Continuous q-Jacobi: Special cases

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Continuous q-Jacobi: Special cases

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Basic hypergeometric representation

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

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

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

Orthogonality relation(s)

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle|\beta|<1}}}$

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

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

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

Recurrence relation

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

Monic recurrence relation

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

q-Difference equation

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

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

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

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

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\delta_{q}C_{n}\!\left(x;\beta\,|\,q\right)=-q^{-\frac{1}{2}n}(1-\beta)({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}})C_{n-1}\!\left(x;\beta q\,|\,q\right)}}}$

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

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

Backward shift operator

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

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

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

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

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

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

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

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

Rodrigues-type formula

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

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

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

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

Generating functions

${\displaystyle{\displaystyle{\displaystyle\left|\frac{\left(\beta{\mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}\right|^{2}=\frac{\left(\beta{\mathrm{e}^{\mathrm{i}\theta}}t,\beta{\mathrm{e}^{-\mathrm{i}\theta}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}\theta}}t,{\mathrm{e}^{-\mathrm{i}\theta}}t;q\right)_{\infty}}=\sum_{n=0}^{\infty}C_{n}\!\left(x;\beta\,|\,q\right)t^{n}}}}$

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

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

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

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

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

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

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

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

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

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

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

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

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

Limit relations

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

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

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

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

Continuous q-ultraspherical / Rogers polynomial to Continuous q-Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow 0}C_{n}\!\left(x;\beta\,|\,q\right)=\frac{H_{n}\!\left(x\,|\,q\right)}{\left(q;q\right)_{n}}}}}$

Continuous q-ultraspherical / Rogers polynomial to Gegenbauer /

${\displaystyle{\displaystyle{\displaystyle\par\lim_{q\rightarrow 1}C_{n}\!\left(x;q^{\lambda}\,|\,q\right)=C^{\lambda}_{n}\left(x\right)}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;\beta\,|\,q\right)=\sum_{k=0}^{n}\frac{\left(\beta;q\right)_{k}\left(\beta;q\right)_{n-k}}{\left(q;q\right)_{k}\left(q;q\right)_{n-k}}{\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\alpha)}_{n}\!\left(x|q\right)\lx@stackrel{{\scriptstyle q^{\alpha+\frac{1}{2}}\rightarrow\beta}}{{\longrightarrow}}\frac{\left(\beta q^{\frac{1}{2}};q\right)_{n}}{\left(\beta^{2};q\right)_{n}}\beta^{\frac{1}{2}n}C_{n}\!\left(x;\beta\,|\,q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;q^{\alpha+\frac{1}{2}}\,|\,q\right)=\frac{\left(q^{2\alpha+1};q\right)_{n}}{\left(q^{\alpha+1};q\right)_{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}P^{(\alpha,\alpha)}_{n}\!\left(x|q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;\beta\,|\,q^{-1}\right)=(\beta q)^{n}C_{n}\!\left(x;\beta^{-1}\,|\,q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;q\,|\,q\right)=\frac{\sin\left(n+1\right)\theta}{\sin\theta}=U_{n}\left(x\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow 1}\frac{1-q^{n}}{2(1-\beta)}C_{n}\!\left(x;\beta\,|\,q\right)=\cos n\theta=T_{n}\left(x\right),\quad x=\cos\theta}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\ldots}}}$

${\displaystyle{\displaystyle{\displaystyle C_{2n}\!\left(x;q^{\lambda}\,|\,q\right)=\frac{\left(q^{\lambda},-q;q\right)_{n}}{\left(q^{\frac{1}{2}},-q^{\frac{1}{2}};q\right)_{n}}q^{-\frac{1}{2}n}P^{(\lambda-\frac{1}{2},-\frac{1}{2})}_{n}\!\left(2x^{2}-1;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C_{2n+1}\!\left(x;q^{\lambda}\,|\,q\right)=\frac{\left(q^{\lambda},-1;q\right)_{n+1}}{\left(q^{\frac{1}{2}},-q^{\frac{1}{2}};q\right)_{n+1}}q^{-\frac{1}{2}n}xP^{(\lambda-\frac{1}{2},\frac{1}{2})}_{n}\!\left(2x^{2}-1;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(x;q^{\frac{1}{2}}\,|\,q\right)=q^{-\frac{1}{4}n}P_{n}\!\left(x|q\right)}}}$

Koornwinder Addendum: Continuous q-ultraspherical / Rogers

Re: (14.10.17)

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

Continuous q-Jacobi: Special cases: Special value

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

Re: (14.10.21)

