Continuous q-Hahn

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Continuous q-Hahn

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Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\frac{(a{\mathrm{e}^{\mathrm{i}\phi}})^{n}p_{n}\!\left(x;a,b,c,d;q\right)}{\left(ab{\mathrm{e}^{2\mathrm{i}\phi}},ac,ad;q\right)_{n}}{}=\qHyperrphis{4}{3}@@{q^{-n},abcdq^{n-1},a{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{ab{\mathrm{e}^{2\mathrm{i}\phi}},ac,ad}{q}{q}}}}$

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

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\frac{1}{4\pi}\int_{-\pi}^{\pi}w(\cos\left(\theta+\phi\right))p_{m}\!\left(\cos\left(\theta+\phi\right);a,b,c,d;q\right)p_{n}\!\left(\cos\left(\theta+\phi\right);a,b,c,d;q\right)\,d\theta{}=\frac{\left(abcdq^{n-1};q\right)_{n}\left(abcdq^{2n};q\right)_{\infty}}{\left(q^{n+1},abq^{n}{\mathrm{e}^{2\mathrm{i}\phi}},acq^{n},adq^{n},bcq^{n},bdq^{n},cdq^{n}{\mathrm{e}^{-2\mathrm{i}\phi}};q\right)_{\infty}}\,\delta_{m,n}}}}$

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

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

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

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle 2x{\tilde{p}}_{n}\!\left(x\right)=A_{n}{\tilde{p}}_{n+1}\!\left(x\right)+\left[a{\mathrm{e}^{\mathrm{i}\phi}}+a^{-1}{\mathrm{e}^{-\mathrm{i}\phi}}-\left(A_{n}+C_{n}\right)\right]{\tilde{p}}_{n}\!\left(x\right)+C_{n}{\tilde{p}}_{n-1}\!\left(x\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{a{\mathrm{e}^{\mathrm{i}\phi}}(1-q^{n})(1-bcq^{n-1})(1-bdq^{n-1})(1-cd{\mathrm{e}^{-2\mathrm{i}\phi}}q^{n-1})}{(1-abcdq^{2n-2})(1-abcdq^{2n-1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-ab{\mathrm{e}^{2\mathrm{i}\phi}}q^{n})(1-acq^{n})(1-adq^{n})(1-abcdq^{n-1})}{a{\mathrm{e}^{\mathrm{i}\phi}}(1-abcdq^{2n-1})(1-abcdq^{2n})}}}}$

${\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}\!\left(x\right):={\tilde{p}}_{n}\!\left(x;a,b,c,d;q\right)=\frac{(a{\mathrm{e}^{\mathrm{i}\phi}})^{n}p_{n}\!\left(x;a,b,c,d;q\right)}{\left(ab{\mathrm{e}^{2\mathrm{i}\phi}},ac,ad;q\right)_{n}}}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{p}}_{n}\!\left(x\right)={\widehat{p}}_{n+1}\!\left(x\right)+\frac{1}{2}\left[a{\mathrm{e}^{\mathrm{i}\phi}}+a^{-1}{\mathrm{e}^{-\mathrm{i}\phi}}-(A_{n}+C_{n})\right]{\widehat{p}}_{n}\!\left(x\right){}+\frac{1}{4}A_{n-1}C_{n}{\widehat{p}}_{n-1}\!\left(x\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{a{\mathrm{e}^{\mathrm{i}\phi}}(1-q^{n})(1-bcq^{n-1})(1-bdq^{n-1})(1-cd{\mathrm{e}^{-2\mathrm{i}\phi}}q^{n-1})}{(1-abcdq^{2n-2})(1-abcdq^{2n-1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-ab{\mathrm{e}^{2\mathrm{i}\phi}}q^{n})(1-acq^{n})(1-adq^{n})(1-abcdq^{n-1})}{a{\mathrm{e}^{\mathrm{i}\phi}}(1-abcdq^{2n-1})(1-abcdq^{2n})}}}}$

${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,c,d;q\right)=2^{n}\left(abcdq^{n-1};q\right)_{n}{\widehat{p}}_{n}\!\left(x\right)}}}$

