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Convergence domains of sedenionic star-power series

Xinyuan Dou, Ming Jin, Guangbin Ren, Irene Sabadini

TL;DR

This work extends the theory of slice regular functions to the sedenions, a non-associative, non-alternative Cayley–Dickson algebra, and analyzes convergence domains for star-power series with sedenionic coefficients. It introduces hyper-solutions and hyper-$\sigma$-balls to handle zero divisors, proving that the domain of convergence is the intersection of a $\sigma$-ball and a hyper-$\sigma$-ball, controlled by two radii $R_a$ and $R_a^{p,J}$. The paper provides explicit characterizations of slice-solutions and hyper-solutions, establishes two types of hyper-$\sigma$-balls as domains of regularity, and develops orthogonal decompositions that underpin the convergence analysis. The results enrich higher-dimensional slice-regular analysis with tools potentially relevant to mathematical physics and the study of Cayley–Dickson algebras in field theories.

Abstract

The theory of slice regular (also called hyperholomorphic) functions is a generalization of complex analysis originally given in the quaternionic framework, and then further extended to Clifford algebras, octonions, and to real alternative algebras. Recently, we have extended this theory to the case of (real) even-dimensional Euclidean space. We provided several tools for studying slice regular functions in this context, such as a path-representation formula, slice-solutions, hyper-solutions and hyper-sigma balls. When adding an algebra structure to Euclidean space we have more properties related to the possibility of multiplying elements in the algebra. In this paper, we focus on exploring the properties specific to the context of the algebra of sedenions which is rather peculiar, being the algebra non associative and non alternative. The consideration of a non alternative algebra is a novelty in the context of slice regularity which opens the way to a number of further progresses. We offer an explicit characterization for slice-solutions and hyper-solutions, and we determine the convergence domain of the power series in the form $q\mapsto\sum_{n\in\mathbb{N}}(q-p)^{*n}a_n$ with sedenionic coefficients. Unlike the case of complex numbers and quaternions, the convergence domain in this context is determined by two convergence radii instead of one. These results are closely tied to the presence of zero divisors in the algebra of sedenions.

Convergence domains of sedenionic star-power series

TL;DR

This work extends the theory of slice regular functions to the sedenions, a non-associative, non-alternative Cayley–Dickson algebra, and analyzes convergence domains for star-power series with sedenionic coefficients. It introduces hyper-solutions and hyper--balls to handle zero divisors, proving that the domain of convergence is the intersection of a -ball and a hyper--ball, controlled by two radii and . The paper provides explicit characterizations of slice-solutions and hyper-solutions, establishes two types of hyper--balls as domains of regularity, and develops orthogonal decompositions that underpin the convergence analysis. The results enrich higher-dimensional slice-regular analysis with tools potentially relevant to mathematical physics and the study of Cayley–Dickson algebras in field theories.

Abstract

The theory of slice regular (also called hyperholomorphic) functions is a generalization of complex analysis originally given in the quaternionic framework, and then further extended to Clifford algebras, octonions, and to real alternative algebras. Recently, we have extended this theory to the case of (real) even-dimensional Euclidean space. We provided several tools for studying slice regular functions in this context, such as a path-representation formula, slice-solutions, hyper-solutions and hyper-sigma balls. When adding an algebra structure to Euclidean space we have more properties related to the possibility of multiplying elements in the algebra. In this paper, we focus on exploring the properties specific to the context of the algebra of sedenions which is rather peculiar, being the algebra non associative and non alternative. The consideration of a non alternative algebra is a novelty in the context of slice regularity which opens the way to a number of further progresses. We offer an explicit characterization for slice-solutions and hyper-solutions, and we determine the convergence domain of the power series in the form with sedenionic coefficients. Unlike the case of complex numbers and quaternions, the convergence domain in this context is determined by two convergence radii instead of one. These results are closely tied to the presence of zero divisors in the algebra of sedenions.

Paper Structure

This paper contains 7 sections, 38 theorems, 365 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Some main properties of sedenions
  4. Hyper-solutions in sedenions
  5. Hyper-$\sigma$-balls in sedenions
  6. Orthogonal decompositions of sedenions
  7. The domain of convergence of power series

Key Result

Theorem 2.13

Let $p\in\mathcal{W}_{\mathfrak{S}}$ and $a=\{a_\ell\}_{\ell\in\mathbb{N}}$ with $a_\ell\in\mathfrak{S}$. Then the power series converges absolutely and it is slice regular in $\Sigma(p,a)$, moreover it diverges in ${\mathop{\mathrm{ext}}}_{\tau_s}\left[\Sigma(p,a)\right]$. Moreover, if $R_a\in(0,+\infty]$ and $\varepsilon\in\left(0,R_a\right)$, then the power series $P$ converges uniformly in

Figures (1)

  • Figure 1:

Theorems & Definitions (99)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 89 more