Skew-orthogonal polynomials for a quartic Freud weight: two classes of quasi-orthogonal polynomials

Abstract

This work is a thorough investigation of skew-orthogonal polynomials with respect to a quartic Freud weight. We provide an explicit method to evaluate skew-orthogonal polynomials of any degree as linear combinations of orthogonal polynomials. The coefficients of these combinations can be evaluated via novel recursive relations. Moreover, we observe that skew-orthogonal polynomials with even and odd degree constitute two families of quasi-orthogonal polynomials with respect to two different semi-classical Laguerre weights, and we provide the first instance of closed recursive relations involving skew-orthogonal polynomials only.