Formula:KLS:09.08:41

<< Formula:KLS:09.08:40

formula in Jacobi: Special cases

Formula:KLS:09.08:42 >>

${\displaystyle{\displaystyle{\displaystyle\frac{2}{n!\Gamma\left(\lambda\right)}\int_{0}^{\infty}H_{n}\left(tx\right)t^{n+2\lambda-1}{\mathrm{e}^{-t^{2}}}dt=C^{\lambda}_{n}\left(x\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : Hermite polynomial ${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : http://dlmf.nist.gov/18.3#T1.t1.r28
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}$ : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5

Bibliography

Equation in Section 9.8 of KLS.

URL links

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<< Formula:KLS:09.08:40

formula in Jacobi: Special cases

Formula:KLS:09.08:42 >>