Error bounds for the asymptotic expansions of the Jacobi polynomials

TL;DR

This paper develops explicit error bounds for the asymptotic expansions of Jacobi polynomials in the oscillatory region, introducing a new technique to handle phase singularities and a recurrence formula for coefficients.

Contribution

It provides the first explicit error bounds and a novel method to address phase singularities in the asymptotic analysis of Jacobi polynomials.

Findings

Derived explicit, computable error bounds for Jacobi polynomial expansions.

Introduced a new technique to handle logarithmic singularities in phase functions.

Developed a recurrence formula for asymptotic expansion coefficients.

Abstract

This paper aims to derive explicit and computable error bounds for the asymptotic expansion of the Jacobi polynomials as their degree approaches infinity, using an integral method. The analysis focuses on the outer or oscillatory region of these polynomials. A novel technique is introduced to address the challenges posed by the logarithmic singularity in the phase function of the integral representation of Jacobi polynomials. A recurrence formula is also developed to compute the coefficients in the asymptotic expansions.