Searcharxiv⌕ Search

arXiv subjects

José Oscar González-Cervantes

Publications and source records attributed to José Oscar González-Cervantes.

At least 19 recordsLinked to original sources
arXiv2026

A connection between quaternionic slice regular and Fueter hyperholormorphic functions

This paper presents some results for the slice regular functions and hyperholomorphic functions theories, as consequences of new relationships between the global operator $G$, Fueter operator $\mathcal D$ and Laplacian operator $Δ_{\mathbb{R}^{4}}$. Although our results are presented in the quaternionic setting, a fine interpretation in terms of vector calculus of the slice regular function theory yields an interesting relationship between this theory and harmonic analysis. Moreover, we introduce and study a submodule of the hyperholomorphic functions associated to the unit ball $\mathbb B^4(0,1)$ given in terms of new series expansions of these functions.

math.CV↗
arXiv2026

A Non-integer Dimensional Space Approach to the Moisil-Teodorescu Operator

The vector calculus in non-integer dimensional space (NIDS), including the NIDS version of the standard vector differential operators (gradient, divergence, and curl) is well-known. A deformation of the quaternionic Moisil-Teodorescu operator, written in terms of NIDS derivatives is the main purpose of this article. Along similar lines, we consider the NIDS reformulation of the quaternionic Bitsadze operator and the Lamé-Navier operator of the three-dimensional elasticity theory. Also, a quaternionic reformulation of a NIDS time-harmonic Maxwell system is introduced, whose solutions are directly related with those of the perturbed NIDS Moisil-Teodorescu operator. Finally, a generalized approach to the study is addressed.

math.CV↗
arXiv2026

Mobius invariant weighted reduced quaternionic Qp-modules

In this paper, we introduce some weighted reduced quaternionic modules induced by a reduced quaternionic Cauchy-Riemann operator associated to arbitrary orthogonal bases of R3. A weighted conformal invariant property of the modules is given.

math.CV↗
arXiv2026

On quaternionic analysis and a certain generalized fractal-fractional $ψ$-Fueter operator

This paper introduce a fractal ψ-Fueter operator in the quaternionic context inspired in the concepts of proportional derivative and Hausdorff derivative (fractal derivative) of a function with respect to a fractal measure. Moreover, we establish the corresponding Stokes and Borel- Pompeiu formulas associated to this generalized fractal ψ-Fueter operator with respect to a vector valued function induced by truncated exponential functions.

math.CV↗
arXiv2025

On slice regular fractal-fractional Dirichlet type spaces

In this paper, we study some families of right modules of quaternionic slice regular functions induced by a generalized fractal-fractional derivative with respect to a truncated quaternionic exponential function on slices. Important Banach spaces of slice regular functions, namely the Bergman and the Dirichlet modules, are two important elements of one of our families.

math.CV↗
arXiv2024

A Borel-Pompeiu formula in a $(q,q')$-model of quaternionic analysis

The study of $ψ-$hyperholomorphic functions defined on domains in $\mathbb R^4$ with values in $\mathbb H$, namely null-solutions of the $ψ-$Fueter operator, is a topic which captured great interest in quaternionic analysis. This class of functions is more general than that of Fueter regular functions. In the setting of $(q,q')-$calculus, also known as post quantum calculus, we introduce a deformation of the $ψ-$Fueter operator written in terms of suitable difference operators, which reduces to a deformed $q$ calculus when $q'=1$. We also prove the Stokes and Borel-Pompeiu formulas in this context. This work is the first investigation of results in quaternionic analysis in the setting of the $(q,q')-$calculus theory.

math.CV↗
arXiv2023

A quaternionic proportional fractional Fueter-type operator calculus

The main goal of this paper is to construct a proportional analogues of the quaternionic fractional Fueter-type operator recently introduced in the literature. We start by establishing a quaternionic version of the well-known proportional fractional integral and derivative with respect to a real-valued function via the Riemann-Liouville fractional derivative. As a main result, we prove a quaternionic proportional fractional Borel-Pompeiu formula based on a quaternionic proportional fractional Stokes formula. This tool in hand allows us to present a Cauchy integral type formula for the introduced quaternionic proportional fractional Fueter-type operator with respect to a real-valued function.

math.CV↗
arXiv2023

A bicomplex proportional fractional $(\vartheta,φ)-$weighted Cauchy-Riemann operator using Riemann-Liouville derivatives with respect to an hyperbolic-valued function

Based on the Riemann-Liouville derivatives with respect to functions taking values in the set of hyperbolic numbers, we consider a novel bicomplex proportional fractional $(\vartheta,φ)-$weighted Cauchy-Riemann operator, involving weights hyperbolic orthogonal bicomplex functions. This operator is defined for the first time here, and its associated fractional Borel-Pompeiu formula is proved as the main result.

math.CV↗
arXiv2023

Fractional slice regular functions of a quaternionic variable

The theory of slice regular functions of a quaternionic variable on the unit ball of the quaternions was introduced by Gentili and Struppa in 2006 and nowadays it is a well established function theory, especially in view of its applications to operator theory. In this paper, we introduce the notion of fractional slice regular functions of a quaternionic variable defined as null-solutions of a fractional Cauchy-Riemann operators. We present a fractional Cauchy-Riemann operator in the sense of Riemann-Liouville and then in the sense of Caputo, with orders associated to an element of $(0,1)\times \mathbb R \times (0,1)\times \mathbb R$ for some axially symmetric slice domains which are new in the literature. We prove a version of the representation theorem, of the splitting lemma and we discuss a series expansion.

