The Fourier - Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving [sub.1]IF/I[sub.2] Hypergeometric Functions That Arise from Them

The Bessel function of the first kind J[sub.N] (kx) is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind I[sub.N] (kx). The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel f...

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Bibliographic Details
Published inAxioms Vol. 13; no. 2
Main Author Straton, Jack C
Format Journal Article
LanguageEnglish
Published MDPI AG 01.02.2024
Subjects
Analysis
Bessel functions
Fourier analysis
Functions, Hypergeometric
Power series
United States
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Summary:The Bessel function of the first kind J[sub.N] (kx) is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind I[sub.N] (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 J[sub.N] (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 [sub.1] F[sub.2] hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1/(2[sup.i] 3[sup.j] 5[sup.k] 7[sup.l] 11[sup.m] 13[sup.n] 17[sup.o] 19[sup.p] ) multiplying powers of the coefficient k.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms13020134