Formula:KLS:09.12:33

<< Formula:KLS:09.12:32

formula in Laguerre

Formula:KLS:09.12:34 >>

${\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}H^{\mu}_{m}\left(x\right)H^{\mu}_{n}\left(x\right)|x|^{2\mu}{\mathrm{e}^{-x^{2}}}dx=0\qquad(m\neq n)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle H_{n}^{\mu}}}}$ : generalized Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:GenHermite
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11

Bibliography

Equation in Section 9.12 of KLS.

URL links

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<< Formula:KLS:09.12:32

formula in Laguerre

Formula:KLS:09.12:34 >>