Generalized Freud’s equation and level densities with polynomial potential

Abstract

We study orthogonal polynomials with weight exp[−N V (x)], where2k /2k is a polynomial of order 2d. We derive the generalised Freud's equations for d = 3, 4 and 5 and using this obtain Rµ = hµ/hµ−1, where hµ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of Rµ, are obtained using Freud's equation and using this, explicit results of level densities as N → ∞ are derived.