Differential equations for discrete Jacobi–Sobolev orthogonal polynomials

Abstract

The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at −1 and/or +1. In particular, we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that when the Jacobi parameters α and β are nonnegative integers the Jacobi-Sobolev orthogonal polynomials are eigenfunctions of a differential operator of finite order (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrices which define the discrete Jacobi-Sobolev bilinear form.

Bispectral Laguerre type polynomials

“…Other examples of bispectral polynomials are the Krall-Sobolev polynomials (see [24,1,9,10]), the exceptional polynomials (see [14,4,5,6,7,13], and references therein) or the Grünbaum and Haine extension of Krall polynomials ( [16]; see also [19,3]). In these cases, the associated operators (in the discrete and continuous variable) have order greater than 2.…”

“…Using the D-operator method, it is proved in [10] (see Theorem 3.1 and the beginning of Section 4 of that paper) that the polynomials (q n ) n are eigenfunctions of a higher-order differential operator (acting on the continuous variable x) of the form…”

Pencils of Semi-Infinite Matrices and Orthogonal Polynomials

“…Discrete Sobolev polynomials from the case C ðÞ may possess higher-order differential equations. This subject has a long history, see historical remarks in recent papers [19,20]. For polynomials (9) some simple general properties of zeros were studied in ref.…”

From Krall discrete orthogonal polynomials to Krall polynomials

“…Orthogonal polynomials (P u,v κ,σ n ) n with respect to (3.24) can be expanded in terms of three consecutive Jacobi polynomials as follows (see Theorem 1.1 in [14]; also [23]) (it is now better to use determinantal notation)…”