q-Laguerre

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q-Laguerre

Basic hypergeometric representation

L n ( α ) ( x ; q ) = ( q α + 1 ; q ) n ( q ; q ) n \qHyperrphis 11 @ @ q - n q α + 1 q - q n + α + 1 x q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 \qHyperrphis 11 @ @ superscript 𝑞 𝑛 superscript 𝑞 𝛼 1 𝑞 superscript 𝑞 𝑛 𝛼 1 𝑥 {\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 \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} }
L n ( α ) ( x ; q ) = 1 ( q ; q ) n \qHyperrphis 21 @ @ q - n , - x 0 q q n + α + 1 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 1 q-Pochhammer-symbol 𝑞 𝑞 𝑛 \qHyperrphis 21 @ @ superscript 𝑞 𝑛 𝑥 0 𝑞 superscript 𝑞 𝑛 𝛼 1 {\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}}}}} {\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)

0 x α ( - x ; q ) L m ( α ) ( x ; q ) L n ( α ) ( x ; q ) 𝑑 x = ( q - α ; q ) ( q ; q ) ( q α + 1 ; q ) n ( q ; q ) n q n Γ ( - α ) Γ ( α + 1 ) δ m , n superscript subscript 0 superscript 𝑥 𝛼 q-Pochhammer-symbol 𝑥 𝑞 q-Laguerre-polynomial-L 𝛼 𝑚 𝑥 𝑞 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 differential-d 𝑥 q-Pochhammer-symbol superscript 𝑞 𝛼 𝑞 q-Pochhammer-symbol 𝑞 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 𝑛 Euler-Gamma 𝛼 Euler-Gamma 𝛼 1 Kronecker-delta 𝑚 𝑛 {\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}}}} {\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): α > - 1 𝛼 1 {\displaystyle{\displaystyle{\displaystyle\alpha>-1}}}


k = - q k α + k ( - c q k ; q ) L m ( α ) ( c q k ; q ) L n ( α ) ( c q k ; q ) = ( q , - c q α + 1 , - c - 1 q - α ; q ) ( q α + 1 , - c , - c - 1 q ; q ) ( q α + 1 ; q ) n ( q ; q ) n q n δ m , n superscript subscript 𝑘 superscript 𝑞 𝑘 𝛼 𝑘 q-Pochhammer-symbol 𝑐 superscript 𝑞 𝑘 𝑞 q-Laguerre-polynomial-L 𝛼 𝑚 𝑐 superscript 𝑞 𝑘 𝑞 q-Laguerre-polynomial-L 𝛼 𝑛 𝑐 superscript 𝑞 𝑘 𝑞 q-Pochhammer-symbol 𝑞 𝑐 superscript 𝑞 𝛼 1 superscript 𝑐 1 superscript 𝑞 𝛼 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑐 superscript 𝑐 1 𝑞 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 𝑛 Kronecker-delta 𝑚 𝑛 {\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}}}} {\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): α > - 1 𝛼 1 {\displaystyle{\displaystyle{\displaystyle\alpha>-1}}} &
c > 0 𝑐 0 {\displaystyle{\displaystyle{\displaystyle c>0}}}


0 x α ( - x ; q ) L m ( α ) ( x ; q ) L n ( α ) ( x ; q ) d q x = 1 - q 2 ( q , - q α + 1 , - q - α ; q ) ( q α + 1 , - q , - q ; q ) ( q α + 1 ; q ) n ( q ; q ) n q n δ m , n superscript subscript 0 superscript 𝑥 𝛼 q-Pochhammer-symbol 𝑥 𝑞 q-Laguerre-polynomial-L 𝛼 𝑚 𝑥 𝑞 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 subscript 𝑑 𝑞 𝑥 1 𝑞 2 q-Pochhammer-symbol 𝑞 superscript 𝑞 𝛼 1 superscript 𝑞 𝛼 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑞 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 𝑛 Kronecker-delta 𝑚 𝑛 {\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}}}} {\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): α > - 1 𝛼 1 {\displaystyle{\displaystyle{\displaystyle\alpha>-1}}}


