Explicit Evaluations and Reciprocity Theorems for Finite Trigonometric Sums

This evaluation can be found in standard tables of series, such as those of E. R. Hansen [44, p. 262, eq. (30.1.2)], L. B. W. Jolly [56, pp. 102–103, eq. (352)], and A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev [70, p. 646, eq. 6]. We do not know who first proved (1.1), but several proofs exist. If we replace the power 2 on the left side by an arbitrary positive even power, finding an explicit evaluation becomes more difficult. Often, in applications, one does not need an explicit evaluation in closed form but only an asymptotic formula; for example, see T. M. Apostol’s paper [4]. The earliest evaluation known to us of a sum of the type (1.1) is by M. Stern [79, p. 155], who in 1861 proved that, for any positive odd integer k,