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**Expanding a function in "Gaussian-Hermites"**

This isn't homework or coursework, but seeing as it's most like a homework problem, I figured this would be the best place to ask.

Note that I'm using the physics

## Homework Statement

I would like to expand a function (let's take a gaussian for example) in terms of this series (very similar to the harmonic oscillator, except for a factor of 2 in the exponential):

[tex]s\left(x\right)=\sum_n\alpha_nH_n\left(x\right)\exp\left(-x^2\right)[/tex]

## Homework Equations

To find the coefficients:

[tex]\alpha_n=\int_{-\infty}^{\infty}F\left(x\right)H_n\left(x\right)\exp\left(-x^2\right)dx/Normalization[/tex]

where [tex]F\left(x\right)=\exp\left(-\frac{x^2}{\sigma^2}\right)[/tex] in my example

## The Attempt at a Solution

I have figured out the normalization of [tex]H_n\left(x\right)\exp\left(-x^2\right)[/tex]

[tex]\alpha_n=\int_{-\infty}^{\infty}\left[H_n\left(x\right)\exp\left(-x^2\right)\right]^2dx=\sqrt{\frac{\pi}{2}}\left(2 n-1\right)![/tex]

but apparently I'm doing something wrong when I write:

[tex]\alpha_n=\frac{\int_{-\infty}^{\infty}\exp\left(-\frac{x^2}{\sigma^2}\right)H_n\left(x\right)\exp\left(-x^2\right)dx}{\sqrt{\sqrt{\frac{\pi}{2}}\left(2 n-1\right)!}}[/tex]

Can you tell me where I'm making my mistake?

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