Classical2-orthogonal polynomials and differential equations

Abstract

We construct the linear differential equations of third order satisfied by the classical2-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=1,4are polynomials whose degrees are, respectively, less than or equal to4,3,2, and1. We also show that the coefficientR4,n(x)can be written asR4,n(x)=F1,n(x)S3(x), whereS3(x)is a polynomial of degree less than or equal to3with coefficients independent ofnanddeg⁡(F1,n(x))≤1. We derive these equations in some cases and we also quote some classical2-orthogonal polynomials, which were the subject of a deep study.