Functional equation for LC-functions with even or odd modulator

TL;DR

This paper proves a functional equation for a class of LC-functions linked to real-analytic functions, especially when their modulators are even or odd, generalizing the Hurwitz zeta function and deriving explicit value formulas.

Contribution

It establishes a functional equation for LC-functions with even or odd modulators, extending the Hurwitz formula and providing explicit value formulas at positive integers.

Findings

Functional equation analogous to Dirichlet L-functions for LC-functions with symmetric modulators.

Explicit formulas for LC-function values at positive integers depending on modulator parity.

Illustrative examples demonstrating the functional equation for different modulator parities.

Abstract

In a recent work, we introduced \textit{LC-functions} $L(s,f)$, associated to a certain real-analytic function $f$ at $0$, extending the concept of the Hurwitz zeta function and its formula. In this paper, we establish the existence of a functional equation for a specific class of LC-functions. More precisely, we demonstrate that if the function $p_f(t):=f(t)(e^t-1)/t$, called the \textit{modulator} of $L(s,f)$, exhibits even or odd symmetry, the \textit{LC-function formula} -- a generalization of the Hurwitz formula -- naturally simplifies to a functional equation analogous to that of the Dirichlet L-function $L(s,\chi)$, associated to a primitive character $\chi$ of inherent parity. Furthermore, using this equation, we derive a general formula for the values of these LC-functions at even or odd positive integers, depending on whether the modulator $p_f$ is even or odd, respectively. Two illustrative examples of the functional equation are provided for distinct parity of modulators.