Recent and new results on octonionic Bergman and Szeg\"o kernels

TL;DR

This paper advances the understanding of octonionic Bergman and Szeg"o kernels by deriving explicit formulas for strip domains and connecting these kernels with periodic octonionic functions, extending recent foundational work.

Contribution

It introduces explicit formulas for Szeg"o and Bergman kernels on octonionic strip domains and relates them via periodic octonionic functions, expanding the class of known kernels.

Findings

Explicit formulas for Szeg"o kernels on octonionic strip domains.

Formulas for Bergman kernels related to periodic octonionic functions.

Connection between kernels and octonionic generalizations of classical trigonometric functions.

Abstract

Very recently one has started to study Bergman and Szeg\"o kernels in the setting of octonionic monogenic functions. In particular, explicit formulas for the Bergman kernel for the octonionic unit ball and for the octonionic right half-space as well as a formula for the Szeg\"o kernel for the octonionic unit ball have been established. In this paper we extend this line of investigation by developing explicit formulas for the Szeg\"o kernel of strip domains of the form ${\cal{S}} := \{z \in \mathbb{O} \mid 0 < \Re(z) < d\}$ from which we derive by a limit argument considering $d \to \infty$ the Szeg\"o kernel of the octonionic right half-space. Additionally, we set up formulas for the Bergman kernel of such strip domains and relate both kernels with each other. In fact, these kernel functions can be expressed in terms of one-fold periodic octonionic monogenic generalizations of the cosecant function and the cotangent function, respectively.