${\displaystyle{\displaystyle{\displaystyle\frac{1-\beta z^{2}}{1-z^{2}}C_{n}\!\left[q^{\frac{1}{2}}z;\beta\,|\,q\right]+\frac{1-\beta z^{-2}}{1-z^{-2}}C_{n}\!\left[q^{-\frac{1}{2}}z;\beta\,|\,q\right]=(q^{-\frac{1}{2}n}+q^{\frac{1}{2}n}\beta)C_{n}\!\left[z;\beta\,|\,q\right]}}}$

Re: (14.10.23)

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left[q^{\frac{1}{2}}z;\beta\,|\,q\right]-C_{n}\!\left[q^{-\frac{1}{2}}z;\beta\,|\,q\right]=q^{-\frac{1}{2}n}(\beta-1)(z-z^{-1})C_{n-1}\!\left[z;q\beta\,|\,q\right]}}}$
${\displaystyle{\displaystyle{\displaystyle(\beta+1)C_{n}\!\left[q^{\frac{1}{2}}z;\beta\,|\,q\right]=(q^{-\frac{1}{2}n}+q^{\frac{1}{2}n}\beta)C_{n}\!\left[z;\beta\,|\,q\right]+q^{-\frac{1}{2}n}(\beta-1)(z-\beta z^{-1})C_{n-1}\!\left[z;q\beta\,|\,q\right]}}}$
${\displaystyle{\displaystyle{\displaystyle(\beta+1)C_{n}\!\left[q^{-\frac{1}{2}}z;\beta\,|\,q\right]=(q^{-\frac{1}{2}n}+q^{\frac{1}{2}n}\beta)C_{n}\!\left[z;\beta\,|\,q\right]+q^{-\frac{1}{2}n}(\beta-1)(z^{-1}-\beta z)C_{n-1}\!\left[z;q\beta\,|\,q\right]}}}$

Trigonometric representation

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(\cos\theta;\beta\,|\,q\right)=\sum_{k=0}^{n}\frac{\left(\beta;q\right)_{k}\left(\beta;q\right)_{n-k}}{\left(q;q\right)_{k}\left(q;q\right)_{n-k}}{\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}}$

Limit for q\downarrow-1

${\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{2m}\!\left(x;-q^{\la}\,|\,-q\right)=C^{\frac{1}{2}(\la+1)}_{m}\left(2x^{2}-1\right)+C^{\frac{1}{2}(\la+1)}_{m-1}\left(2x^{2}-1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{2m+1}\!\left(x;-q^{\la}\,|\,-q\right)=2xC^{\frac{1}{2}(\la+1)}_{m}\left(2x^{2}-1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{2m}\!\left(x;-q^{\lambda}\,|\,-q\right)=\frac{{\left(\lambda\right)_{m}}}{{\left(\frac{1}{2}\lambda\right)_{m}}}P^{(\frac{1}{2}\lambda,\frac{1}{2}\lambda-1)}_{m}\left(2x^{2}-1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{2m+1}\!\left(x;-q^{\lambda}\,|\,-q\right)=2\frac{{\left(\lambda+1\right)_{m}}}{{\left(\frac{1}{2}\lambda+1\right)_{m}}}xP^{(\frac{1}{2}\lambda,\frac{1}{2}\lambda)}_{m}\left(2x^{2}-1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{n}\!\left(x;-q^{\lambda}\,|\,-q\right)=\mathrm{c}onstS^{(\frac{1}{2}\lambda,\frac{1}{2}\lambda-1)}_{n}\left(x\right)}}}$

Basic hypergeometric representation

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

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

Orthogonality relation(s)

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

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

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

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

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle 2(1-q^{n+\frac{1}{2}})xP_{n}\!\left(x|q\right)=q^{-\frac{1}{4}}(1-q^{n+1})P_{n+1}\!\left(x|q\right)+q^{\frac{1}{4}}(1-q^{n})P_{n-1}\!\left(x|q\right)}}}$

Monic recurrence relation

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

q-Difference equation

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=P_{n}\!\left(x|q\right)}}}$ &

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

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

Rodrigues-type formula

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

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

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

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

Generating functions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Limit relations

Continuous q-Legendre polynomial to Legendre / Spherical polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P_{n}\!\left(x|q\right)=\LegendrePoly{n}@{x}}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q\right)=q^{\frac{1}{4}n}\sum_{k=0}^{n}\frac{\left(q^{\frac{1}{2}};q\right)_{k}\left(q^{\frac{1}{2}};q\right)_{n-k}}{\left(q;q\right)_{k}\left(q;q\right)_{n-k}}{\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q^{2}\right)=P_{n}\!\left(x;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q^{-1}\right)=P_{n}\!\left(x|q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q\right)=q^{\frac{1}{4}n}C_{n}\!\left(x;q^{\frac{1}{2}}\,|\,q\right)}}}$

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