L^{\alpha}_{n}\left(0\right)}}}}$

Meixner polynomial to Charlier polynomial

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

Remarks

${\displaystyle{\displaystyle{\displaystyle\frac{{\left(\beta\right)_{n}}}{n!}M_{n}\!\left(x;\beta,c\right)=P^{(\beta-1,-n-\beta-x)}_{n}\left((2-c\right)c^{-1})}}}$
${\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(x;p,N\right)=M_{n}\!\left(x;-N,(p-1)^{-1}p\right)}}}$

Koornwinder Addendum: Meixner

<< Pseudo Jacobi

Meixner

Krawtchouk >>