Some Neumann-Bessel series and the Laplacian on polygons

TL;DR

This paper generalizes the Neumann expansion for the Laplacian eigenstates in regular polygons, evaluating sums of Neumann series with Bessel and trigonometric functions as finite trigonometric sums.

Contribution

It introduces a new generalization of the Neumann expansion for Laplacian eigenstates in regular polygons and evaluates related sums explicitly.

Findings

Finite sums of Neumann series with Bessel and trigonometric functions are expressed as trigonometric sums.

The work extends the understanding of Laplacian eigenstates in polygonal domains.

Explicit formulas for these sums are provided.

Abstract

Several sums of Neumann series with Bessel and trigonometric functions are evaluated, as finite sums of trigonometric functions. They arise from a generalization of the Neumann expansion of the eigenstates of the Laplacian in regular polygons.