Big q-Laguerre
Big q-Laguerre
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqLaguerre{n}@{x}{a}{b}{q}=\qHyperrphis{3}{2}@@{q^{-n},0,x}{aq,bq}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqLaguerre{n}@{x}{a}{b}{q}=\frac{1}{\qPochhammer{b^{-1}q^{-n}}{q}{n}}\,\qHyperrphis{2}{1}@@{q^{-n},aqx^{-1}}{aq}{q}{\frac{x}{b}} }}
Orthogonality relation(s)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{bq}^{aq}\frac{\qPochhammer{a^{-1}x,b^{-1}x}{q}{\infty}}{\qPochhammer{x}{q}{\infty}} \bigqLaguerre{m}@{x}{a}{b}{q}\bigqLaguerre{n}@{x}{a}{b}{q}\,d_qx {}=aq(1-q)\frac{\qPochhammer{q,a^{-1}b,ab^{-1}q}{q}{\infty}} {\qPochhammer{aq,bq}{q}{\infty}}\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{aq,bq}{q}{n}}(-abq^2)^nq^{\binomial{n}{2}}\,\Kronecker{m}{n} }}
Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (x-1)\bigqLaguerre{n}@{x}{a}{b}{q}=A_n\bigqLaguerre{n+1}@{x}{a}{b}{q}-\left(A_n+C_n\right)\bigqLaguerre{n}@{x}{a}{b}{q} {}+C_n\bigqLaguerre{n-1}@{x}{a}{b}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=(1-aq^{n+1})(1-bq^{n+1})}}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicbigqLaguerre{n}@@{x}{a}{b}{q}=\monicbigqLaguerre{n+1}@@{x}{a}{b}{q}+\left[1-(A_n+C_n)\right]\monicbigqLaguerre{n}@@{x}{a}{b}{q} {}-abq^{n+1}(1-q^n)(1-aq^n)(1-bq^n)\monicbigqLaguerre{n-1}@@{x}{a}{b}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=(1-aq^{n+1})(1-bq^{n+1})}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqLaguerre{n}@{x}{a}{b}{q}=\frac{1}{\qPochhammer{aq,bq}{q}{n}}\monicbigqLaguerre{n}@@{x}{a}{b}{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)x^2y(x)=B(x)y(qx)-\left[B(x)+D(x)\right]y(x)+D(x)y(q^{-1}x) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=abq(1-x)}}
&
Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqLaguerre{n}@{x}{a}{b}{q}-\bigqLaguerre{n}@{qx}{a}{b}{q}=\frac{q^{-n+1}(1-q^n)} {(1-aq)(1-bq)}x\bigqLaguerre{n-1}@{qx}{aq}{bq}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\bigqLaguerre{n}@{x}{a}{b}{q}=\frac{q^{-n+1}(1-q^n)} {(1-q)(1-aq)(1-bq)}\bigqLaguerre{n-1}@{qx}{aq}{bq}{q} }}
Backward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (x-a)(x-b)\bigqLaguerre{n}@{x}{a}{b}{q}-ab(1-x)\bigqLaguerre{n}@{qx}{a}{b}{q} {}=(1-a)(1-b)x\bigqLaguerre{n+1}@{x}{aq^{-1}}{bq^{-1}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qderiv{q}\left[w(x;a,b;q)\bigqLaguerre{n}@{x}{a}{b}{q}\right] {}=\frac{(1-a)(1-b)}{ab(1-q)}w(x;aq^{-1},bq^{-1};q)\bigqLaguerre{n+1}@{x}{aq^{-1}}{bq^{-1}}{q} }}
Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a,b;q)\bigqLaguerre{n}@{x}{a}{b}{q} {}=\frac{a^nb^nq^{n(n+1)}(1-q)^n} {\qPochhammer{aq,bq}{q}{n}}\left(\qderiv{q}\right)^n\left[w(x;aq^n,bq^n;q)\right] }}
Generating functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{bqt}{q}{\infty}\cdot\qHyperrphis{2}{1}@@{aqx^{-1},0}{aq}{q}{xt} =\sum_{n=0}^{\infty}\frac{\qPochhammer{bq}{q}{n}}{\qPochhammer{q}{q}{n}}\bigqLaguerre{n}@{x}{a}{b}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{aqt}{q}{\infty}\cdot\qHyperrphis{2}{1}@@{bqx^{-1},0}{bq}{q}{xt} =\sum_{n=0}^{\infty}\frac{\qPochhammer{aq}{q}{n}}{\qPochhammer{q}{q}{n}}\bigqLaguerre{n}@{x}{a}{b}{q}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{3}{2}@@{0,0,x}{aq,bq}{q}{t}=\sum_{n=0}^{\infty} \frac{(-1)^nq^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}\bigqLaguerre{n}@{x}{a}{b}{q}t^n }}
Limit relations
Big q-Jacobi polynomial to Big q-Laguerre polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \bigqJacobi{n}@{x}{a}{0}{c}{q}=\bigqLaguerre{n}@{x}{a}{c}{q} }}
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} }}
Big q-Laguerre polynomial to Al-Salam-Carlitz I polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{a\rightarrow 0}\frac{\bigqLaguerre{n}@{aqx}{a}{ab}{q}}{a^n}=q^n\AlSalamCarlitzI{b}{n}@{x}{q} }}
Big q-Laguerre polynomial to Laguerre polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\pseudoJacobi{n}@{x}{q^{\alpha}}{(1-q}^{-1}q^{\beta};q)= \frac{\Laguerre[\alpha]{n}@{x-1}}{\Laguerre[\alpha]{n}@{0}} }}
Remark
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AffqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\bigqLaguerre{n}@{q^{-x}}{p}{q^{-N-1}}{q} }}