Infinite series representations for Bessel functions of the first kind of integer order

TL;DR

This paper introduces three simple, elementary-function-based infinite series representations for Bessel functions of the first kind with integer orders, derived via Fourier series, which can be adapted by parameter choices and used to generate new trigonometric series.

Contribution

The paper presents novel non-power infinite series for Bessel functions, derived as Fourier series, with adjustable parameters and practical truncations that preserve key behaviors.

Findings

Series contain only elementary functions

Truncated series mimic Bessel functions at large x

New series expansions for trigonometric functions

Abstract

We have discovered three non-power infinite series representations for Bessel functions of the first kind of integer orders and real arguments. These series contain only elementary functions and are remarkably simple. Each series was derived as a Fourier series of a certain function that contains Bessel function. The series contain parameter $b$ by setting which to specific values one can change specific form of series. Truncated series retain qualitatively behaviour of Bessel functions at large $x$: they have sine-like shape with decreasing amplitude. Derived series allow to obtain new series expansions for trigonometric functions.