arXiv subjects

Lahcen Lamgouni

Publications and source records attributed to Lahcen Lamgouni.

3 recordsLinked to original sources

arXiv2025

Functional equation for LC-functions with even or odd modulator

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.

math.NT↗

arXiv2023

Spherical-vectors and geometric interpretation of unit quaternions

In this article, we introduce and study the concept of $\textit{spherical-vectors}$, which can be perceived as a natural extension of the arguments of complex numbers in the context of quaternions. We initially establish foundational properties of these distinct vectors, followed by a demonstration, through the transfer of structure, that spherical-vectors constitute a non-abelian additive group, isomorphic to the group of unit quaternions. This identification facilitates the presentation of a novel polar form of quaternions, highlighting its algebraic properties, as well as the algebraic properties of the exponential writing. Furthermore, it enables the depiction of unit quaternions on the unit sphere of $\mathbb{R}^3$, allowing for a geometric interpretation of their multiplication.

math.RA↗

arXiv2023

C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function

Let $f(t)=\sum_{n=0}^{+\infty}\frac{C_{f,n}}{n!}t^n$ be an analytic function at $0$, and let $C_{f, n}(x)=\sum_{k=0}^{n}\binom{n}{k}C_{f,k} x^{n-k}$ be the sequence of Appell polynomials, referred to as $\textit{C-polynomials associated to f}$, constructed from the sequence of coefficients $C_{f,n}$. We also define $P_{f,n}(x)$ as the sequence of C-polynomials associated to the function $p_{f}(t)=f(t)(e^t-1)/t$, called $\textit{P-polynomials associated to f}$. This work investigates three main topics. Firstly, we examine the properties of C-polynomials and P-polynomials and the underlying features that connect them. Secondly, drawing inspiration from the definition of P-polynomials and subject to an additional condition on $f$, we introduce and study the complex-variable function $P_{f}(s,z)=\sum_{k=0}^{+\infty}\binom{z}{k}P_{f,k}s^{z-k}$, which generalizes the $s^z$ function and is denoted by $s^{(z,f)}$. Thirdly, the paper's significant contribution is the generalization of the Hurwitz zeta function and its fundamental properties, most notably Hurwitz's formula, by constructing a novel class of functions defined by $L(z,f)=\sum_{n=n_{f}}^{+\infty}n^{(-z,f)}$, which are intrinsically linked to C-polynomials and referred to as $\textit{LC-functions associated to f}$ (the constant $n_{f}$ is a positive integer dependent on the choice of $f$). This research offers a detailed analysis of C-polynomials, P-polynomials, and LC-functions associated to a given analytic function $f$, thoroughly examining their interrelations and introducing unexplored research directions for a novel and expansive class of LC-functions possessing a functional equation equivalent to that of the Riemann zeta function, thereby highlighting the potential applications and implications of the findings.

math.NT↗