q-Laguerre

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q-Laguerre

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qLaguerre[\alpha]{n}@{x}{q}=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\ \qHyperrphis{1}{1}@@{q^{-n}}{q^{\alpha+1}}{q}{-q^{n+\alpha+1}x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qLaguerre[\alpha]{n}@{x}{q}=\frac{1}{\qPochhammer{q}{q}{n}}\,\qHyperrphis{2}{1}@@{q^{-n},-x}{0}{q}{q^{n+\alpha+1}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{0}^{\infty}\frac{x^{\alpha}}{\qPochhammer{-x}{q}{\infty}}\qLaguerre[\alpha]{m}@{x}{q}\qLaguerre[\alpha]{n}@{x}{q}\,dx {}=\frac{\qPochhammer{q^{-\alpha}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} \frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}q^n}\EulerGamma@{-\alpha}\EulerGamma@{\alpha+1}\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \alpha>-1}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{k=-\infty}^{\infty}\frac{q^{k\alpha+k}}{\qPochhammer{-cq^k}{q}{\infty}}\qLaguerre[\alpha]{m}@{cq^k}{q}\qLaguerre[\alpha]{n}@{cq^k}{q} {}=\frac{\qPochhammer{q,-cq^{\alpha+1},-c^{-1}q^{-\alpha}}{q}{\infty}} {\qPochhammer{q^{\alpha+1},-c,-c^{-1}q}{q}{\infty}}\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}q^n}\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \alpha>-1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle c>0}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_0^{\infty}\frac{x^{\alpha}}{\qPochhammer{-x}{q}{\infty}}\qLaguerre[\alpha]{m}@{x}{q}\qLaguerre[\alpha]{n}@{x}{q}\,d_qx {}=\frac{1-q}{2}\,\frac{\qPochhammer{q,-q^{\alpha+1},-q^{-\alpha}}{q}{\infty}} {\qPochhammer{q^{\alpha+1},-q,-q}{q}{\infty}}\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}q^n}\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \alpha>-1}}


Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -q^{2n+\alpha+1}x\qLaguerre[\alpha]{n}@{x}{q}=(1-q^{n+1})\qLaguerre[\alpha]{n+1}@{x}{q} {}-\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]\qLaguerre[\alpha]{n}@{x}{q} {}+q(1-q^{n+\alpha})\qLaguerre[\alpha]{n-1}@{x}{q} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicqLaguerre[\alpha]{n}@@{x}{q}=\monicqLaguerre[\alpha]{n+1}@@{x}{q}+q^{-2n-\alpha-1}\left[(1-q^{n+1})+q(1-q^{n+\alpha})\right]\monicqLaguerre[\alpha]{n}@@{x}{q} {}+q^{-4n-2\alpha+1}(1-q^n)(1-q^{n+\alpha})\monicqLaguerre[\alpha]{n-1}@@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qLaguerre[\alpha]{n}@{x}{q}=\frac{(-1)^nq^{n(n+\alpha)}}{\qPochhammer{q}{q}{n}}\monicqLaguerre[\alpha]{n}@@{x}{q} }}

q-Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\qLaguerre[\alpha]{n}@{x}{q}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qLaguerre[\alpha]{n}@{x}{q}-\qLaguerre[\alpha]{n}@{qx}{q}=-q^{\alpha+1}x\qLaguerre[\alpha+1]{n-1}@{qx}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\qLaguerre[\alpha]{n}@{x}{q}=-\frac{q^{\alpha+1}}{1-q}\qLaguerre[\alpha+1]{n-1}@{qx}{q} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qLaguerre[\alpha]{n}@{x}{q}-q^{\alpha}(1+x)\qLaguerre[\alpha]{n}@{qx}{q}=(1-q^{n+1})\qLaguerre[\alpha-1]{n+1}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\left[w(x;\alpha;q)\qLaguerre[\alpha]{n}@{x}{q}\right]= \frac{1-q^{n+1}}{1-q}w(x;\alpha-1;q)\qLaguerre[\alpha-1]{n+1}@{x}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;\alpha;q)=\frac{x^{\alpha}}{\qPochhammer{-x}{q}{\infty}}}}


Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;\alpha;q)\qLaguerre[\alpha]{n}@{x}{q}= \frac{(1-q)^n}{\qPochhammer{q}{q}{n}}\left(\qderiv{q}\right)^n\left[w(x;\alpha+n;q)\right] }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;\alpha;q)=\frac{x^{\alpha}}{\qPochhammer{-x}{q}{\infty}}}}


Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{1}{1}@@{-x}{0}{q}{q^{\alpha+1}t} =\sum_{n=0}^{\infty}\qLaguerre[\alpha]{n}@{x}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{0}{1}@@{-}{q^{\alpha+1}}{q}{-q^{\alpha+1}xt} =\sum_{n=0}^{\infty}\frac{\qLaguerre[\alpha]{n}@{x}{q}}{\qPochhammer{q^{\alpha+1}}{q}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{t}{q}{\infty}\cdot\qHyperrphis{0}{2}@@{-}{q^{\alpha+1},t}{q}{-q^{\alpha+1}xt} =\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\qLaguerre[\alpha]{n}@{x}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{\gamma t}{q}{\infty}}{\qPochhammer{t}{q}{\infty}}\,\qHyperrphis{1}{2}@@{\gamma}{q^{\alpha+1},\gamma t}{q}{-q^{\alpha+1}xt} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\qLaguerre[\alpha]{n}@{x}{q}t^n }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


Limit relations

Little q-Jacobi polynomial to q-Laguerre polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{c\rightarrow\infty}\qMeixner{n}@{cq^{\alpha}x}{q^{\alpha}}{c}{q}= \frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\qLaguerre[\alpha]{n}@{x}{q} }}

q-Laguerre polynomial to Stieltjes-Wigert polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\alpha\rightarrow\infty}\qLaguerre[\alpha]{n}@{xq^{-\alpha}}{q}=\StieltjesWigert{n}@{x}{q} }}

q-Laguerre polynomial to Laguerre / Charlier polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\Laguerre[\alpha]{n}@{(1-q}x;q)=\Laguerre[\alpha]{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\,\qPochhammer{q}{q}{n}\qLaguerre[\alpha]{n}@{-q^{-x}}{q}=\Charlier{n}@{x}{a} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q^{\alpha}=\frac{1}{a(q-1)}\quad\textrm{or}\quad\alpha=-\frac{\ln@{q-1}a}{\ln@@{q}} }}

Remarks

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qLaguerre[\alpha]{n}@{x}{q^{-1}}=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}q^{n\alpha}}\littleqLaguerre{n}@{-x}{q^{\alpha}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qBesselPoly{n}@{q^x}{a}{q}}{\qPochhammer{q}{q}{n}}=\qLaguerre[x-n]{n}@{aq^n}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qCharlier{n}@{-x}{-q^{-\alpha}}{q}}{\qPochhammer{q}{q}{n}}=\qLaguerre[\alpha]{n}@{x}{q} }}

Koornwinder Addendum: q-Laguerre

Orthogonality relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_n=q^{-\frac12\alpha(\alpha+1)} \qPochhammer{q}{q}{\alpha} \log(q^{-1})\qquad(\alpha\in\ZZ_{\ge0})}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{h_n}{h_0} =- \frac{\qPochhammer{q^{-\alpha}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} \frac\cpi{\sin@{\cpi\alpha}}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_n=q^{-\frac12\alpha(\alpha+1)} \qPochhammer{q}{q}{\alpha} \log(q^{-1})\qquad(\alpha\in\ZZ_{\ge0})}}


Expansion of x^n

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x^n=q^{-\frac12 n(n+2\alpha+1)} \qPochhammer{q^{\alpha+1}}{q}{n} \sum_{k=0}^n\frac{\qPochhammer{q^{-n}}{q}{k}}{\qPochhammer{q^{\alpha+1}}{q}{k}} q^k \qLaguerre[\alpha]{k}@{x}{q} }}

Quadratic transformations

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\qLaguerre[1/2]{n}@{x^2}{q^2}= \frac{(-1)^n q^{2n^2+n}}{\qPochhammer{q^2}{q^2}{n}} \discrqHermiteII{2n+1}@{x}{q} }}

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