Recurrence relation

- q 2 n + α + 1 x L n ( α ) ( x ; q ) = ( 1 - q n + 1 ) L n + 1 ( α ) ( x ; q ) - [ ( 1 - q n + 1 ) + q ( 1 - q n + α ) ] L n ( α ) ( x ; q ) + q ( 1 - q n + α ) L n - 1 ( α ) ( x ; q ) superscript 𝑞 2 𝑛 𝛼 1 𝑥 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 1 superscript 𝑞 𝑛 1 q-Laguerre-polynomial-L 𝛼 𝑛 1 𝑥 𝑞 delimited-[] 1 superscript 𝑞 𝑛 1 𝑞 1 superscript 𝑞 𝑛 𝛼 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 𝑞 1 superscript 𝑞 𝑛 𝛼 q-Laguerre-polynomial-L 𝛼 𝑛 1 𝑥 𝑞 {\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)}}} {\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

x L ^ n ( α ) ( x ) = L ^ n + 1 ( α ) ( x ) + q - 2 n - α - 1 [ ( 1 - q n + 1 ) + q ( 1 - q n + α ) ] L ^ n ( α ) ( x ) + q - 4 n - 2 α + 1 ( 1 - q n ) ( 1 - q n + α ) L ^ n - 1 ( α ) ( x ) 𝑥 q-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑞 q-Laguerre-polynomial-monic-p 𝛼 𝑛 1 𝑥 𝑞 superscript 𝑞 2 𝑛 𝛼 1 delimited-[] 1 superscript 𝑞 𝑛 1 𝑞 1 superscript 𝑞 𝑛 𝛼 q-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑞 superscript 𝑞 4 𝑛 2 𝛼 1 1 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝛼 q-Laguerre-polynomial-monic-p 𝛼 𝑛 1 𝑥 𝑞 {\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 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} }
L n ( α ) ( x ; q ) = ( - 1 ) n q n ( n + α ) ( q ; q ) n L ^ n ( α ) ( x ) q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 superscript 1 𝑛 superscript 𝑞 𝑛 𝑛 𝛼 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Laguerre-polynomial-monic-p 𝛼 𝑛 𝑥 𝑞 {\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)}}} {\displaystyle \qLaguerre[\alpha]{n}@{x}{q}=\frac{(-1)^nq^{n(n+\alpha)}}{\qPochhammer{q}{q}{n}}\monicqLaguerre[\alpha]{n}@@{x}{q} }

q-Difference equation

- q α ( 1 - q n ) x y ( x ) = q α ( 1 + x ) y ( q x ) - [ 1 + q α ( 1 + x ) ] y ( x ) + y ( q - 1 x ) superscript 𝑞 𝛼 1 superscript 𝑞 𝑛 𝑥 𝑦 𝑥 superscript 𝑞 𝛼 1 𝑥 𝑦 𝑞 𝑥 delimited-[] 1 superscript 𝑞 𝛼 1 𝑥 𝑦 𝑥 𝑦 superscript 𝑞 1 𝑥 {\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)}}} {\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): y ( x ) = L n ( α ) ( x ; q ) 𝑦 𝑥 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=L^{(\alpha)}_{n}\!\left(x;q% \right)}}}


