Formula:KLS:14.21:30

<< Formula:KLS:14.21:29

formula in q-Laguerre

Formula:KLS:14.21:31 >>

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

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle h_{n}=q^{-\frac{1}{2}\alpha(\alpha+1)}\left(q;q\right)_{\alpha}\log(q^{-1})\qquad(\alpha\in\ZZ_{\geq 0})}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:qLaguerre
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4
${\displaystyle{\displaystyle{\displaystyle\mathrm{log}}}}$ : principle branch of logarithm logarithm : http://dlmf.nist.gov/4.2#E2

Bibliography

Equation in Section 14.21 of KLS.

URL links

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<< Formula:KLS:14.21:29

formula in q-Laguerre

Formula:KLS:14.21:31 >>