On some classical type Sobolev orthogonal polynomials

Abstract

In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: 2F2(−n,1;α+1,κ+1;x) and 3F2(−n,n+α+β+1,1;α+1,κ+1;x) (α,β,κ>−1, n=0,1,…), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: Ly+λnDy=0, where L,D are linear differential operators with polynomial coefficients not depending on n. For nonnegative integer values of the parameter κ these polynomials are Sobolev orthogonal polynomials with some explicitly given measures. Some basic properties of these polynomials, including recurrence relations, are obtained. Some generalizations of these polynomials are discussed as well.