Continuous q-Laguerre

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Continuous q-Laguerre

Little q-Laguerre / Wall >>

Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\ctsqLaguerre{\alpha}{n}@{x}{q}=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\,{{}_{3}\phi_{2}}\!\left({q^{-n},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-\iunit\theta}\atop q^{\alpha+1},0};q,q\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\ctsqLaguerre{\alpha}{n}@{x}{q}=\frac{\qPochhammer{q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{-\iunit\theta}}{q}{n}}{\qPochhammer{q}{q}{n}}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}\expe^{\iunit n\theta}{}{{}_{2}\phi_{1}}\!\left({q^{-n},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta}\atop q^{-\frac{1}{2}\alpha+\frac{1}{4}-n}\expe^{\iunit\theta}};q,q^{-\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-\iunit\theta}\right)}}}$

Orthogonality relation(s)

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

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

${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}}$ &

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

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle 2x\ctsqLaguerre{\alpha}{n}@{x}{q}=q^{-\frac{1}{2}\alpha-\frac{1}{4}}(1-q^{n+1})\ctsqLaguerre{\alpha}{n+1}@{x}{q}{}+q^{n+\frac{1}{2}\alpha+\frac{1}{4}}(1+q^{\frac{1}{2}})\ctsqLaguerre{\alpha}{n}@{x}{q}{}+q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{n+\alpha})\ctsqLaguerre{\alpha}{n-1}@{x}{q}}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x\monicctsqLaguerre{\alpha}{n}@@{x}{q}=\monicctsqLaguerre{\alpha}{n+1}@@{x}{q}+\frac{1}{2}q^{n+\frac{1}{2}\alpha+\frac{1}{4}}(1+q^{\frac{1}{2}})\monicctsqLaguerre{\alpha}{n}@@{x}{q}{}+\frac{1}{4}(1-q^{n})(1-q^{n+\alpha})\monicctsqLaguerre{\alpha}{n-1}@@{x}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\ctsqLaguerre{\alpha}{n}@{x}{q}=\frac{2^{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}{\qPochhammer{q}{q}{n}}\monicctsqLaguerre{\alpha}{n}@@{x}{q}}}}$

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

${\displaystyle{\displaystyle{\displaystyle y(x)=\ctsqLaguerre{\alpha}{n}@{x}{q}}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;q^{\alpha}|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta}q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta}}{q}{\infty}}\right|^{2}=\left|\frac{\qPochhammer{\expe^{\iunit\theta},-\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}}{\qPochhammer{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}}}$ &

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

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\delta_{q}\ctsqLaguerre{\alpha}{n}@{x}{q}=-q^{-n+\frac{1}{2}\alpha+\frac{3}{4}}({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}})\ctsqLaguerre{\alpha+1}{n-1}@{x}{q}}}}$

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

${\displaystyle{\displaystyle{\displaystyle D_{q}\ctsqLaguerre{\alpha}{n}@{x}{q}=\frac{2q^{-n+\frac{1}{2}\alpha+\frac{5}{4}}}{1-q}\ctsqLaguerre{\alpha+1}{n-1}@{x}{q}}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;q^{\alpha}|q)\ctsqLaguerre{\alpha}{n}@{x}{q}\right]{}=q^{-\frac{1}{2}\alpha-\frac{1}{4}}(1-q^{n+1})(\expe^{\iunit\theta}-\expe^{-\iunit\theta}){}{\tilde{w}}(x;q^{\alpha-1}|q)\ctsqLaguerre{\alpha-1}{n+1}@{x}{q}}}}$

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

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

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

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

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

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

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

Rodrigues-type formula

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

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

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

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

Generating functions

${\displaystyle{\displaystyle{\displaystyle\frac{\qPochhammer{q^{\alpha+\frac{1}{2}}t,q^{\alpha+1}t}{q}{\infty}}{\qPochhammer{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta}t,q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-\iunit\theta}t}{q}{\infty}}=\sum_{n=0}^{\infty}\ctsqLaguerre{\alpha}{n}@{x}{q}t^{n}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\,{{}_{2}\phi_{1}}\!\left({q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta}q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta}\atop q^{\alpha+1}};q,\expe^{-\iunit\theta}t\right){}=\sum_{n=0}^{\infty}\frac{\ctsqLaguerre{\alpha}{n}@{x}{q}t^{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{t}{q}{\infty}\cdot{{}_{2}\phi_{1}}\!\left({q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta}q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-\iunit\theta}\atop q^{\alpha+1}};q,t\right){}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\ctsqLaguerre{\alpha}{n}@{x}{q}t^{n}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\frac{\qPochhammer{\gamma\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ {{}_{3}\phi_{2}}\!\left({\gamma,q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta}\atop q^{\alpha+1},\gamma\expe^{\iunit\theta}t};q,\expe^{-\iunit\theta}t\right){}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\frac{\ctsqLegendre{n}@{x}{q}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}t^{n}}}}$

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

Limit relations

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

${\displaystyle{\displaystyle{\displaystyle\AlSalamChihara{n}@{x}{q^{\frac{1}{2}\alpha+\frac{1}{4}}}{q^{\frac{1}{2}\alpha+\frac{3}{4}}}{q}=\frac{\qPochhammer{q}{q}{n}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}\ctsqLaguerre{\alpha}{n}@{x}{q}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\qMeixnerPollaczek{n}@{\cos\!\left(\theta+\phi\right)}{q^{\frac{1}{2}\alpha+\frac{1}{2}}}{q}=q^{-(\frac{1}{2}\alpha+\frac{1}{4})n}\ctsqLaguerre{\alpha}{n}@{\cos\theta}{q}}}}$

Continuous q-Jacobi polynomial to Continuous q-Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow\infty}\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}=\ctsqLaguerre{\alpha}{n}@{x}{q}}}}$

Continuous q-Laguerre polynomial to Continuous q-Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\frac{\ctsqLaguerre{\alpha}{n}@{x}{q}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}=\frac{\ctsqHermite{n}@{x}{q}}{\qPochhammer{q}{q}{n}}}}}$

Continuous q-Laguerre polynomial to Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\ctsqLaguerre{\alpha}{n}@{q^{x}}{q}=\Laguerre[\alpha]{n}@{2x}}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle\MeixnerPollaczek{\alpha}{n}@{x}{q}=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\,{{}_{3}\phi_{2}}\!\left({q^{-n},q^{\frac{1}{2}}\expe^{\iunit\theta},q^{\frac{1}{2}}\expe^{-\iunit\theta}\atop q^{\alpha+1},-q};q,q\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\ctsqLaguerre{\alpha}{n}@{x}{q^{2}}=q^{n\alpha}\MeixnerPollaczek{\alpha}{n}@{x}{q}}}}$

Reference

<< Continuous big q-Hermite

Continuous q-Laguerre

Little q-Laguerre / Wall >>