Formula:KLS:14.21:03

<< Formula:KLS:14.21:02

formula in q-Laguerre

Formula:KLS:14.21:04 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\alpha>-1}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:qLaguerre
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.21 of KLS.

URL links

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<< Formula:KLS:14.21:02

formula in q-Laguerre

Formula:KLS:14.21:04 >>