# q-Meixner-Pollaczek

## Basic hypergeometric representation

$\displaystyle {\displaystyle \qMeixnerPollaczek{n}@{x}{a}{q}=a^{-n}\expe^{-\iunit n\phi}\frac{\qPochhammer{a^2}{q}{n}}{\qPochhammer{q}{q}{n}}\ \qHyperrphis{3}{2}@@{q^{-n},a\expe^{\iunit(\theta+2\phi)},a\expe^{-\iunit\theta}}{a^2,0}{q}{q} }$
$\displaystyle {\displaystyle \qMeixnerPollaczek{n}@{x}{a}{q}=\frac{\qPochhammer{a\expe^{-\iunit\theta}}{q}{n}}{\qPochhammer{q}{q}{n}}\expe^{\iunit n(\theta+\phi)}\ \qHyperrphis{2}{1}@@{q^{-n},a\expe^{\iunit\theta}}{a^{-1}q^{-n+1}\expe^{\iunit\theta}}{q}{qa^{-1}\expe^{-\iunit(\theta+2\phi)}} }$

## Orthogonality relation(s)

$\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-\cpi}^{\cpi}w(\cos@{\theta+\phi};a|q) \qMeixnerPollaczek{m}@{\cos@{\theta+\phi}}{a}{q}\qMeixnerPollaczek{n}@{\cos@{\theta+\phi}}{a}{q}\,d\theta {}=\frac{\,\Kronecker{m}{n}}{\qPochhammer{q}{q}{n}\qPochhammer{q,a^2q^n}{q}{\infty}} }$

Constraint(s): $\displaystyle {\displaystyle 0

Substitution(s): $\displaystyle {\displaystyle w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit(\theta+2\phi)},a\expe^{\iunit\theta}}{q}{\infty}}\right|^2= \frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a\expe^{\iunit\phi})h(x,a\expe^{-\iunit\phi})}}$ &

$\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit(\theta+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{q}{\infty}}$ &

$\displaystyle {\displaystyle x=\cos@{\theta+\phi}}$

## Recurrence relation

$\displaystyle {\displaystyle 2x\qMeixnerPollaczek{n}@{x}{a}{q}=(1-q^{n+1})\qMeixnerPollaczek{n+1}@{x}{a}{q}+2aq^n\cos@@{\phi} \qMeixnerPollaczek{n}@{x}{a}{q} {}+(1-a^2q^{n-1})\qMeixnerPollaczek{n-1}@{x}{a}{q} }$

## Monic recurrence relation

$\displaystyle {\displaystyle x\monicqMeixnerPollaczek{n}@@{x}{a}{q}=\monicqMeixnerPollaczek{n+1}@@{x}{a}{q}+aq^n\cos@@{\phi}\,\monicqMeixnerPollaczek{n}@@{x}{a}{q}+\frac{1}{4}(1-q^n)(1-a^2q^{n-1})\monicqMeixnerPollaczek{n-1}@@{x}{a}{q} }$
$\displaystyle {\displaystyle \qMeixnerPollaczek{n}@{x}{a}{q}=\frac{2^n}{\qPochhammer{q}{q}{n}}\monicqMeixnerPollaczek{n}@@{x}{a}{q} }$

## q-Difference equation

$\displaystyle {\displaystyle (1-q)^2D_q\left[{\tilde w}(x;aq^{\frac{1}{2}}|q)D_qy(x)\right] +4q^{-n+1}(1-q^n){\tilde w}(x;a|q)y(x)=0 }$

Substitution(s): $\displaystyle {\displaystyle {\tilde w}(x;a|q):=\frac{w(x;a|q)}{\sqrt{1-x^2}}}$ &

$\displaystyle {\displaystyle y(x)=\qMeixnerPollaczek{n}@{x}{a}{q}}$ &
$\displaystyle {\displaystyle w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit(\theta+2\phi)},a\expe^{\iunit\theta}}{q}{\infty}}\right|^2= \frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a\expe^{\iunit\phi})h(x,a\expe^{-\iunit\phi})}}$ &
$\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit(\theta+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{q}{\infty}}$ &

$\displaystyle {\displaystyle x=\cos@{\theta+\phi}}$

## Forward shift operator

$\displaystyle {\displaystyle \delta_q\qMeixnerPollaczek{n}@{x}{a}{q}=-q^{-\frac{1}{2}n}(\expe^{\iunit(\theta+\phi)}-\expe^{-\iunit(\theta+\phi)}) \qMeixnerPollaczek{n-1}@{x}{aq^{\frac{1}{2}}}{q} }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@{\theta+\phi}}$

$\displaystyle {\displaystyle D_q\qMeixnerPollaczek{n}@{x}{a}{q}=\frac{2q^{-\frac{1}{2}(n-1)}}{1-q}\qMeixnerPollaczek{n-1}@{x}{aq^{\frac{1}{2}}}{q} }$

## Backward shift operator

$\displaystyle {\displaystyle \delta_q\left[{\tilde w}(x;a|q)\qMeixnerPollaczek{n}@{x}{a}{q}\right] {}=q^{-\frac{1}{2}(n+1)}(1-q^{n+1})(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {} {\tilde w}(x;aq^{-\frac{1}{2}}|q)\qMeixnerPollaczek{n+1}@{x}{aq^{-\frac{1}{2}}}{q} }$

