Discrete Complex Analysis in Split Quaternions

TL;DR: In this paper, the Taylor series of a discrete holomorphic function is shown to converge to itself in the whole grid in the setting of a new kind of discrete holomorph function on the square grid with values in split quaternions.

Abstract: A natural question in discrete complex analysis is whether the Taylor series of a discrete holomorphic function is convergent to itself in the whole grid \({\mathbb {Z}}_h^2\). In this paper we answer this question in the affirmative in the setting of a new kind of discrete holomorphic function on the square grid \({\mathbb {Z}}_h^2\) with values in split quaternions based on the methods of Sheffer sequences. On the other hand, we also establish the integral theory for this new kind of discrete holomorphic functions, including the discrete Green theorem and the Cauchy integral formula. In contrast to the discrete Clifford analysis, we obtain a new version of the discrete Cauchy integral formula without the extra error term.