Asymptotic Bessel-function expansions for Legendre and Jacobi functions

TL;DR

This paper introduces new asymptotic series for Legendre and Jacobi functions using Bessel functions, enhancing scattering problem calculations and providing correction terms to existing approximations.

Contribution

It presents a novel method based on Barnes-type representations to derive asymptotic expansions for these special functions, improving upon previous results.

Findings

New asymptotic series for Legendre and Jacobi functions

Enhanced accuracy in scattering calculations

Method based on Barnes-type representations

Abstract

We present new asymptotic series for the Legendre and Jacobi functions of the first and second kinds in terms of Bessel functions with appropriate arguments. The results are useful in the context of scattering problems, improve on known limiting results, and allow the calculation of corrections to the leading Bessel-function approximations for these functions. Our derivations of these series are based on Barnes-type representations of the Legendre, Jacobi, and Bessel functions; our method appears to be new. We use the results, finally, to obtain asymptotic Bessel function expansions for the rotation functions needed to describe the scattering of particles with spin.