Substitution(s): $\displaystyle {\displaystyle {\tilde w}(x;a|q):=\frac{w(x;a|q)}{\sqrt{1-x^2}}}$ &

$\displaystyle {\displaystyle w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit(\theta+2\phi)},a\expe^{\iunit\theta}}{q}{\infty}}\right|^2= \frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a\expe^{\iunit\phi})h(x,a\expe^{-\iunit\phi})}}$ &
$\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit(\theta+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{q}{\infty}}$ &

$\displaystyle {\displaystyle x=\cos@{\theta+\phi}}$

$\displaystyle {\displaystyle D_q\left[{\tilde w}(x;a|q)\qMeixnerPollaczek{n}@{x}{a}{q}\right]= -2q^{-\frac{1}{2}n}\frac{1-q^{n+1}}{1-q}{\tilde w}(x;aq^{-\frac{1}{2}}|q)\qMeixnerPollaczek{n+1}@{x}{aq^{-\frac{1}{2}}}{q} }$

Substitution(s): $\displaystyle {\displaystyle {\tilde w}(x;a|q):=\frac{w(x;a|q)}{\sqrt{1-x^2}}}$ &

$\displaystyle {\displaystyle w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit(\theta+2\phi)},a\expe^{\iunit\theta}}{q}{\infty}}\right|^2= \frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a\expe^{\iunit\phi})h(x,a\expe^{-\iunit\phi})}}$ &
$\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit(\theta+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{q}{\infty}}$ &

$\displaystyle {\displaystyle x=\cos@{\theta+\phi}}$

## Rodrigues-type formula

$\displaystyle {\displaystyle {\tilde w}(x;a|q)\qMeixnerPollaczek{n}@{x}{a}{q}=\left(\frac{q-1}{2}\right)^n q^{\frac{1}{4}n(n-1)}\frac{1}{\qPochhammer{q}{q}{n}}\left(D_q\right)^n\left[{\tilde w}(x;aq^{\frac{1}{2}n}|q)\right] }$

Substitution(s): $\displaystyle {\displaystyle {\tilde w}(x;a|q):=\frac{w(x;a|q)}{\sqrt{1-x^2}}}$ &

$\displaystyle {\displaystyle w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit(\theta+2\phi)},a\expe^{\iunit\theta}}{q}{\infty}}\right|^2= \frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,a\expe^{\iunit\phi})h(x,a\expe^{-\iunit\phi})}}$ &
$\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit(\theta+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{q}{\infty}}$ &

$\displaystyle {\displaystyle x=\cos@{\theta+\phi}}$

## Generating functions

$\displaystyle {\displaystyle \left|\frac{\qPochhammer{a\expe^{\iunit\phi}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit(\theta+\phi)}t}{q}{\infty}}\right|^2 {}=\frac{\qPochhammer{a\expe^{\iunit\phi}t,a\expe^{-\iunit\phi}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit(\theta+\phi)}t,\expe^{-\iunit(\theta+\phi)}t}{q}{\infty}} }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@{\theta+\phi}}$

$\displaystyle {\displaystyle \left|\frac{\qPochhammer{a\expe^{\iunit\phi}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit(\theta+\phi)}t}{q}{\infty}}\right|^2 {}=\sum_{n=0}^{\infty}\qMeixnerPollaczek{n}@{x}{a}{q}t^n }$
$\displaystyle {\displaystyle \frac{1}{\qPochhammer{\expe^{\iunit(\theta+\phi)}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{a\expe^{\iunit(\theta+2\phi)},a\expe^{\iunit\theta}}{a^2}{q}{\expe^{-\iunit(\theta+\phi)}t} {}=\sum_{n=0}^{\infty}\frac{\qMeixnerPollaczek{n}@{x}{a}{q}}{\qPochhammer{a^2}{q}{n}}t^n }$

Substitution(s): $\displaystyle {\displaystyle x=\cos@{\theta+\phi}}$

## Limit relations

### Continuous q-Hahn polynomial to q-Meixner-Pollaczek polynomial

$\displaystyle {\displaystyle \frac{\ctsqHahn{n}@{\cos@{\theta+\phi}}{a}{0}{0}{a}{q}}{\qPochhammer{q}{q}{n}}=\qMeixnerPollaczek{n}@{\cos@{\theta+\phi}}{a}{q} }$

### q-Meixner-Pollaczek polynomial to Continuous q-ultraspherical /

$\displaystyle {\displaystyle \qMeixnerPollaczek{n}@{\cos@@{\phi}}{\beta}{q}=\ctsqUltra{n}@{\cos@@{\phi}}{\beta}{q} }$

### q-Meixner-Pollaczek polynomial to Continuous

$\displaystyle {\displaystyle \qMeixnerPollaczek{n}@{\cos@{\theta+\phi}}{q^{\frac{1}{2}\alpha+\frac{1}{2}}}{q}= q^{-(\frac{1}{2}\alpha+\frac{1}{4})n}\ctsqLaguerre{\alpha}{n}@{\cos@@{\theta}}{q} }$

### q-Meixner-Pollaczek polynomial to Meixner-Pollaczek polynomial

$\displaystyle {\displaystyle \lim_{q\rightarrow 1}\qMeixnerPollaczek{n}@{\cos@{\ln@@{q^{-x}}+\phi}}{q^{\lambda}}{q} =\MeixnerPollaczek{\lambda}{n}@{x}{-\phi} }$