q-Meixner-Pollaczek

<< Al-Salam-Chihara

q-Meixner-Pollaczek

Continuous q-Jacobi >>

Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\qMeixnerPollaczek{n}@{x}{a}{q}=a^{-n}{\mathrm{e}^{-\mathrm{i}n\phi}}\frac{\left(a^{2};q\right)_{n}}{\left(q;q\right)_{n}}\ \qHyperrphis{3}{2}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{a^{2},0}{q}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\qMeixnerPollaczek{n}@{x}{a}{q}=\frac{\qPochhammer{a\expe^{-\iunit\theta}}{q}{n}}{\qPochhammer{q}{q}{n}}\expe^{\iunit n(\theta+\phi)}\ {{}_{2}\phi_{1}}\!\left({q^{-n},a\expe^{\iunit\theta}\atop a^{-1}q^{-n+1}\expe^{\iunit\theta}};q,qa^{-1}\expe^{-\iunit(\theta+2\phi)}\right)}}}$

Orthogonality relation(s)

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<a<1}}}$

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

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

${\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}$

Recurrence relation

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

Monic recurrence relation

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

q-Difference equation

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

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

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

${\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}$

Forward shift operator

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

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

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

Backward shift operator

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

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

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

${\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}$

Rodrigues-type formula

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

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

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

${\displaystyle{\displaystyle{\displaystyle x=\cos\left(\theta+\phi\right)}}}$

Generating functions

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

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

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

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

Limit relations

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

${\displaystyle{\displaystyle{\displaystyle\frac{\ctsqHahn{n}@{\cos\!\left(\theta+\phi\right)}{a}{0}{0}{a}{q}}{\qPochhammer{q}{q}{n}}=\qMeixnerPollaczek{n}@{\cos\!\left(\theta+\phi\right)}{a}{q}}}}$

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

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

q-Meixner-Pollaczek polynomial to Continuous

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

q-Meixner-Pollaczek polynomial to Meixner-Pollaczek polynomial

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

<< Al-Salam-Chihara

q-Meixner-Pollaczek

Continuous q-Jacobi >>