Closed-form summation of some trigonometric series

Abstract:

The problem of numerical evaluation of the classical trigonometric series \[ {S_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\sin (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}}\quad {\text {and}}\quad } {C_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\cos (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}},} \] where $\nu > 1$ in the case of ${S_{2n}}(\alpha )$ and ${C_{2n + 1}}(\alpha )$ with $n = 1,2,3, \ldots$ has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when $\alpha$ is equal to a rational multiple of $2\pi$, these series can in the general case be summed in closed form in terms of known constants and special functions. General formulae giving ${C_\nu }(\alpha )$ and ${S_\nu }(\alpha )$ in terms of the generalized Riemann zeta function and the cosine and sine functions, respectively, are derived. Some simpler variants of these formulae are obtained, and various special results are established.