Asymptotics (p→∞) of L_p-norms of hypergeometric orthogonal polynomials

摘要

The determination of the weighted L p norms of the real orthogonal polynomials of hypergeometric type {yn(x)} is not only a very important problem per se in the theory of special functions, but also because of their recent entropic characterization and applications in quantum chemistry, quantum physics and information theory. Indeed, they essentially describe the pth-order Rényi and Tsallis entropies of the numerous quantum systems whose wavefunctions are controlled by these polynomials.Moreover, for different values of p, up to a constant factor, these norms characterize various fundamental and experimentally accessible quantities of manyelectron systems. As well, the L p norms have been used to develop and interpret all energy components in the density-functional theory of the ground-state of atoms and molecules. The asymptotics of these quantities when n →∞and p > 0 have been recently calculated for Hermite polynomials, although not yet for Laguerre and Jacobipolynomials. Here, we determine the asymptotics (p→∞, n fixed) of the weighted L p norms for general orthogonal polynomials in terms of the weight function and the coefficients of the second-order hypergeometric differential equation that they satisfy,and we apply it to the three classical families of real orthogonal polynomials. Moreover we analyse and discuss the monotonicity of this asymptotics, and we carry out a detailed numerical study of it.