Asymptotics of orthogonal polynomials generated by a Geronimus perturbation of the Laguerre measure

TL;DR

This paper investigates the asymptotic behavior of orthogonal polynomials generated by a Geronimus perturbation of the Laguerre measure, revealing their strong and relative asymptotics as the degree tends to infinity.

Contribution

It provides a detailed analysis of the asymptotics of orthogonal polynomials under a specific Geronimus spectral transformation of the Laguerre measure, including new asymptotic formulas.

Findings

Asymptotic formulas for the orthogonal polynomials as degree increases.

Comparison between perturbed and classical Laguerre polynomial asymptotics.

Impact of the parameter N and shift c on asymptotic behavior.

Abstract

This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure: \[ \frac{1}{x-c}x^{\alpha }e^{-x}dx+N\delta (x-c), \] for $x\in[0,\infty)$, $\alpha>-1$, a free parameter $N\in \mathbb{R}_{+}$ and a shift $c<0$. We analyze the asymptotic behavior (both strong and relative to classical Laguerre polynomials) of these orthogonal polynomials as $n$ tends to infinity.