Continuous q-Jacobi

<< q-Meixner-Pollaczek

Continuous q-Jacobi

Continuous q-Jacobi: Special cases >>

Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x|q\right){}=\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}}\ \qHyperrphis{4}{3}@@{q^{-n},q^{n+\alpha+\beta+1},q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}{q^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{q}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x)}{\sqrt{1-x^{2}}}P^{(\alpha,\beta)}_{m}\!\left(x|q\right)P^{(\alpha,\beta)}_{n}\!\left(x|q\right)\,dx{}=\frac{\left(q^{\frac{1}{2}(\alpha+\beta+2)},q^{\frac{1}{2}(\alpha+\beta+3)};q\right)_{\infty}}{\left(q,q^{\alpha+1},q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+1)}-q^{\frac{1}{2}(\alpha+\beta+2)};q\right)_{\infty}}\,\frac{1-q^{\alpha+\beta+1}}{1-q^{2n+\alpha+\beta+1}}{}\frac{\left(q^{\alpha+1},q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+3)};q\right)_{n}}{\left(q,q^{\alpha+\beta+1},-q^{\frac{1}{2}(\alpha+\beta+1)};q\right)_{n}}q^{(\alpha+\frac{1}{2})n}\,\delta_{m,n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{3}{4}})}}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle 2x{\tilde{P}}^{(n,\beta)}_{n}\!\left(x|q\right)=A_{n}{\tilde{P}}^{(n+1,\beta)}_{n}\!\left(x|q\right)+\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+q^{-\frac{1}{2}\alpha-\frac{1}{4}}-\left(A_{n}+C_{n}\right)\right]{\tilde{P}}^{(n,\beta)}_{n}\!\left(x|q\right){}+C_{n}{\tilde{P}}^{(n-1,\beta)}_{n}\!\left(x|q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{n})(1-q^{n+\beta})(1+q^{n+\frac{1}{2}(\alpha+\beta)})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})}{(1-q^{2n+\alpha+\beta})(1-q^{2n+\alpha+\beta+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+\alpha+1})(1-q^{n+\alpha+\beta+1})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})(1+q^{n+\frac{1}{2}(\alpha+\beta+2)})}{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{2n+\alpha+\beta+1})(1-q^{2n+\alpha+\beta+2})}}}}$

${\displaystyle{\displaystyle{\displaystyle{\tilde{P}}^{(n,\beta)}_{n}\!\left(x|q\right):={\tilde{P}}^{(\alpha,\beta)}_{n}\!\left(x|q\right)=\frac{\left(q;q\right)_{n}}{\left(q^{\alpha+1};q\right)_{n}}P^{(\alpha,\beta)}_{n}\!\left(x|q\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}^{(\alpha,\beta)}_{n}\!\left(x\right)={\widehat{P}}^{(\alpha,\beta)}_{n+1}\!\left(x\right)+\frac{1}{2}\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+q^{-\frac{1}{2}\alpha-\frac{1}{4}}-(A_{n}+C_{n})\right]{\widehat{P}}^{(\alpha,\beta)}_{n}\!\left(x\right){}+\frac{1}{4}A_{n-1}C_{n}{\widehat{P}}^{(\alpha,\beta)}_{n-1}\!\left(x\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{n})(1-q^{n+\beta})(1+q^{n+\frac{1}{2}(\alpha+\beta)})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})}{(1-q^{2n+\alpha+\beta})(1-q^{2n+\alpha+\beta+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+\alpha+1})(1-q^{n+\alpha+\beta+1})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})(1+q^{n+\frac{1}{2}(\alpha+\beta+2)})}{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{2n+\alpha+\beta+1})(1-q^{2n+\alpha+\beta+2})}}}}$

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x|q\right)=\frac{2^{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}\left(q^{n+\alpha+\beta+1};q\right)_{n}}{\left(q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)};q\right)_{n}}{\widehat{P}}^{(\alpha,\beta)}_{n}\!\left(x\right)}}}$

