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Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.

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Random variables with density distributions given by squared Hermite polynomials

I was wondering whether anything is known on the following: Let $h_k (x)= (-1)^k e^{x^2/2} \frac {d^k}{dx^k} \, e^{-x^2/2}$, $k \geq 0$, be the classical Hermite polynomials ($h_0(x) = 1$, $h_1(x) = x …