Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

Abstract

The determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states. For the D-dimensional hydrogenic and oscillator-like systems, the wavefunctions of the corresponding bound states are controlled by the Laguerre (L_α^m(x)) and Gegenbauer (C_α^m(x)) polynomials in both position and momentum spaces, where the parameter α linearly depends on D. In this work we study the asymptotic behavior as α→∞ of the associated entropy-like integral functionals of these two families of hypergeometric polynomials.