Closed-form summation of some trigonometric series

Abstract

has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when a is equal to a rational multiple of 2k , these series can in the general case be summed in closed form in terms of known constants and special functions. General formulae giving Cv(a) and Sv(a) 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.

The structural elucidation of Eisenstein’s formula

“…[22,Egs. (8), (9), p. 342]) follow, and they are equivalent to Cvijović-Klinowski results [7] (cf. [6]).…”