Sums of Powers of Sine and Generalized Bernoulli 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.