Forward shift operator

L n ( α ) ( x ; q ) - L n ( α ) ( q x ; q ) = - q α + 1 x L n - 1 ( α + 1 ) ( q x ; q ) q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 q-Laguerre-polynomial-L 𝛼 𝑛 𝑞 𝑥 𝑞 superscript 𝑞 𝛼 1 𝑥 q-Laguerre-polynomial-L 𝛼 1 𝑛 1 𝑞 𝑥 𝑞 {\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 \qLaguerre[\alpha]{n}@{x}{q}-\qLaguerre[\alpha]{n}@{qx}{q}=-q^{\alpha+1}x\qLaguerre[\alpha+1]{n-1}@{qx}{q} }
𝒟 q L n ( α ) ( x ; q ) = - q α + 1 1 - q L n - 1 ( α + 1 ) ( q x ; q ) q-derivative 𝑞 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 superscript 𝑞 𝛼 1 1 𝑞 q-Laguerre-polynomial-L 𝛼 1 𝑛 1 𝑞 𝑥 𝑞 {\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)}}} {\displaystyle \qderiv{q}\qLaguerre[\alpha]{n}@{x}{q}=-\frac{q^{\alpha+1}}{1-q}\qLaguerre[\alpha+1]{n-1}@{qx}{q} }

Backward shift operator

L n ( α ) ( x ; q ) - q α ( 1 + x ) L n ( α ) ( q x ; q ) = ( 1 - q n + 1 ) L n + 1 ( α - 1 ) ( x ; q ) q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 superscript 𝑞 𝛼 1 𝑥 q-Laguerre-polynomial-L 𝛼 𝑛 𝑞 𝑥 𝑞 1 superscript 𝑞 𝑛 1 q-Laguerre-polynomial-L 𝛼 1 𝑛 1 𝑥 𝑞 {\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 \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} }
𝒟 q [ w ( x ; α ; q ) L n ( α ) ( x ; q ) ] = 1 - q n + 1 1 - q w ( x ; α - 1 ; q ) L n + 1 ( α - 1 ) ( x ; q ) q-derivative 𝑞 𝑤 𝑥 𝛼 𝑞 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 1 superscript 𝑞 𝑛 1 1 𝑞 𝑤 𝑥 𝛼 1 𝑞 q-Laguerre-polynomial-L 𝛼 1 𝑛 1 𝑥 𝑞 {\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)}}} {\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): w ( x ; α ; q ) = x α ( - x ; q ) 𝑤 𝑥 𝛼 𝑞 superscript 𝑥 𝛼 q-Pochhammer-symbol 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha;q)=\frac{x^{\alpha}}{% \left(-x;q\right)_{\infty}}}}}


Rodrigues-type formula

w ( x ; α ; q ) L n ( α ) ( x ; q ) = ( 1 - q ) n ( q ; q ) n ( 𝒟 q ) n [ w ( x ; α + n ; q ) ] 𝑤 𝑥 𝛼 𝑞 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 superscript 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript q-derivative 𝑞 𝑛 delimited-[] 𝑤 𝑥 𝛼 𝑛 𝑞 {\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]}}} {\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): w ( x ; α ; q ) = x α ( - x ; q ) 𝑤 𝑥 𝛼 𝑞 superscript 𝑥 𝛼 q-Pochhammer-symbol 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle w(x;\alpha;q)=\frac{x^{\alpha}}{% \left(-x;q\right)_{\infty}}}}}