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle(1-q)^{2}D_{q}\left[{\tilde{w}}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}}cq^{\frac{1}{2}},dq^{\frac{1}{2}};q)D_{q}y(x)\right]{}+\lambda_{n}{\tilde{w}}(x;a,b,c,d;q)y(x)=0}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=p_{n}\!\left(x;a,b,c,d;q\right)}}}$ &

${\displaystyle{\displaystyle{\displaystyle\lambda_{n}=4q^{-n+1}(1-q^{n})(1-abcdq^{n-1})}}}$ &
${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c,d;q):=\frac{w(x;a,b,c,d;q)}{\sqrt{1-x^{2}}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c,d;q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}},b{\mathrm{e}^{\mathrm{i}(\theta+2\phi)}}c{\mathrm{e}^{\mathrm{i}\theta}},d{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a{\mathrm{e}^{\mathrm{i}\phi}})h(x,b{\mathrm{e}^{\mathrm{i}\phi}})h(x,c{\mathrm{e}^{-\mathrm{i}\phi}})h(x,d{\mathrm{e}^{-\mathrm{i}\phi}})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}(\theta+\phi)}},\alpha{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}};q\right)_{\infty}}}}$ &

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

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\delta_{q}p_{n}\!\left(x;a,b,c,d;q\right){}=-q^{-\frac{1}{2}n}(1-q^{n})(1-abcdq^{n-1})({\mathrm{e}^{\mathrm{i}(\theta+\phi)}}-{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}}){}p_{n-1}\!\left(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}},cq^{\frac{1}{2}},dq^{\frac{1}{2}};q\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle D_{q}p_{n}\!\left(x;a,b,c,d;q\right)=2q^{-\frac{1}{2}(n-1)}\frac{(1-q^{n})(1-abcdq^{n-1})}{1-q}{}p_{n-1}\!\left(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}},cq^{\frac{1}{2}},dq^{\frac{1}{2}};q\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;a,b,c,d;q)p_{n}\!\left(x;a,b,c,d;q\right)\right]{}=q^{-\frac{1}{2}(n+1)}({\mathrm{e}^{\mathrm{i}(\theta+\phi)}}-{\mathrm{e}^{-\mathrm{i}(\theta+\phi)}}){\tilde{w}}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}};q){}p_{n+1}\!\left(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}};q\right)}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x;a,b,c,d;q)p_{n}\!\left(x;a,b,c,d;q\right)\right]{}=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde{w}}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}};q){}p_{n+1}\!\left(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}};q\right)}}}$

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

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

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

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c,d;q)p_{n}\!\left(x;a,b,c,d;q\right){}=\left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}\left(D_{q}\right)^{n}\left[{\tilde{w}}(x;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n},cq^{\frac{1}{2}n},dq^{\frac{1}{2}n};q)\right]}}}$

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

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

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

Generating functions

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

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

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

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

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

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

Limit relations

Askey-Wilson polynomial to Continuous q-Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(\cos\left(\theta+\phi\right);a{\mathrm{e}^{\mathrm{i}\phi}},b{\mathrm{e}^{\mathrm{i}\phi}},c{\mathrm{e}^{-\mathrm{i}\phi}},d{\mathrm{e}^{-\mathrm{i}\phi}}\,|\,q\right)=p_{n}\!\left(\cos\left(\theta+\phi\right);a,b,c,d;q\right)}}}$

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

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

Continuous q-Hahn polynomial to Continuous Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{p_{n}\!\left(\cos\left(\ln q^{-x}+\phi\right);q^{a},q^{b},q^{c},q^{d};q\right)}{(1-q)^{n}\left(q;q\right)_{n}}=(-2\sin\phi)^{n}p_{n}\!\left(x;a,b,c,d\right)}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}\!\left(x;a,b,c,d;q^{-1}\right)={\tilde{p}}_{n}\!\left(x;a^{-1},b^{-1},c^{-1},d^{-1};q\right)}}}$

Koornwinder Addendum: Continuous q-Hahn

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

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Continuous q-Hahn

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