A note on trigonometric approximations of Bessel functions of the first kind and trigonometric power sums

TL;DR

This paper explores improved methods for approximating Bessel functions using finite trigonometric sums, leveraging recent series evaluations to enhance efficiency and derive new parametric sums involving powers of sine and cosine.

Contribution

It introduces a novel approach to approximate Bessel functions with optimized trigonometric sums and derives new parametric sums based on recent series evaluations.

Findings

Efficient trigonometric approximations of Bessel functions.

New parametric sums involving powers of sine and cosine.

Simplified evaluations of Neumann-Bessel series.

Abstract

I reconsider the approximation of Bessel functions with finite sums of trigonometric functions, in the light of recent evaluations of Neumann-Bessel series with trigonometric coefficients. A proper choice of the angle allows for an efficient choice of the trigonometric sum. Based on these series, I also obtain straightforward non-standard evaluations of new parametric sums with powers of cosine and sine functions.