Sums of Powers of Sine and Generalized Bernoulli Polynomials
Leon D. Fairbanks
TL;DR
The paper analyzes sums of inverse powers of sine evaluated at odd multiples of $\frac{\pi}{2^n}$, linking them to the Riemann zeta function through limiting processes and expressing them via generalized Bernoulli and Euler polynomials. It introduces a matrix-analytic framework that encodes negative odd powers of sine, enabling explicit finite-sum representations for $S(2m+1,n)$ and related even- and odd-power sums, and derives an integral representation for $\zeta(2j+1)$ in terms of Euler polynomials. For negative even powers, the authors identify leading asymptotic terms tied to $\zeta(2k)$ and describe a zero-sum structure in the associated matrices, providing computationally efficient alternatives to direct summation. Overall, the work reveals deep connections between trigonometric sums, Bernoulli/Euler polynomial theory, and zeta-values, with potential implications for efficient zeta evaluations and analytic number theory.
Abstract
We produce formulas for $$\sum_{j=1}^{2^{n-2}}\frac{1}{\sin^s\left(\frac{(2j-1)π}{2^n}\right)}$$ 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.