# Generalized Cosecant Numbers and the Hurwitz Zeta Function

## TL;DR

This paper introduces generalized cosecant numbers as coefficients in a power series expansion of a fundamental function, revealing their polynomial nature in the parameter and linking them to the Hurwitz zeta function for even powers.

## Contribution

It develops a partition method for calculating these coefficients for all complex values of c
hoe, and establishes their polynomial form and connection to symmetric polynomials and the Hurwitz zeta function.

## Key findings

- Generalized cosecant numbers are polynomials in c
hoe of degree k.
- Explicit formulas for highest order coefficients are provided.
- Hurwitz zeta function values for even powers are expressed in terms of these numbers.

## Abstract

This announcement paper summarises recent development concerning the generalized cosecant numbers $c_{\rho,k}$, which represent the coefficients of the power series expansion for the important fundamental function $z^{\rho}/\sin^{\rho} z$. These coefficients are obtained for all, including complex, values of $\rho$ via the partition method for a power series expansion, which is more versatile than the standard Taylor series approach, but yields the same results as the latter when both can be applied, though in a different form. Surprisingly, the generalized cosecant numbers are polynomials in $\rho$ of degree $k$, where $k$ is the power of $z$. General formulas for the coefficients of the highest order terms in the generalized cosecant numbers are presented. It is then shown how the generalized cosecant numbers are related to the specific symmetric polynomials 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.