q-Meixner-Pollaczek

<< Al-Salam-Chihara

q-Meixner-Pollaczek

Continuous q-Jacobi >>

Basic hypergeometric representation

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

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}w(\cos\left(\theta+\phi\right);a|q)P_{m}\!\left(\cos\left(\theta+\phi\right);a|q\right)P_{n}\!\left(\cos\left(\theta+\phi\right);a|q\right)\,d\theta{}=\frac{\,\delta_{m,n}}{\left(q;q\right)_{n}\left(q,a^{2}q^{n};q\right)_{\infty}}}}}$

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

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

${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}(\theta+\phi)}},\alpha{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}}}$ &

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

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle 2xP_{n}\!\left(x;a|q\right)=(1-q^{n+1})P_{n+1}\!\left(x;a|q\right)+2aq^{n}\cos\phi P_{n}\!\left(x;a|q\right){}+(1-a^{2}q^{n-1})P_{n-1}\!\left(x;a|q\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\!\left(x\right)={\widehat{P}}_{n+1}\!\left(x\right)+aq^{n}\cos\phi\,{\widehat{P}}_{n}\!\left(x\right)+\frac{1}{4}(1-q^{n})(1-a^{2}q^{n-1}){\widehat{P}}_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a|q\right)=\frac{2^{n}}{\left(q;q\right)_{n}}{\widehat{P}}_{n}\!\left(x\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle y(x)=P_{n}\!\left(x;a|q\right)}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x;a|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}},a{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a{\mathrm{e}^{\mathrm{i}\phi}})h(x,a{\mathrm{e}^{-\mathrm{i}\phi}})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}(\theta+\phi)}},\alpha{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}}}$ &

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

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\delta_{q}P_{n}\!\left(x;a|q\right)=-q^{-\frac{1}{2}n}({\mathrm{e}^{\mathrm{i}(\theta+\phi)}}-{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}})P_{n-1}\!\left(x;aq^{\frac{1}{2}}|q\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle D_{q}P_{n}\!\left(x;a|q\right)=\frac{2q^{-\frac{1}{2}(n-1)}}{1-q}P_{n-1}\!\left(x;aq^{\frac{1}{2}}|q\right)}}}$

Backward shift operator

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

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

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

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

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

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

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

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

Rodrigues-type formula

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

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

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

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

Generating functions

${\displaystyle{\displaystyle{\displaystyle\left|\frac{\left(a{\mathrm{e}^{\mathrm{i}\phi}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}(\theta+\phi)}}t;q\right)_{\infty}}\right|^{2}{}=\frac{\left(a{\mathrm{e}^{\mathrm{i}\phi}}t,a{\mathrm{e}^{-\mathrm{i}\phi}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}(\theta+\phi)}}t,{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}}t;q\right)_{\infty}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\left|\frac{\left(a{\mathrm{e}^{\mathrm{i}\phi}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}(\theta+\phi)}}t;q\right)_{\infty}}\right|^{2}{}=\sum_{n=0}^{\infty}P_{n}\!\left(x;a|q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{1}{\left({\mathrm{e}^{\mathrm{i}(\theta+\phi)}}t;q\right)_{\infty}}\ \qHyperrphis{2}{1}@@{a{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}},a{\mathrm{e}^{\mathrm{i}\theta}}}{a^{2}}{q}{{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}}t}{}=\sum_{n=0}^{\infty}\frac{P_{n}\!\left(x;a|q\right)}{\left(a^{2};q\right)_{n}}t^{n}}}}$

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

Limit relations

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

${\displaystyle{\displaystyle{\displaystyle\frac{p_{n}\!\left(\cos\left(\theta+\phi\right);a,0,0,a;q\right)}{\left(q;q\right)_{n}}=P_{n}\!\left(\cos\left(\theta+\phi\right);a|q\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\par P_{n}\!\left(\cos\phi;\beta|q\right)=C_{n}\!\left(\cos\phi;\beta\,|\,q\right)}}}$

q-Meixner-Pollaczek polynomial to Continuous

${\displaystyle{\displaystyle{\displaystyle\par P_{n}\!\left(\cos\left(\theta+\phi\right);q^{\frac{1}{2}\alpha+\frac{1}{2}}|q\right)=q^{-(\frac{1}{2}\alpha+\frac{1}{4})n}P^{(\alpha)}_{n}\!\left(\cos\theta|q\right)}}}$

q-Meixner-Pollaczek polynomial to Meixner-Pollaczek polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(\cos\left(\ln q^{-x}+\phi\right);q^{\lambda}|q\right)=P^{(\lambda)}_{n}\!\left(x;-\phi\right)}}}$

<< Al-Salam-Chihara

q-Meixner-Pollaczek

Continuous q-Jacobi >>