Sums of Powers of Sine and Generalized Bernoulli Polynomials

TL;DR

This paper derives formulas for sums involving powers of sine functions using Generalized Bernoulli and Euler polynomials, and applies one to obtain an integral representation of the Riemann zeta function.

Contribution

It introduces new formulas connecting sine power sums with Bernoulli and Euler polynomials and provides an integral representation of the Riemann zeta function.

Findings

Formulas for sums of powers of sine functions in terms of Bernoulli and Euler polynomials

A new integral representation of the Riemann zeta function

Connections between trigonometric sums and special polynomials

Abstract

We produce formulas for $$\sum_{j=1}^{2^{n-2}}\frac{1}{\sin^s\left(\frac{(2j-1)\pi}{2^n}\right)}$$ in terms of Generalized Bernoulli and Euler polynomials and use one of the formulas to produce a nice integral representation of the Riemann zeta function.