# Meixner

## Contents

- 1 Meixner
- 2 Hypergeometric representation
- 3 Orthogonality relation(s)
- 4 Recurrence relation
- 5 Monic recurrence relation
- 6 Difference equation
- 7 Forward shift operator
- 8 Backward shift operator
- 9 Rodrigues-type formula
- 10 Generating functions
- 11 Limit relations
- 12 Remarks
- 13 Koornwinder Addendum: Meixner

## Meixner

## Hypergeometric representation

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \Meixner{n}@{x}{\beta}{c}=\HyperpFq{2}{1}@@{-n,-x}{\beta}{1-\frac{1}{c}} }}**

## Orthogonality relation(s)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \sum_{x=0}^{\infty}\frac{\pochhammer{\beta}{x}}{x!}c^x\Meixner{m}@{x}{\beta}{c}\Meixner{n}@{x}{\beta}{c} {}=\frac{c^{-n}n!}{\pochhammer{\beta}{n}(1-c)^{\beta}}\,\Kronecker{m}{n} }}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \beta > 0\quad 0 < c < 1}}**

## Recurrence relation

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle (c-1)x\Meixner{n}@{x}{\beta}{c}=c(n+\beta)\Meixner{n+1}@{x}{\beta}{c} {}-\left[n+(n+\beta)c\right]\Meixner{n}@{x}{\beta}{c}+n\Meixner{n-1}@{x}{\beta}{c} }}**

## Monic recurrence relation

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle x\monicMeixner{n}@@{x}{\beta}{c}=\monicMeixner{n+1}@@{x}{\beta}{c}+\frac{n+(n+\beta)c}{1-c}\monicMeixner{n}@@{x}{\beta}{c}+ \frac{n(n+\beta-1)c}{(1-c)^2}\monicMeixner{n-1}@@{x}{\beta}{c} }}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \Meixner{n}@{x}{\beta}{c}=\frac{1}{\pochhammer{\beta}{n}}\left(\frac{c-1}{c}\right)^n\monicMeixner{n}@@{x}{\beta}{c} }}**

## Difference equation

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1) }}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle y(x)=\Meixner{n}@{x}{\beta}{c}}}**

## Forward shift operator

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \Meixner{n}@{x+1}{\beta}{c}-\Meixner{n}@{x}{\beta}{c}= \frac{n}{\beta}\left(\frac{c-1}{c}\right)\Meixner{n-1}@{x}{\beta+1}{c} }}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \Delta \Meixner{n}@{x}{\beta}{c}=\frac{n}{\beta}\left(\frac{c-1}{c}\right)\Meixner{n-1}@{x}{\beta+1}{c} }}**

## Backward shift operator

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle c(\beta+x-1)\Meixner{n}@{x}{\beta}{c}-x\Meixner{n}@{x-1}{\beta}{c}=c(\beta-1)\Meixner{n+1}@{x}{\beta-1}{c} }}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \nabla\left[\frac{\pochhammer{\beta}{x}c^x}{x!}\Meixner{n}@{x}{\beta}{c}\right]= \frac{\pochhammer{\beta-1}{x}c^x}{x!}\Meixner{n+1}@{x}{\beta-1}{c} }}**

## Rodrigues-type formula

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \frac{\pochhammer{\beta}{x}c^x}{x!}\Meixner{n}@{x}{\beta}{c}=\nabla^n\left[\frac{\pochhammer{\beta+n}{x}c^x}{x!}\right] }}**

## Generating functions

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= \sum_{n=0}^{\infty}\frac{\pochhammer{\beta}{n}}{n!}\Meixner{n}@{x}{\beta}{c}t^n }}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \expe^t\,\HyperpFq{1}{1}@@{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= \sum_{n=0}^{\infty}\frac{\Meixner{n}@{x}{\beta}{c}}{n!}t^n }}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle (1-t)^{-\gamma}\,\HyperpFq{2}{1}@@{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} =\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{n!}\Meixner{n}@{x}{\beta}{c}t^n }}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \gamma}}**arbitrary

## Limit relations

### Hahn polynomial to Meixner polynomial

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \lim_{c\rightarrow 1} \Meixner{n}@{(1-c)^{-1}x}{\alpha+1}{c}=\frac{\Laguerre[\alpha]{n}@{x}}{\Laguerre[\alpha]{n}@{0}} }}**

### Meixner polynomial to Charlier polynomial

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \lim_{\beta\rightarrow\infty} \Meixner{n}@{x}{\beta}{(a+\beta)^{-1}a}=\Charlier{n}@{x}{a} }}**

## Remarks

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \frac{\pochhammer{\beta}{n}}{n!}\Meixner{n}@{x}{\beta}{c}=\Jacobi{\beta-1}{-n-\beta-x}{n}@{(2-c}c^{-1}) }}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\Meixner{n}@{x}{-N}{(p-1)^{-1}p} }}**