Generating functions

1 ( t ; q ) \qHyperrphis 11 @ @ - x 0 q q α + 1 t = n = 0 L n ( α ) ( x ; q ) t n 1 q-Pochhammer-symbol 𝑡 𝑞 \qHyperrphis 11 @ @ 𝑥 0 𝑞 superscript 𝑞 𝛼 1 𝑡 superscript subscript 𝑛 0 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 superscript 𝑡 𝑛 {\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 \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 }
1 ( t ; q ) \qHyperrphis 01 @ @ - q α + 1 q - q α + 1 x t = n = 0 L n ( α ) ( x ; q ) ( q α + 1 ; q ) n t n 1 q-Pochhammer-symbol 𝑡 𝑞 \qHyperrphis 01 @ @ superscript 𝑞 𝛼 1 𝑞 superscript 𝑞 𝛼 1 𝑥 𝑡 superscript subscript 𝑛 0 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 superscript 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 \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 }
( t ; q ) \qHyperrphis 02 @ @ - q α + 1 , t q - q α + 1 x t = n = 0 ( - 1 ) n q \binomial n 2 ( q α + 1 ; q ) n L n ( α ) ( x ; q ) t n q-Pochhammer-symbol 𝑡 𝑞 \qHyperrphis 02 @ @ superscript 𝑞 𝛼 1 𝑡 𝑞 superscript 𝑞 𝛼 1 𝑥 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 superscript 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 \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 }
( γ t ; q ) ( t ; q ) \qHyperrphis 12 @ @ γ q α + 1 , γ t q - q α + 1 x t = n = 0 ( γ ; q ) n ( q α + 1 ; q ) n L n ( α ) ( x ; q ) t n q-Pochhammer-symbol 𝛾 𝑡 𝑞 q-Pochhammer-symbol 𝑡 𝑞 \qHyperrphis 12 @ @ 𝛾 superscript 𝑞 𝛼 1 𝛾 𝑡 𝑞 superscript 𝑞 𝛼 1 𝑥 𝑡 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝛾 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 superscript 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}% }}} {\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): γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Limit relations

Little q-Jacobi polynomial to q-Laguerre polynomial

lim b - p n ( - b - 1 q - 1 x ; q α , b ; q ) = ( q ; q ) n ( q α + 1 ; q ) n L n ( α ) ( x ; q ) subscript 𝑏 little-q-Jacobi-polynomial-p 𝑛 superscript 𝑏 1 superscript 𝑞 1 𝑥 superscript 𝑞 𝛼 𝑏 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{b\rightarrow-\infty}p_{n}\!% \left(-b^{-1}q^{-1}x;q^{\alpha},b;q\right)=\frac{\left(q;q\right)_{n}}{\left(q% ^{\alpha+1};q\right)_{n}}L^{(\alpha)}_{n}\!\left(x;q\right)}}} {\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

lim c M n ( c q α x ; q α , c ; q ) = ( q ; q ) n ( q α + 1 ; q ) n L n ( α ) ( x ; q ) subscript 𝑐 q-Meixner-polynomial-M 𝑛 𝑐 superscript 𝑞 𝛼 𝑥 superscript 𝑞 𝛼 𝑐 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 {\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)}}} {\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

lim α L n ( α ) ( x q - α ; q ) = S n ( x ; q ) subscript 𝛼 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript 𝑞 𝛼 𝑞 Stieltjes-Wigert-polynomial-S 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}L^{(% \alpha)}_{n}\!\left(xq^{-\alpha};q\right)=S_{n}\!\left(x;q\right)}}} {\displaystyle \lim_{\alpha\rightarrow\infty}\qLaguerre[\alpha]{n}@{xq^{-\alpha}}{q}=\StieltjesWigert{n}@{x}{q} }

q-Laguerre polynomial to Laguerre / Charlier polynomial

lim q 1 L n α ( ( 1 - q ) x ; q ) = L n α ( x ) fragments subscript 𝑞 1 generalized-Laguerre-polynomial-L 𝛼 𝑛 fragments ( 1 q x ; q ) generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}L^{\alpha}_{n}% \left((1-q\right)x;q)=L^{\alpha}_{n}\left(x\right)}}} {\displaystyle \lim_{q\rightarrow 1}\Laguerre[\alpha]{n}@{(1-q}x;q)=\Laguerre[\alpha]{n}@{x} }
lim q 1 ( q ; q ) n L n ( α ) ( - q - x ; q ) = C n ( x ; a ) subscript 𝑞 1 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Laguerre-polynomial-L 𝛼 𝑛 superscript 𝑞 𝑥 𝑞 Charlier-polynomial-C 𝑛 𝑥 𝑎 {\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 \lim_{q\rightarrow 1}\,\qPochhammer{q}{q}{n}\qLaguerre[\alpha]{n}@{-q^{-x}}{q}=\Charlier{n}@{x}{a} }
q α = 1 a ( q - 1 ) or α = - ln ( q - 1 ) a ln q formulae-sequence superscript 𝑞 𝛼 1 𝑎 𝑞 1 or 𝛼 𝑞 1 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle q^{\alpha}=\frac{1}{a(q-1)}\quad% \textrm{or}\quad\alpha=-\frac{\ln\left(q-1\right)a}{\ln q}}}} {\displaystyle q^{\alpha}=\frac{1}{a(q-1)}\quad\textrm{or}\quad\alpha=-\frac{\ln@{q-1}a}{\ln@@{q}} }

