Classical 2-orthogonal polynomials and differential equations

Abstract

We construct the linear differential equations of third order satisfied by the classical 2-orthogonal polynomials. We show that these differential equations have the following form: R4,n(x)Pn+3(3)(x)+R3,n(x)P″n+3(x)+R2,n(x)P′n+3(x)+R1,n(x)Pn+3(x)=0, where the coefficients {Rk,n(x)}k=1,4 are polynomials whose degrees are, respectively, less than or equal to 4, 3, 2, and 1. We also show that the coefficient R4,n(x) can be written as R4,n(x)=F1,n(x)S3(x), where S3(x) is a polynomial of degree less than or equal to 3 with coefficients independent of n and deg⁡(F1,n(x))≤1. We derive these equations in some cases and we also quote some classical 2-orthogonal polynomials, which were the subject of a deep study.