On sums of powers of cosecs

TL;DR

This paper revisits and expands on historical methods for summing powers of cosecants, highlighting their relevance in physics and mathematics, and providing new generating functions with potential applications in field theory and entropy calculations.

Contribution

It re-exposes and generalizes early 19th-century results on sums of powers of cosec, emphasizing their ongoing importance and introducing new generating functions with theoretical significance.

Findings

Reproduction of Jeffery's 19th-century results

Introduction of simple trigonometric generating functions

Comments on applications to Rénnyi entropies

Abstract

The finite sums of powers of cosecs occur in numerous situations, both physical and mathematical, examples being the Casimir effect, Renyi entropy, Verlinde's formula and Dedekind sums. I here present some further discussion which consists mainly of a reprise of early work by H.M.Jeffery in 1862-64 which has fallen by the wayside and whose results are being reproduced up to the present day. The motivation is partly historical justice and partly that, because of the continuing appearance of the sums, his particular methods deserve re--exposure. For example, simple trigonometric generating functions are found and these have a field theoretic, Green function significance and I make a few comments in the topic of R\'enyi entropies.