Formula:KLS:09.12:02

<< Formula:KLS:09.12:01

formula in Laguerre

Formula:KLS:09.12:03 >>

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

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}$ : Laguerre (or generalized Laguerre) polynomial : http://dlmf.nist.gov/18.3#T1.t1.r27
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : 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 9.12 of KLS.

URL links

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<< Formula:KLS:09.12:01

formula in Laguerre

Formula:KLS:09.12:03 >>