math.FA↗
arXiv2022

A fractional Borel-Pompeiu type formula for holomorphic functions of two complex variables

The present paper is a continuation of our work [11], where we introduced a fractional operator calculus related to a fractional $ψ-$Fueter operator in the one-dimensional Riemann-Liouville derivative sense in each direction of the quaternionic structure, that depends on an additional vector of complex parameters with fractional real parts. This allowed us also to study a pair of lower order fractional operators and prove the associated analogues of both Stokes and Borel-Pompieu formulas for holomorphic functions in two complex variables.

math.CV↗
arXiv2021

On fiber bundles and quaternionic slice regular functions

The papers \cite{O1,O2} are the first works to apply the theory of fiber bundles in the study of the quaternionic slice regular functions. The main goal of the present work is to extend the results given in \cite{O1}, where the quaternionic right linear space of quaternionic slice regular functions was presented as the base space of a fiber bundle. When the quaternionic right linear space of quaternionic slice regular functions is associated to certain domains then this paper shows that the elements of total space, given in \cite{O1}, are defined from a pair of harmonic functions and a pair of orthogonal vectors. Simplifying the computations presented in \cite{O1}, where each element of the total space is formed by two pair of conjugate harmonic functions and a pair of orthogonal unit vectors. This work also gives some interpretations of the behavior of the zero sets of some quaternionic slice regular polynomials in terms of the theory of fiber bundles.

math.CV↗
arXiv2021

On some quaternionic generalized slice regular functions

The quaternionic valued functions of a quaternionic variable, often referred to as slice regular functions has been studied extensively due to the large number of generali\-zed results of the theory of one complex variable, see \cite{cgs,CSS,GSC,GS2,gssbook,gp,gpr,GS} and the references given there. Recently, several global properties of these functions has been found of the study of a differential operator, see \cite{GlobalOp,GP_2, G, GG1,GG2}. Particularly, given a structural set $ψ$ the Borel-Pompieu formula induced by the operator ${}^ψG$ and its consequences in the slice regular function theory were studied in \cite{GG1}. The aim of this paper is to present some global and local properties of a kind of quaternionic generalized slice regular functions. We shall see that the global properties are consequences of the study of the perturbed global-type operator: \begin{align*} {}^ψG_v [f] := {}^ψ G [f] -\frac{{\bf x}_ψ}{ 2} ({\bf x}_ψ v + v {\bf x}_ψ ) f , \end{align*} where $v$ is a quaternionic constant and $f$ is a quaternionic-valued continuously differentiable function with domain in $\mathbb H$ since our generalized slice regular function space coincides with $\textrm{Ker} {}^{ψ_{\textrm{st}}}G_v$ associated to an axially symmetric s-domain, where the $ψ_{\textrm{st}}$ is standard structural set. Among the local properties studied in this work are the versions of Splitting Lemma and Representation Theorem that show us a deep relationship between this generalized slice regular function space with a complex generalized analytic function space on each slice.

math.CV↗
arXiv2021

A quaternionic perturbed fractional $ψ-$Fueter operator calculus

Quaternionic analysis offers a function theory focused on the concept of $ψ-$hyperholomorphic functions defined as null solutions of the $ψ-$Fueter operator, where $ψ$ is an arbitrary orthogonal base (called structural set) of $\mathbb H^4$. The main goal of the present paper is to extend the results given in \cite{BG2}, where a fractional $ψ-$hyperholomorphic function theory was developed. We introduce a quaternionic perturbed fractional $ψ-$Fueter operator calculus, where Stokes and Borel-Pompeiu formulas in this perturbed fractional $ψ-$Fueter setting are presented.

math.CV↗
arXiv2021

On hyperholomorphic Bergman type spaces in domains of $\mathbb C^2$

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field $\mathbb H$. In this work we deals with a well-known $(θ, u)-$hyperholomorphic $\mathbb H-$valued functions class related to elements of the kernel of the Helmholtz operator with a parameter $u \in\mathbb H$, just in the same way as the usual quaternionic analysis is related to the set of the harmonic functions. Given a domain $Ω\subset\mathbb H\cong \mathbb C^2$, we define and study a Bergman spaces theory for $(θ, u)-$ hyperholomorphic quaternion-valued functions introduced as elements of the kernel of ${}^θ_{u}\mathcal D [f]= {}^θ \mathcal D [f] + u f$ with $u\in \mathbb H$ defined in $C^1(Ω, \mathbb H)$, where \[ {}^θ\mathcal D:= \frac{\partial}{\partial \bar z_1} + ie^{iθ}\frac{\partial}{\partial z_2}j = \frac{\partial}{\partial \bar z_1} + ie^{iθ}j\frac{\partial}{\partial \bar z_2},\hspace{0.5cm} θ\in[0,2π). \] Using as a guiding fact that $(θ, u)-$hyperholomorphic functions includes, as a proper subset, all complex valued holomorphic functions of two complex variables we obtain some assertions for the theory of Bergman spaces and Bergman operators in domains of $\mathbb C^2$, in particular, existence of a reproducing kernel, its projection and their covariant and invariant properties of certain objects.

math.CV↗