Asymptotic behavior of varying discrete Jacobi--Sobolev orthogonal polynomials

TL;DR

This paper investigates the asymptotic properties and zero behavior of Jacobi--Sobolev orthogonal polynomials with a varying discrete inner product, highlighting differences from classical Jacobi polynomials through Mehler-Heine type formulas.

Contribution

It provides new asymptotic results and zero approximations for Jacobi--Sobolev orthogonal polynomials, extending recent literature findings.

Findings

Derivation of Mehler-Heine type formulas for Sobolev polynomials

Asymptotic descriptions of zeros in terms of special functions

Generalization of recent results in the literature

Abstract

In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions. We generalize some results appeared in the literature very recently.