MHB Hermite Function: Writing x^2r in Polynomial Form

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To express x^{2r} in Hermite polynomial form, one can utilize the orthogonality of Hermite polynomials. The coefficients a_n can be calculated using the integral a_n = ∫_{-∞}^{∞} x^{2r} (H_n(x))^2 e^{-x^2} dx. This method allows for the determination of the series term by term. For a more general expression, employing the generating function is recommended to derive the (H_n(x))^2 term. The discussion emphasizes the importance of these mathematical tools in achieving the desired polynomial representation.
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How can $$x^{2r}$$ be written in hermite polynomial form?
 
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Another said:
How can $$x^{2r}$$ be written in hermite polynomial form?
The Hermite polynomials are an orthogonal set so if you are looking for [math]x^{2r} = a_0 H_0(x) + a_1 H_1 (x) + \text{ ...}[/math], then
[math]a_n = \int_{-\infty}^{\infty} x^{2r} \left ( H_n (x) \right ) ^2 e^{-x^2} ~ dx[/math]

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 [math]\left ( H_n (x) \right )^2[/math] expression.

-Dan
 
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