Generalized cosecant numbers and trigonometric inverse power sums

Abstract

The generalized cosecant numbers denoted here by c?,k represent the coefficients of the power series expansion or generating function of the fundamental function x?= sin?x. In actual fact, these interesting numbers are polynomials in ? of degree k, whose coefficients are only dependent upon k. In this paper we show how they emerge in the calculation of trigonometric inverse power sums. After introducing the generalized cosecant numbers we present a novel and elegant integral approach for computing the Gardner-Fisher trigonometric inverse power sum, which is given by Sv,2(m) = (?/2m)2v ?m-1,k=1 cos-2v (k?/2m), where m and v are positive integers. This method not only confirms the solutions obtained earlier by an empirical method, but it is also much more expedient from a computational point of view. By comparing the formulas from both methods, we derive several new and interesting number-theoretical results involving symmetric polynomials over the set of quadratic powers up to (v-1)2 and the generalized cosecant numbers.