q-Laguerre

<< Little q-Laguerre / Wall

q-Laguerre

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Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle L^{(\alpha)}_{n}\!\left(x;q\right)=\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}}\ \qHyperrphis{1}{1}@@{q^{-n}}{q^{\alpha+1}}{q}{-q^{n+\alpha+1}x}}}}$
${\displaystyle{\displaystyle{\displaystyle L^{(\alpha)}_{n}\!\left(x;q\right)=\frac{1}{\left(q;q\right)_{n}}\,\qHyperrphis{2}{1}@@{q^{-n},-x}{0}{q}{q^{n+\alpha+1}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}\frac{x^{\alpha}}{\left(-x;q\right)_{\infty}}L^{(\alpha)}_{m}\!\left(x;q\right)L^{(\alpha)}_{n}\!\left(x;q\right)\,dx{}=\frac{\left(q^{-\alpha};q\right)_{\infty}}{\left(q;q\right)_{\infty}}\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}}\Gamma\left(-\alpha\right)\Gamma\left(\alpha+1\right)\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\alpha>-1}}}$

${\displaystyle{\displaystyle{\displaystyle\sum_{k=-\infty}^{\infty}\frac{q^{k\alpha+k}}{\left(-cq^{k};q\right)_{\infty}}L^{(\alpha)}_{m}\!\left(cq^{k};q\right)L^{(\alpha)}_{n}\!\left(cq^{k};q\right){}=\frac{\left(q,-cq^{\alpha+1},-c^{-1}q^{-\alpha};q\right)_{\infty}}{\left(q^{\alpha+1},-c,-c^{-1}q;q\right)_{\infty}}\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\alpha>-1}}}$ &
${\displaystyle{\displaystyle{\displaystyle c>0}}}$

${\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}\frac{x^{\alpha}}{\left(-x;q\right)_{\infty}}L^{(\alpha)}_{m}\!\left(x;q\right)L^{(\alpha)}_{n}\!\left(x;q\right)\,d_{q}x{}=\frac{1-q}{2}\,\frac{\left(q,-q^{\alpha+1},-q^{-\alpha};q\right)_{\infty}}{\left(q^{\alpha+1},-q,-q;q\right)_{\infty}}\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\alpha>-1}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle-q^{2n+\alpha+1}xL^{(\alpha)}_{n}\!\left(x;q\right)=(1-q^{n+1})L^{(\alpha)}_{n+1}\!\left(x;q\right){}-\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]L^{(\alpha)}_{n}\!\left(x;q\right){}+q(1-q^{n+\alpha})L^{(\alpha)}_{n-1}\!\left(x;q\right)}}}$

Monic recurrence relation

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

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle-q^{\alpha}(1-q^{n})xy(x)=q^{\alpha}(1+x)y(qx)-\left[1+q^{\alpha}(1+x)\right]y(x)+y(q^{-1}x)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=L^{(\alpha)}_{n}\!\left(x;q\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle L^{(\alpha)}_{n}\!\left(x;q\right)-L^{(\alpha)}_{n}\!\left(qx;q\right)=-q^{\alpha+1}xL^{(\alpha+1)}_{n-1}\!\left(qx;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}L^{(\alpha)}_{n}\!\left(x;q\right)=-\frac{q^{\alpha+1}}{1-q}L^{(\alpha+1)}_{n-1}\!\left(qx;q\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle L^{(\alpha)}_{n}\!\left(x;q\right)-q^{\alpha}(1+x)L^{(\alpha)}_{n}\!\left(qx;q\right)=(1-q^{n+1})L^{(\alpha-1)}_{n+1}\!\left(x;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}\left[w(x;\alpha;q)L^{(\alpha)}_{n}\!\left(x;q\right)\right]=\frac{1-q^{n+1}}{1-q}w(x;\alpha-1;q)L^{(\alpha-1)}_{n+1}\!\left(x;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;\alpha;q)=\frac{x^{\alpha}}{\left(-x;q\right)_{\infty}}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;\alpha;q)L^{(\alpha)}_{n}\!\left(x;q\right)=\frac{(1-q)^{n}}{\left(q;q\right)_{n}}\left(\mathcal{D}_{q}\right)^{n}\left[w(x;\alpha+n;q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;\alpha;q)=\frac{x^{\alpha}}{\left(-x;q\right)_{\infty}}}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(t;q\right)_{\infty}}\,\qHyperrphis{1}{1}@@{-x}{0}{q}{q^{\alpha+1}t}=\sum_{n=0}^{\infty}L^{(\alpha)}_{n}\!\left(x;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(t;q\right)_{\infty}}\,\qHyperrphis{0}{1}@@{-}{q^{\alpha+1}}{q}{-q^{\alpha+1}xt}=\sum_{n=0}^{\infty}\frac{L^{(\alpha)}_{n}\!\left(x;q\right)}{\left(q^{\alpha+1};q\right)_{n}}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot\qHyperrphis{0}{2}@@{-}{q^{\alpha+1},t}{q}{-q^{\alpha+1}xt}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left(q^{\alpha+1};q\right)_{n}}L^{(\alpha)}_{n}\!\left(x;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\left(\gamma t;q\right)_{\infty}}{\left(t;q\right)_{\infty}}\,\qHyperrphis{1}{2}@@{\gamma}{q^{\alpha+1},\gamma t}{q}{-q^{\alpha+1}xt}{}=\sum_{n=0}^{\infty}\frac{\left(\gamma;q\right)_{n}}{\left(q^{\alpha+1};q\right)_{n}}L^{(\alpha)}_{n}\!\left(x;q\right)t^{n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

