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Functional equation for LC-functions with even or odd modulator

Lahcen Lamgouni

TL;DR

The paper shows that LC-functions generalizing the Hurwitz zeta admit a Dirichlet-L-function–type functional equation when their modulators are even or odd. By exploiting parity, the LC-formula simplifies to a compact equation involving F(s, f_{(2iπ)}) and a cos/sin factor, yielding explicit evaluations at even or odd positive integers. It also establishes conditions under which the LC-series converges in the half-plane σ>0 and provides concrete examples with cosine and hyperbolic sine modulators to illustrate the theory. These results deepen the analogy between LC-functions and Dirichlet L-functions, and they offer computable formulas for special values via Eulerian polynomials and related contour integrals.

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 $L(s,χ)$, associated to a primitive character $χ$ 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.

Functional equation for LC-functions with even or odd modulator

TL;DR

The paper shows that LC-functions generalizing the Hurwitz zeta admit a Dirichlet-L-function–type functional equation when their modulators are even or odd. By exploiting parity, the LC-formula simplifies to a compact equation involving F(s, f_{(2iπ)}) and a cos/sin factor, yielding explicit evaluations at even or odd positive integers. It also establishes conditions under which the LC-series converges in the half-plane σ>0 and provides concrete examples with cosine and hyperbolic sine modulators to illustrate the theory. These results deepen the analogy between LC-functions and Dirichlet L-functions, and they offer computable formulas for special values via Eulerian polynomials and related contour integrals.

Abstract

In a recent work, we introduced \textit{LC-functions} , associated to a certain real-analytic function at , 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 , called the \textit{modulator} of , 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 , associated to a primitive character 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 is even or odd, respectively. Two illustrative examples of the functional equation are provided for distinct parity of modulators.
Paper Structure (14 sections, 4 theorems, 108 equations, 4 figures)

This paper contains 14 sections, 4 theorems, 108 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background material
  3. Generalized complex exponentiation
  4. LC-functions
  5. FC-functions
  6. LC-formula
  7. The Main Theorem
  8. Functional equation for LC-functions with even or odd modulator
  9. Riemann zeta function as an LC-function with even modulator
  10. Specific values of LC-functions and FC-functions with even or odd modulators
  11. Series representation in the half-plane sigma > 0 for LC-functions with modulator vanishing at 0
  12. Applications
  13. Example of an LC-function with an even modulator
  14. Example of an LC-function with an odd modulator

Key Result

Theorem 1.1

Let $L(s,f)$ be an LC-function with modulator $p_f$ of even or odd parity. For all $s\in\mathbb{C}\setminus\{0\}$, Here $\kappa=0$ if $p_f$ is even and $\kappa=1$ if $p_f$ is odd.

Figures (4)

  • Figure 1: Let $s\in \mathbb{C}$ be fixed. The generalized complex exponentiation function $z\mapsto z^{(s,f)}$ is well-defined on $\Omega_f$, particularly for all integers $n\geq n_f$.
  • Figure 2: The Hankel contour $\mu_f$ is oriented counterclockwise around the negative real axis and the closed disk $\left\lvert z\right\rvert\leq r_f$, so as not to encircle any points of discontinuity of the function $1/(e^{-z} - 1)$ that lie within the region $\Omega_f$. The integer $m_f=\lfloor r_f/(2\pi)\rfloor+1$ is defined as the smallest positive integer $m$ such that $i2m\pi\in\Omega_f$.
  • Figure 3: This figure illustrates the Hankel contour $\mu_{f_{(2i\pi)}}$, shown to be equivalent to $\mu_{f_{(-2i\pi)}}$.
  • Figure 4: The Hankel contour $\mu_{\varphi_{(2i\pi)}}$ reduces to a circle $\mathcal{C}_r$ with radius $r$ such that $2\pi\left\lvert w\right\rvert<r<2\pi$.

Theorems & Definitions (9)

  • Theorem 1.1: cf. Theorem \ref{['t:Mai_The_v2']}
  • Remark 2.1
  • Theorem 3.1
  • Remark 3.1
  • proof : Proof of the Main Theroem
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof