`International Journal of Mathematics and Mathematical SciencesVolume 2006 (2006), Article ID 12640, 32 pagesdoi:10.1155/IJMMS/2006/12640`

Boukhemis Ammar and Zerouki Ebtissem

Department of Mathematics, Faculty of Sciences, University of Annaba, BP 12, Annaba 23000, Algeria

Received 16 May 2005; Revised 17 April 2006; Accepted 25 April 2006

Copyright © 2006 Boukhemis Ammar and Zerouki Ebtissem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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)Pn+3(x)+R2,n(x)Pn+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.