Limit relations

Little q-Jacobi polynomial to q-Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{b\rightarrow-\infty}\littleqJacobi{n}@{-b^{-1}q^{-1}x}{q^{\alpha}}{b}{q}=\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\qLaguerre[\alpha]{n}@{x}{q}}}}$

q-Meixner polynomial to q-Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow\infty}M_{n}\!\left(cq^{\alpha}x;q^{\alpha},c;q\right)=\frac{\left(q;q\right)_{n}}{\left(q^{\alpha+1};q\right)_{n}}L^{(\alpha)}_{n}\!\left(x;q\right)}}}$

q-Laguerre polynomial to Stieltjes-Wigert polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}L^{(\alpha)}_{n}\!\left(xq^{-\alpha};q\right)=S_{n}\!\left(x;q\right)}}}$

q-Laguerre polynomial to Laguerre / Charlier polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}L^{\alpha}_{n}\left((1-q\right)x;q)=L^{\alpha}_{n}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\,\left(q;q\right)_{n}L^{(\alpha)}_{n}\!\left(-q^{-x};q\right)=C_{n}\!\left(x;a\right)}}}$
${\displaystyle{\displaystyle{\displaystyle q^{\alpha}=\frac{1}{a(q-1)}\quad\textrm{or}\quad\alpha=-\frac{\ln\left(q-1\right)a}{\ln q}}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle\qLaguerre[\alpha]{n}@{x}{q^{-1}}=\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n\alpha}}\littleqLaguerre{n}@{-x}{q^{\alpha}}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\qBesselPoly{n}@{q^{x}}{a}{q}}{\left(q;q\right)_{n}}=\qLaguerre[x-n]{n}@{aq^{n}}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\qCharlier{n}@{-x}{-q^{-\alpha}}{q}}{\qPochhammer{q}{q}{n}}=\qLaguerre[\alpha]{n}@{x}{q}}}}$

Koornwinder Addendum: q-Laguerre

Orthogonality relation

${\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}\qLaguerre[\alpha]{m}@{x}{q}\qLaguerre[\alpha]{n}@{x}{q}\frac{x^{\alpha}}{\qPochhammer{-x}{q}{\infty}}dx=h_{n}\Kronecker{m}{n}\qquad(\alpha>-1)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle h_{n}=q^{-\frac{1}{2}\alpha(\alpha+1)}\qPochhammer{q}{q}{\alpha}\log(q^{-1})\qquad(\alpha\in\ZZ_{\geq 0})}}}$

${\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}q^{n}},\qquad h_{0}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=-\frac{\left(q^{-\alpha};q\right)_{\infty}}{\left(q;q\right)_{\infty}}\frac{\pi}{\sin\left(\pi\alpha\right)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h_{n}=q^{-\frac{1}{2}\alpha(\alpha+1)}\qPochhammer{q}{q}{\alpha}\log(q^{-1})\qquad(\alpha\in\ZZ_{\geq 0})}}}$

Expansion of x^n

${\displaystyle{\displaystyle{\displaystyle x^{n}=q^{-\frac{1}{2}n(n+2\alpha+1)}\left(q^{\alpha+1};q\right)_{n}\sum_{k=0}^{n}\frac{\left(q^{-n};q\right)_{k}}{\left(q^{\alpha+1};q\right)_{k}}q^{k}\qLaguerre[\alpha]{k}@{x}{q}}}}$

Quadratic transformations

${\displaystyle{\displaystyle{\displaystyle\qLaguerre[-1/2]{n}@{x^{2}}{q^{2}}=\frac{(-1)^{n}q^{2n^{2}-n}}{\qPochhammer{q^{2}}{q^{2}}{n}}\discrqHermiteII{2n}@{x}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle x\qLaguerre[1/2]{n}@{x^{2}}{q^{2}}=\frac{(-1)^{n}q^{2n^{2}+n}}{\left(q^{2};q^{2}\right)_{n}}\discrqHermiteII{2n+1}@{x}{q}}}}$

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q-Laguerre

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