The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving ₁F₂ Hypergeometric Functions That Arise from Them

Authors

Jack C. Straton, Portland State UniversityFollow

Published In

Axioms

Document Type

Article

Publication Date

2-2024

Subjects

Quantum physics, Bessel functions

Abstract

The Bessel function of the first kind JN^(kx) is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind IN^(kx). The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for JN^(kx) useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving ₁F₂ hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1/ (2ⁱ3^j5k7^l11^m13ⁿ17^o19^p) multiplying powers of the coefficient k.

Rights

Copyright: © 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Description

Locate the Document

https://doi.org/10.3390/ axioms13020134

DOI

10.3390/ axioms13020134

Persistent Identifier

https://archives.pdx.edu/ds/psu/41216

Citation Details

Straton, J.C. The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving ₁ F ₂ Hypergeometric Functions That Arise from Them. Axioms 2024, 13, 134. https://doi.org/10.3390/ axioms13020134