The integration shown here appears in the stationary solution of the perturbed non linear oscillators in Physics. Is there any direct way to perform the definite integration of the form shown here $\int^{+\infty}_{-\infty} x^r \mathrm{Exp}[-x^2] \mathrm{H}^2_n[x] \mathrm{d}x$. I want the solution of the integral for $r \geq 4$. I hope there maybe some techniques which can be used to calculate the integration of above integral. Please i need suggestion from this forum to solve this integration. Highly appreciated!
2026-03-25 11:09:54.1774436994
Integration involving hermite polynomials
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Not a full answer (My reputation's too low to make a comment), but here's maybe some helpful information for solving the case when r = 1. (I know you need r>=4, but it might be a good start).
Here it's shown that: $$\int e^{- 2 z^2} z H_n(z)^2 \, dz=-\frac{2 n H_{n-1}(z){}^2+H_n(z){}^2}{4 e^{2 z^2}}$$
Maybe with some clever substitutions with the identities given on the website, you can get the cases when r>4