Remarks

L n ( α ) ( x ; q - 1 ) = ( q α + 1 ; q ) n ( q ; q ) n q n α p n ( - x ; q α ; q ) q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 superscript 𝑞 1 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 𝑛 𝛼 little-q-Laguerre-Wall-polynomial-p 𝑛 𝑥 superscript 𝑞 𝛼 𝑞 {\displaystyle{\displaystyle{\displaystyle L^{(\alpha)}_{n}\!\left(x;q^{-1}% \right)=\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n\alpha}% }p_{n}\!\left(-x;q^{\alpha};q\right)}}} {\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} }
y n ( q x ; a ; q ) ( q ; q ) n = L n ( x - n ) ( a q n ; q ) q-Bessel-polynomial-y 𝑛 superscript 𝑞 𝑥 𝑎 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Laguerre-polynomial-L 𝑥 𝑛 𝑛 𝑎 superscript 𝑞 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{y_{n}\!\left(q^{x};a;q\right)}% {\left(q;q\right)_{n}}=L^{(x-n)}_{n}\!\left(aq^{n};q\right)}}} {\displaystyle \frac{\qBesselPoly{n}@{q^x}{a}{q}}{\qPochhammer{q}{q}{n}}=\qLaguerre[x-n]{n}@{aq^n}{q} }
C n ( - x ; - q - α ; q ) ( q ; q ) n = L n ( α ) ( x ; q ) q-Charlier-polynomial-C 𝑛 𝑥 superscript 𝑞 𝛼 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{C_{n}\!\left(-x;-q^{-\alpha};q% \right)}{\left(q;q\right)_{n}}=L^{(\alpha)}_{n}\!\left(x;q\right)}}} {\displaystyle \frac{\qCharlier{n}@{-x}{-q^{-\alpha}}{q}}{\qPochhammer{q}{q}{n}}=\qLaguerre[\alpha]{n}@{x}{q} }

Koornwinder Addendum: q-Laguerre

Orthogonality relation

0 L m ( α ) ( x ; q ) L n ( α ) ( x ; q ) x α ( - x ; q ) d x = h n δ m , n    ( α > - 1 ) fragments superscript subscript 0 q-Laguerre-polynomial-L 𝛼 𝑚 𝑥 𝑞 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 superscript 𝑥 𝛼 q-Pochhammer-symbol 𝑥 𝑞 d x subscript 𝑛 Kronecker-delta 𝑚 𝑛 italic-   fragments ( α 1 ) {\displaystyle{\displaystyle{\displaystyle\int_{0}^{\infty}L^{(\alpha)}_{m}\!% \left(x;q\right)L^{(\alpha)}_{n}\!\left(x;q\right)\frac{x^{\alpha}}{\left(-x;q% \right)_{\infty}}dx=h_{n}\delta_{m,n}\qquad(\alpha>-1)}}} {\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): h n = q - 1 2 α ( α + 1 ) ( q ; q ) α log ( q - 1 )    ( α \ZZ 0 ) fragments subscript 𝑛 superscript 𝑞 1 2 𝛼 𝛼 1 q-Pochhammer-symbol 𝑞 𝑞 𝛼 fragments ( superscript 𝑞 1 ) italic-   fragments ( α subscript \ZZ absent 0 ) {\displaystyle{\displaystyle{\displaystyle h_{n}=q^{-\frac{1}{2}\alpha(\alpha+% 1)}\left(q;q\right)_{\alpha}\log(q^{-1})\qquad(\alpha\in\ZZ_{\geq 0})}}}


