Asymptotics for Laguerre-Sobolev type ortogonal polynomials modified within their oscillatory regime

TL;DR

This paper investigates the asymptotic behavior of Laguerre-Sobolev orthogonal polynomials with perturbations inside their oscillatory region, providing new insights into their recurrence relations and asymptotics.

Contribution

It introduces a detailed analysis of Laguerre-Sobolev orthogonal polynomials with internal perturbations, including their recurrence coefficients and asymptotic properties.

Findings

Derived five-term recurrence relation coefficients.

Established asymptotic behavior of these coefficients.

Analyzed outer relative asymptotics of the polynomials.

Abstract

In this paper we consider sequences of polynomials orthogonal with respect to certain discrete Laguerre-Sobolev inner product, with two perturbations (involving derivatives) located inside the oscillatory region for the classical Laguerre polynomials. We focus our attention on the representation of these polynomials in terms of the classical Laguerre polynomials and deduce the coefficients of their corresponding five-term recurrence relation, as well as the asymptotic behavior of these coefficients when the degree of the polynomials tends to infinity. Also, the outer relative asymptotics of orthogonal polynomials with respect to this discrete Sobolev inner product is analyzed.