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

TL;DR

This paper investigates skew-orthogonal polynomials related to a quartic Freud weight, introducing explicit evaluation methods, recursive relations, and identifying two families of quasi-orthogonal polynomials with novel properties.

Contribution

It provides an explicit method to evaluate skew-orthogonal polynomials as linear combinations of orthogonal polynomials with recursive coefficients and identifies two classes of quasi-orthogonal polynomials.

Findings

Explicit evaluation method for skew-orthogonal polynomials.

Recursive relations for coefficients of polynomial combinations.

Identification of two families 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.