Abstract
The fundamental information-theoretic measures (the R\'enyi R_p[ρ] and Tsallis T_p[ρ] entropies, p>0) of the highly-excited (Rydberg) quantum states of the D-dimensional (D>1) hydrogenic systems, which include the Shannon entropy (p → 1) and the disequilibrium (p = 2), are analytically determined by use of the strong asymptotics of the Laguerre orthogonal polynomials which control the wavefunctions of these states. We first realize that these quantities are derived from the entropic moments of the quantum-mechanical probability ρ(r) densities associated to the Rydberg hydrogenic wavefunctions Ψ_n,l,{μ}(r), which are closely connected to the L_p-norms of the associated Laguerre polynomials. Then, we determine the (n→∞)-asymptotics of these norms in terms of the basic parameters of our system (the dimensionality D, the nuclear charge and the hyperquantum numbers (n,l,{μ}) of the state) by use of recent techniques of approximation theory. Finally, these three entropic quantities are analytically and numerically discussed in terms of the basic parameters of the system for various particular states.