International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 12640, 32 pages
doi:10.1155/IJMMS/2006/12640
Classical 2-orthogonal polynomials and differential equations
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)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.