Szeg\"o kernels and asymptotic expansions for Legendre polynomials

TL;DR

This paper introduces a geometric method using Szeg"o kernels to derive comprehensive asymptotic expansions for Legendre polynomials across expanding subintervals of their domain.

Contribution

It presents a novel geometric approach leveraging Szeg"o kernels of Fermat quadrics to analyze Legendre polynomial asymptotics.

Findings

Complete asymptotic expansions on expanding subintervals

New geometric perspective on Legendre polynomial asymptotics

Asymptotic formulas derived using Szeg"o kernel techniques

Abstract

We present a geometric approach to the asymptotics of the Legendre polynomials $P_{k,n+1}$, based on the Szeg\"o kernel of the Fermat quadric hypersurface, and leading to complete asymptotic expansions holding on expanding subintervals of $[-1,1]$.