Asymptotics of $L_p$-norms of Hermite polynomials

The asymptotic of the weighted $L_p$-norms of Hermite polynomials is determined for $n \to \infty$ and $p>0$. The result is motivated by calculations the Rényi entropy of the quantum-mechanical probability density of the highly-excited (Rydberg) states of the isotropic oscillator.