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle(1-q)^{2}D_{q}\left[{\tilde{w}}(x;q^{\alpha+1},q^{\beta+1}|q)D_{q}y(x)\right]+\lambda_{n}{\tilde{w}}(x;q^{\alpha},q^{\beta}|q)y(x)=0}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\lambda_{n}=4q^{-n+1}(1-q^{n})(1-q^{n+\alpha+\beta+1})}}}$ &

${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;q^{\alpha},q^{\beta}|q):=\frac{w(x;q^{\alpha},q^{\beta}|q)}{\sqrt{1-x^{2}}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle y(x)=P^{(\alpha,\beta)}_{n}\!\left(x|q\right)}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{3}{4}})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\delta_{q}P^{(\alpha,\beta)}_{n}\!\left(x|q\right)=-\frac{q^{-n+\frac{1}{2}\alpha+\frac{3}{4}}(1-q^{n+\alpha+\beta+1})({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}})}{(1+q^{\frac{1}{2}(\alpha+\beta+1)})(1+q^{\frac{1}{2}(\alpha+\beta+2)})}{}P_{n-1}^{(\alpha+1,\beta+1)}(x|q)}}}$

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

${\displaystyle{\displaystyle{\displaystyle D_{q}P^{(\alpha,\beta)}_{n}\!\left(x|q\right)=\frac{2q^{-n+\frac{1}{2}\alpha+\frac{5}{4}}(1-q^{n+\alpha+\beta+1})}{(1-q)(1+q^{\frac{1}{2}(\alpha+\beta+1)})(1+q^{\frac{1}{2}(\alpha+\beta+2)})}{}P_{n-1}^{(\alpha+1,\beta+1)}(x|q)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;q^{\alpha},q^{\beta}|q)P^{(\alpha,\beta)}_{n}\!\left(x|q\right)\right]{}=q^{-\frac{1}{2}\alpha-\frac{1}{4}}(1-q^{n+1})(1+q^{\frac{1}{2}(\alpha+\beta-1)})(1+q^{\frac{1}{2}(\alpha+\beta)})({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}}){}{\tilde{w}}(x;q^{\alpha-1},q^{\beta-1}|q)P_{n+1}^{(\alpha-1,\beta-1)}(x|q)}}}$

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

${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{3}{4}})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

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

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

${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{3}{4}})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;q^{\alpha},q^{\beta}|q)P^{(\alpha,\beta)}_{n}\!\left(x|q\right){}=\left(\frac{q-1}{2}\right)^{n}\frac{q^{\frac{1}{4}n^{2}+\frac{1}{2}n\alpha}}{\left(q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)};q\right)_{n}}{}\left(D_{q}\right)^{n}\left[{\tilde{w}}(x;q^{\alpha+n},q^{\beta+n}|q)\right]}}}$

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

${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-q^{\frac{1}{2}\beta+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}{\left(q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{3}{4}})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}}}{q^{\alpha+1}}{q}{{\mathrm{e}^{-\mathrm{i}\theta}}t}\ \qHyperrphis{2}{1}@@{-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{3}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}{q^{\beta+1}}{q}{{\mathrm{e}^{\mathrm{i}\theta}}t}{}=\sum_{n=0}^{\infty}\frac{\left(-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)};q\right)_{n}}{\left(q^{\alpha+1},q^{\beta+1};q\right)_{n}}\frac{P^{(\alpha,\beta)}_{n}\!\left(x|q\right)}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}t^{n}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}}{-q^{\frac{1}{2}(\alpha+\beta+1)}}{q}{{\mathrm{e}^{-\mathrm{i}\theta}}t}\ \qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{3}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}{-q^{\frac{1}{2}(\alpha+\beta+3)}}{q}{{\mathrm{e}^{\mathrm{i}\theta}}t}{}=\sum_{n=0}^{\infty}\frac{\left(-q^{\frac{1}{2}(\alpha+\beta+2)};q\right)_{n}}{\left(-q^{\frac{1}{2}(\alpha+\beta+3)};q\right)_{n}}\frac{P^{(\alpha,\beta)}_{n}\!\left(x|q\right)}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}t^{n}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}}}{-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{{\mathrm{e}^{-\mathrm{i}\theta}}t}\ \qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{3}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}-q^{\frac{1}{2}\beta+\frac{1}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}{-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{{\mathrm{e}^{\mathrm{i}\theta}}t}{}=\sum_{n=0}^{\infty}\frac{\left(-q^{\frac{1}{2}(\alpha+\beta+1)};q\right)_{n}}{\left(-q^{\frac{1}{2}(\alpha+\beta+2)};q\right)_{n}}\frac{P^{(\alpha,\beta)}_{n}\!\left(x|q\right)}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}t^{n}}}}$