h n h 0 = ( q α + 1 ; q ) n ( q ; q ) n q n , h 0 subscript 𝑛 subscript 0 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑞 𝑛 subscript 0 {\displaystyle{\displaystyle{\displaystyle\frac{h_{n}}{h_{0}}=\frac{\left(q^{% \alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}},\qquad h_{0}}}} {\displaystyle \frac{h_n}{h_0}=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n} q^n},\qquad h_0 }

Substitution(s): h n h 0 = - ( q - α ; q ) ( q ; q ) π sin ( π α ) subscript 𝑛 subscript 0 q-Pochhammer-symbol superscript 𝑞 𝛼 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝛼 {\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)}}}} &
h n = q - 1 2 α ( α + 1 ) ( q ; q ) α log ( q - 1 )    ( α \ZZ 0 ) fragments subscript 𝑛 superscript 𝑞 1 2 𝛼 𝛼 1 q-Pochhammer-symbol 𝑞 𝑞 𝛼 fragments ( superscript 𝑞 1 ) italic-   fragments ( α subscript \ZZ absent 0 ) {\displaystyle{\displaystyle{\displaystyle h_{n}=q^{-\frac{1}{2}\alpha(\alpha+% 1)}\left(q;q\right)_{\alpha}\log(q^{-1})\qquad(\alpha\in\ZZ_{\geq 0})}}}


Expansion of x^n

x n = q - 1 2 n ( n + 2 α + 1 ) ( q α + 1 ; q ) n k = 0 n ( q - n ; q ) k ( q α + 1 ; q ) k q k L k ( α ) ( x ; q ) superscript 𝑥 𝑛 superscript 𝑞 1 2 𝑛 𝑛 2 𝛼 1 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 superscript subscript 𝑘 0 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑛 𝑞 𝑘 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑘 superscript 𝑞 𝑘 q-Laguerre-polynomial-L 𝛼 𝑘 𝑥 𝑞 {\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}L^{(\alpha)}_{k}\!\left(x;q\right)}}} {\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

L n ( - 1 / 2 ) ( x 2 ; q 2 ) = ( - 1 ) n q 2 n 2 - n ( q 2 ; q 2 ) n h ~ 2 n ( x ; q ) q-Laguerre-polynomial-L 1 2 𝑛 superscript 𝑥 2 superscript 𝑞 2 superscript 1 𝑛 superscript 𝑞 2 superscript 𝑛 2 𝑛 q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 𝑛 discrete-q-Hermite-polynomial-II-h-tilde 2 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle L^{(-1/2)}_{n}\!\left(x^{2};q^{2}% \right)=\frac{(-1)^{n}q^{2n^{2}-n}}{\left(q^{2};q^{2}\right)_{n}}\tilde{h}_{2n% }\!\left(x;q\right)}}} {\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} }
x L n ( 1 / 2 ) ( x 2 ; q 2 ) = ( - 1 ) n q 2 n 2 + n ( q 2 ; q 2 ) n h ~ 2 n + 1 ( x ; q ) 𝑥 q-Laguerre-polynomial-L 1 2 𝑛 superscript 𝑥 2 superscript 𝑞 2 superscript 1 𝑛 superscript 𝑞 2 superscript 𝑛 2 𝑛 q-Pochhammer-symbol superscript 𝑞 2 superscript 𝑞 2 𝑛 discrete-q-Hermite-polynomial-II-h-tilde 2 𝑛 1 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle xL^{(1/2)}_{n}\!\left(x^{2};q^{2}% \right)=\frac{(-1)^{n}q^{2n^{2}+n}}{\left(q^{2};q^{2}\right)_{n}}\tilde{h}_{2n% +1}\!\left(x;q\right)}}} {\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} }