Generalized Cosecant Numbers and the Hurwitz Zeta Function

Abstract

This paper presents recent developments concerning the generalized cosecant numbers c ρ,k , which emerge as the coefficients of the power series expansion for the important fundamental function z ρ / sin ρ z. These coefficients can be computed for all, including complex, values of ρ by using the relatively novel graphical method known as the partition method for a power series expansion or by using intrinsic routines in Mathematica. In fact, they represent polynomials in ρ of degree k, where k is the power of z.In addition, though related to the Bernoulli numbers, they possess more properties and do not diverge like the former. The partition method for a power series expansion has the advantage that it yields the k-behaviour of the highest order coefficients. Thus, general formulas for such coefficients are derived by considering the properties of the highest part partitions. It is then shown how the generalized cosecant numbers are related to the specific symmetric polynomials that arise from summing over quadratic powers of integers. Consequently, integral values of the Hurwitz zeta function for even powers are expressed for the first time ever in terms of ratios of the generalized cosecant numbers.

On a finite trigonometric sum related to Dedekind sum

“…From which the cosecant numbers are extracted, as the rational coefficients of it's power series expansion, a generalization of these numbers is given in the paper [6] of V. Kowalenko.…”