Asymptotics of Orthogonal Polynomials via Recurrence Relations

TL;DR

This paper introduces a systematic method for deriving asymptotic formulas for orthogonal polynomials using recurrence relations, demonstrated through Legendre and Hermite polynomials, and addresses a recent open problem by Ismail.

Contribution

The paper presents a new, simple technique for asymptotic analysis of orthogonal polynomials based on recurrence relations, applicable to various polynomial families.

Findings

Derived asymptotic formulas for Legendre and Hermite polynomials

Provided a solution to a recent open problem by M. E. H. Ismail

Demonstrated the effectiveness of the recurrence relation method

Abstract

We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this technique is to provide a solution to a problem recently raised by M. E. H. Ismail.