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

Limit relations

Askey-Wilson polynomial to Continuous q-Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle\frac{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}p_{n}\!\left(x;q^{\frac{1}{2}\alpha+\frac{1}{4}},q^{\frac{1}{2}\alpha+\frac{3}{4}},-q^{\frac{1}{2}\beta+\frac{1}{4}},-q^{\frac{1}{2}\beta+\frac{3}{4}}\,|\,q\right)}{\left(q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)};q\right)_{n}}=P^{(\alpha,\beta)}_{n}\!\left(x|q\right)}}}$

Continuous q-Jacobi polynomial to Continuous q-Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow\infty}P^{(\alpha,\beta)}_{n}\!\left(x|q\right)=P^{(\alpha)}_{n}\!\left(x|q\right)}}}$

Continuous q-Jacobi polynomial to Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P^{(\alpha,\beta)}_{n}\!\left(x|q\right)=P^{(\alpha,\beta)}_{n}\left(x\right)}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x;q\right)=\frac{\left(q^{\alpha+1},-q^{\beta+1};q\right)_{n}}{\left(q,-q;q\right)_{n}}{}\qHyperrphis{4}{3}@@{q^{-n},q^{n+\alpha+\beta+1},q^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{1}{2}}{\mathrm{e}^{-\mathrm{i}\theta}}}{q^{\alpha+1},-q^{\beta+1},-q}{q}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x|q^{2}\right)=\frac{\left(-q;q\right)_{n}}{\left(-q^{\alpha+\beta+1};q\right)_{n}}q^{n\alpha}P^{(\alpha,\beta)}_{n}\!\left(x;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C_{2n}\!\left(x;q^{\lambda}\,|\,q\right)=\frac{\left(q^{\lambda},-q;q\right)_{n}}{\left(q^{\frac{1}{2}},-q^{\frac{1}{2}};q\right)_{n}}q^{-\frac{1}{2}n}P^{(\lambda-\frac{1}{2},-\frac{1}{2})}_{n}\!\left(2x^{2}-1;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C_{2n+1}\!\left(x;q^{\lambda}\,|\,q\right)=\frac{\left(q^{\lambda},-1;q\right)_{n+1}}{\left(q^{\frac{1}{2}},-q^{\frac{1}{2}};q\right)_{n+1}}q^{-\frac{1}{2}n}xP^{(\lambda-\frac{1}{2},\frac{1}{2})}_{n}\!\left(2x^{2}-1;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x|q^{-1}\right)=q^{-n\alpha}P^{(\alpha,\beta)}_{n}\!\left(x|q\right)\quad\textrm{and}\quad P^{(\alpha,\beta)}_{n}\!\left(x;q^{-1}\right)=q^{-n(\alpha+\beta)}P^{(\alpha,\beta)}_{n}\!\left(x;q\right)}}}$

Koornwinder Addendum: q-Meixner-Pollaczek

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a|q\right):=\frac{1}{\left(q;q\right)_{n}}p_{n}\!\left(x;a{\mathrm{e}^{\mathrm{i}\phi}},0,a{\mathrm{e}^{-\mathrm{i}\phi}},0\,|\,q\right)(x=\cos\left(\theta+\phi\right))}}}$

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Continuous q-Jacobi

Continuous q-Jacobi: Special cases >>