Closed-form summation of some trigonometric series

Abstract

The problem of numerical evaluation of the classical trigonometric series \[ S ν ( α ) = ∑ k = 0 ∞ sin ⁡ ( 2 k + 1 ) α ( 2 k + 1 ) ν and C ν ( α ) = ∑ k = 0 ∞ cos ⁡ ( 2 k + 1 ) α ( 2 k + 1 ) ν , {S_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\sin (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}}\quad {\text {and}}\quad } {C_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\cos (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}},} \] where ν > 1 \nu > 1 in the case of S 2 n ( α ) {S_{2n}}(\alpha ) and C 2 n + 1 ( α ) {C_{2n + 1}}(\alpha ) with n = 1 , 2 , 3 , … n = 1,2,3, \ldots has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when α \alpha is equal to a rational multiple of 2 π 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 ν ( α ) {C_\nu }(\alpha ) and S ν ( α ) {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.