On some properties of polycosecant numbers and polycotangent numbers

TL;DR

This paper explores algebraic properties of polycosecant and polycotangent numbers, level two analogues of poly-Bernoulli numbers, and generalizes duality formulas, highlighting areas for future research such as the zeta function interpolation.

Contribution

It introduces algebraic properties of polycosecant and polycotangent numbers and extends duality formulas, advancing understanding of these lesser-known mathematical objects.

Findings

Derived algebraic properties of polycosecant numbers

Extended duality formulas for polycosecant and polycotangent numbers

Identified open problems like zeta function interpolation

Abstract

Polycosecant numbers and polycotangent numbers are introduced as level two analogues of poly-Bernoulli numbers. It is shown that polycosecant numbers and polycotangent numbers satisfy many formulas similar to those of poly-Bernoulli numbers. However, there is much unknown about polycotangent numbers. For example, the zeta function interpolating them at non-positive integers has not yet been constructed. In this paper, we show some algebraic properties of polycosecant numbers and polycotangent numbers. Also, we generalize duality formulas for polycosecant numbers which essentially include those for polycotangent numbers.