Askey-Wilson

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Askey-Wilson

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Basic hypergeometric representation

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

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

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x)}{\sqrt{1-x^{2}}}p_{m}\!\left(x;a,b,c,d\,|\,q\right)p_{n}\!\left(x;a,b,c,d\,|\,q\right)\,dx=h_{n}\,\delta_{m,n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle h_{n}=\frac{\left(abcdq^{n-1};q\right)_{n}\left(abcdq^{2n};q\right)_{\infty}}{\left(q^{n+1},abq^{n},acq^{n},adq^{n},bcq^{n},bdq^{n},cdq^{n};q\right)_{\infty}}}}}$ &

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

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

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x)}{\sqrt{1-x^{2}}}p_{m}\!\left(x;a,b,c,d\,|\,q\right)p_{n}\!\left(x;a,b,c,d\,|\,q\right)\,dx{}+\sum_{\begin{array}[]{c}\scriptstyle k\\ \scriptstyle 1<aq^{k}\leq a\end{array}}w_{k}p_{m}\!\left(x_{k};a,b,c,d\,|\,q\right)p_{n}\!\left(x_{k};a,b,c,d\,|\,q\right)=h_{n}\,\delta_{m,n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w_{k}=\frac{\left(a^{-2};q\right)_{\infty}}{\left(q,ab,ac,ad,a^{-1}b,a^{-1}c,a^{-1}d;q\right)_{\infty}}{}\frac{(1-a^{2}q^{2k})\left(a^{2},ab,ac,ad;q\right)_{k}}{(1-a^{2})\left(q,ab^{-1}q,ac^{-1}q,ad^{-1}q;q\right)_{k}}\left(\frac{q}{abcd}\right)^{k}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x_{k}=\frac{aq^{k}+\left(aq^{k}\right)^{-1}}{2}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h_{n}=\frac{\left(abcdq^{n-1};q\right)_{n}\left(abcdq^{2n};q\right)_{\infty}}{\left(q^{n+1},abq^{n},acq^{n},adq^{n},bcq^{n},bdq^{n},cdq^{n};q\right)_{\infty}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c,d|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}c{\mathrm{e}^{\mathrm{i}\theta}},d{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)h(x,d)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

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

Recurrence relation

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{a(1-q^{n})(1-bcq^{n-1})(1-bdq^{n-1})(1-cdq^{n-1})}{(1-abcdq^{2n-2})(1-abcdq^{2n-1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-abq^{n})(1-acq^{n})(1-adq^{n})(1-abcdq^{n-1})}{a(1-abcdq^{2n-1})(1-abcdq^{2n})}}}}$

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

Monic recurrence relation

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{a(1-q^{n})(1-bcq^{n-1})(1-bdq^{n-1})(1-cdq^{n-1})}{(1-abcdq^{2n-2})(1-abcdq^{2n-1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-abq^{n})(1-acq^{n})(1-adq^{n})(1-abcdq^{n-1})}{a(1-abcdq^{2n-1})(1-abcdq^{2n})}}}}$

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

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

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

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

${\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})(1-abcdq^{n-1}){\mathcal{P}}_{n}(z){}=A(z){\mathcal{P}}_{n}(qz)-\left[A(z)+A(z^{-1})\right]{\mathcal{P}}_{n}(z)+A(z^{-1}){\mathcal{P}}_{n}(q^{-1}z)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle A(z)=\frac{(1-az)(1-bz)(1-cz)(1-dz)}{(1-z^{2})(1-qz^{2})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle{\mathcal{P}}_{n}(z):=\frac{\left(ab,ac,ad;q\right)_{n}}{a^{n}}\,\qHyperrphis{4}{3}@@{q^{-n},abcdq^{n-1},az,az^{-1}}{ab,ac,ad}{q}{q}}}}$

Forward shift operator

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

Backward shift operator

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

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

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

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

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

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

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

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

Rodrigues-type formula

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

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

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

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

Generating functions

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

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

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

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

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

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

Limit relations

Askey-Wilson polynomial to Continuous dual q-Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,c,0\,|\,q\right)=p_{n}\!\left(x;a,b,c|q\right)}}}$

Askey-Wilson polynomial to Continuous q-Hahn polynomial

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

Askey-Wilson polynomial to Big q-Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}\!\left(x;a,b,c,d\,|\,q\right)=\frac{a^{n}p_{n}\!\left(x;a,b,c,d\,|\,q\right)}{\left(ab,ac,ad;q\right)_{n}}}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow 0}{\tilde{p}}_{n}\!\left(\textstyle\frac{1}{2}a^{-1}x;a,a^{-1}\alpha q,a^{-1}\gamma q,a\beta\gamma^{-1}\,|\,q\right)=P_{n}\!\left(x;\alpha,\beta,\gamma;q\right)}}}$

Askey-Wilson polynomial to Continuous q-Jacobi polynomial

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

Askey-Wilson polynomial to Continuous q-ultraspherical / Rogers polynomial

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

Askey-Wilson polynomial to Wilson polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{p_{n}\!\left(\frac{1}{2}\left(q^{\mathrm{i}x}+q^{-\mathrm{i}x}\right);q^{a},q^{b},q^{c},q^{d}\,|\,q\right)}{(1-q)^{3n}}=W_{n}\!\left(x^{2};a,b,c,d\right)}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right){}=\frac{(\gamma\delta q)^{\frac{1}{2}n}p_{n}\!\left(\nu(x);\gamma^{\frac{1}{2}}\delta^{\frac{1}{2}}q^{\frac{1}{2}},\alpha\gamma^{-\frac{1}{2}}\delta^{-\frac{1}{2}}q^{\frac{1}{2}},\beta\gamma^{-\frac{1}{2}}\delta^{\frac{1}{2}}q^{\frac{1}{2}},\gamma^{\frac{1}{2}}\delta^{-\frac{1}{2}}q^{\frac{1}{2}}\,|\,q\right)}{\left(\alpha q,\beta\delta q,\gamma q;q\right)_{n}}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\nu(x)=\textstyle\frac{1}{2}\gamma^{\frac{1}{2}}\delta^{\frac{1}{2}}q^{x+\frac{1}{2}}+\textstyle\frac{1}{2}\gamma^{-\frac{1}{2}}\delta^{-\frac{1}{2}}q^{-x-\frac{1}{2}}}}}$

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

Koornwinder Addendum: Askey-Wilson

<< Orthogonal Polynomials

Askey-Wilson

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