Asymptotics of $L_p$-norms of Laguerre polynomials and entropic moments of $D$-dimensional oscillator

The asymptotics of the weighted $L_p$-norms of the Laguerre polynomials is determined for $n\to\infty$ and $p>0$. The result is motivated by calculations the entropic moments of the quantum-mechanical probability density of the highly-excited (Rydberg) states of $D$-dimensional oscillator.