Sums of the Even Integral Powers ... Sums of the Even Integral Powers of the Cosecant and Secant

Special finite sums of the even powers of the cosecant and of the secant are studied, ∑ k csc (kπ/N) and ∑ k sec (kπ/N), with positive integers N ≥ 3,m and 1 ≤ k < N/2 . The main result of this article is that these power sums are even polynomials in N , of order 2m, whose coefficients are rational. The approach is based on new differential identities for the functions cscz and secz. The Mittag-Leffler expansions for these functions are invoked and the corresponding infinite series are summed to give closed form expressions for the desired sums. Specific polynomial coefficients are obtained, for 1 ≤ m ≤ 6 and for all N ≥ 3, to illustrate the method. Similar sums involving the cotangent and the tangent are also examined.