Hermite Function: Writing x^2r in Polynomial Form

[Another1]

How can \(\displaystyle x^{2r}\) be written in hermite polynomial form?

 

[MountEvariste]

Are you looking for these expressions?

 

[topsquark]

How can \(\displaystyle x^{2r}\) be written in hermite polynomial form?

The Hermite polynomials are an orthogonal set so if you are looking for \(\displaystyle x^{2r} = a_0 H_0(x) + a_1 H_1 (x) + \text{ ...}\), then
\(\displaystyle a_n = \int_{-\infty}^{\infty} x^{2r} \left ( H_n (x) \right ) ^2 e^{-x^2} ~ dx\)

That will get you the series term by term. If you want a more general expression you'll likely have to use the generating function to get the \(\displaystyle \left ( H_n (x) \right )^2\) expression.

-Dan