The second secant zeta function and its anti-periodization evaluated at $1/\sqrt{n}$

TL;DR

This paper investigates the 2-periodic secant zeta function and its anti-periodization, providing explicit evaluations at reciprocal square roots of positive integers through reformulating identities as Abel equations.

Contribution

It introduces a novel approach to evaluate the secant zeta function and its anti-periodization at specific points using Abel equation reformulations, linked to fluid dynamics applications.

Findings

Explicit values of the functions at 1/√n for positive integers n.

Reformulation of identities as Abel equations.

Application to fluid dynamics problems.

Abstract

The 2-periodic secant zeta function $\psi(r)$ and its $1$-antiperiodization $f(r)=(\psi(r)-\psi(r+1))/2$, arising from a fluid dynamics application, are investigated. In particular, their values at $1/\sqrt{n}$ for positive integers $n$ are determined, using a reformulation of an identity satisfied by $\psi$ as an Abel equation.