Continuous q-Laguerre

<< Continuous big q-Hermite

Continuous q-Laguerre

Little q-Laguerre / Wall >>

Basic hypergeometric representation

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

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

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

Orthogonality relation(s)

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

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

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

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

Recurrence relation

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

Monic recurrence relation

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

q-Difference equations

${\displaystyle{\displaystyle{\displaystyle(1-q)^{2}D_{q}\left[{\tilde{w}}(x;q^{\alpha+1}|q)D_{q}y(x)\right]+4q^{-n+1}(1-q^{n}){\tilde{w}}(x;q^{\alpha}|q)y(x)=0}}}$

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

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

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

Forward shift operator

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

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

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

Backward shift operator

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

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

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

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

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

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

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

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

Rodrigues-type formula

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

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

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

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

Generating functions

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

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

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

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

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

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

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

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

Limit relations

Al-Salam-Chihara polynomial to Continuous q-Laguerre polynomial

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

q-Meixner-Pollaczek polynomial to Continuous q-Laguerre polynomial

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

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-Laguerre polynomial to Continuous q-Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\frac{P^{(\alpha)}_{n}\!\left(x|q\right)}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}=\frac{H_{n}\!\left(x\,|\,q\right)}{\left(q;q\right)_{n}}}}}$

Continuous q-Laguerre polynomial to Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}P^{(\alpha)}_{n}\!\left(q^{x}|q\right)=L^{\alpha}_{n}\left(2x\right)}}}$

Remark

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

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

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}\!\left(x|q^{2}\right)=q^{n\alpha}P^{(\alpha)}_{n}\!\left(x;q\right)}}}$

Reference

<< Continuous big q-Hermite

Continuous q-Laguerre

Little q-Laguerre / Wall >>