Strong and ratio asymptotics for Laguerre polynomials revisited

TL;DR

This paper revisits the strong asymptotic behavior of Laguerre polynomials in the complex plane, introducing an alternative Bessel function expansion that simplifies the computation of higher order coefficients and enhances asymptotic analysis.

Contribution

It proposes using Buchholz's Bessel function expansion for Laguerre polynomials, enabling straightforward calculation of higher order terms and improving asymptotic ratio analysis.

Findings

Derived extra terms in Laguerre polynomial ratio asymptotics

Simplified computation of higher order coefficients

Enhanced understanding of Laguerre polynomial asymptotics

Abstract

In this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex plane. The leading behavior is well known from Perron and Mehler-Heine formulas, but higher order coefficients, which are important in the context of Krall-Laguerre or Laguerre-Sobolev-type orthogonal polynomials, are notoriously difficult to compute. In this paper, we propose the use of an alternative expansion, due to Buchholz, in terms of Bessel functions of the first kind. The coefficients in this expansion can be obtained in a straightforward way using symbolic computation. As an application, we derive extra terms in the asymptotic expansion of ratios of Laguerre polynomials in $C\[0,\infty)$.