Little q-Laguerre / Wall

<< Continuous q-Laguerre

Little q-Laguerre / Wall

q-Laguerre >>

Basic hypergeometric representation

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

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{k=0}^{\infty}\frac{(aq)^k}{\qPochhammer{q}{q}{k}}\littleqLaguerre{m}@{q^k}{a}{q}\littleqLaguerre{n}@{q^k}{a}{q}= \frac{(aq)^n}{\qPochhammer{aq}{q}{\infty}}\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{aq}{q}{n}}\,\Kronecker{m}{n} }}

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

Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -x\littleqLaguerre{n}@{x}{a}{q}=A_n\littleqLaguerre{n+1}@{x}{a}{q}-\left(A_n+C_n\right)\littleqLaguerre{n}@{x}{a}{q}+C_n\littleqLaguerre{n-1}@{x}{a}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=aq^n(1-q^n)}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=q^n(1-aq^{n+1})}}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\moniclittleqLaguerre{n}@@{x}{a}{q}=\moniclittleqLaguerre{n+1}@@{x}{a}{q}+(A_n+C_n)\moniclittleqLaguerre{n}@@{x}{a}{q}+aq^{2n-1}(1-q^n)(1-aq^n)\moniclittleqLaguerre{n-1}@@{x}{a}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=aq^n(1-q^n)}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=q^n(1-aq^{n+1})}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqLaguerre{n}@{x}{a}{q}=\frac{(-1)^nq^{-\binomial{n}{2}}}{\qPochhammer{aq}{q}{n}}\moniclittleqLaguerre{n}@@{x}{a}{q} }}

q-Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -q^{-n}(1-q^n)xy(x)=ay(qx)+(x-a-1)y(x)+(1-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)=\littleqLaguerre{n}@{x}{a}{q}}}

Forward shift operator

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

Backward shift operator

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

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

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;\alpha|q)\littleqLaguerre{n}@{x}{q^{\alpha}}{q}=\frac{q^{n\alpha+\binomial{n}{2}}(1-q)^n} {\qPochhammer{q^{\alpha+1}}{q}{n}}\left(\mathcal{D}_{q^{-1}}\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)=\qPochhammer{qx}{q}{\infty}x^{\alpha}}}

Generating function

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

Limit relations

Big q-Laguerre polynomial to Little q-Laguerre / Wall polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{b\rightarrow -\infty}\bigqLaguerre{n}@{bqx}{a}{b}{q}=\littleqLaguerre{n}@{x}{a}{q} }}

Little q-Jacobi polynomial to Little q-Laguerre / Wall polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqJacobi{n}@{x}{a}{0}{q}=\littleqLaguerre{n}@{x}{a}{q} }}

Affine q-Krawtchouk polynomial to Little q-Laguerre / Wall polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty}\AffqKrawtchouk{n}@{q^{x-N}}{p}{N}{q}=\littleqLaguerre{n}@{q^x}{p}{q} }}

Little q-Laguerre / Wall 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}\littleqLaguerre{n}@{(1-q)x}{q^{\alpha}}{q}=\frac{\Laguerre[\alpha]{n}@{x}}{\Laguerre[\alpha]{n}@{0}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\frac{\littleqLaguerre{n}@{q^x}{(q-1)a}{q}}{(1-q)^n}= \frac{\Charlier{n}@{x}{a}}{a^n} }}

Remark

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

Koornwinder Addendum: Little q-Laguerre / Wall

Re: (14.20.11)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{0,0}{c}{q}{z}=\frac1{\qPochhammer{z}{q}{\infty}} \qHyperrphis{0}{1}@@{-}{c}{q}{cz} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{t}{q}{\infty} \qHyperrphis{2}{1}@@{0,0}{aq}{q}{xt} =\sum_{n=0}^\infty\frac{(-1)^n q^{\frac12 n(n-1)}}{\qPochhammer{q}{q}{n}} \littleqLaguerre{n}@{x}{a}{q} t^n\qquad(|xt|<1) }}

Expansion of x^n

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x^n=\qPochhammer{a}{q}{n} \sum_{k=0}^n \frac{\qPochhammer{q^{-n}}{q}{k}}{\qPochhammer{q}{q}{k}} q^{nk} \littleqLaguerre{k}@{x}{a}{q} }}

Quadratic transformations

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqLaguerre{n}@{x^2}{q^{-1}}{q^2}= \frac{(-1)^n q^{-n(n-1)}}{\qPochhammer{q}{q^2}{n}} \discrqHermiteI{2n}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\littleqLaguerre{n}@{x^2}{q}{q^2}= \frac{(-1)^n q^{-n(n-1)}}{\qPochhammer{q^3}{q^2}{n}} \discrqHermiteI{2n+1}@{x}{q} }}

<< Continuous q-Laguerre

Little q-Laguerre / Wall

q-Laguerre >>