Properties of a class of generalized Freud polynomials

Abstract

Semiclassical orthogonal polynomials are polynomials orthogonal with respect to semiclassical weights. The fascinating link between semiclassical orthogonal polynomials and discrete integrable equations can be traced back to the work of Shohat and Freud and later by Bonan and Nevai; orthogonal polynomials with Freud-type exponential weights have three-term recurrence coe cients that satisfy nonlinear second order difference equations. Fokas, Its and Kitaev identi ed these equations as discrete Painlev e equations. Magnus related the recurrence coe cients of orthogonal polynomials with respect to the Freud weight and classical solutions of the fourth Painlev e equation. We extend Magnus's results for Freud weight, by considering polynomials orthogonal with respect to a generalized Freud weight, by studying the theory of Painlev e equations. These generalized Freud polynomials arise from a symmetrization of semiclassical Laguerre polynomials. We prove that the coe cients in the three-term recurrence relation associated with a generalized Freud weight can be expressed in terms of Wronskians of parabolic cylinder functions that appear in the description of special function solutions of the fourth Painlev e equation. This closed form expression for the recurrence coe cients allows the investigation of certain properties of the generalized Freud polynomials. We obtain an explicit formulation for the generalized Freud polynomials in terms of the recurrence coe cients, investigate the higher order moments, as well as the Pearson equation satis ed by the generalized Freud weight. We also derive a second-order linear ordinary di erential equation and a di erential-di erence equation satis ed by the generalized Freud polynomials and we use the di erential equation to study some properties of the zeros of generalized Freud polynomials. Furthermore, we obtain limit relations for the recurrence coe cients of the generalized Freud polynomials using Freud's Kunstgri method. We verify the existence of an asymptotic series for the recurrence coe cient using an extension of the result by Bleher and Its [17] and we provide an asymptotic expansion for the recurrence coe cients of the three-term recurrence relation satis ed by monic generalized Freud polynomials.