Fueter's theorem in discrete Clifford analysis

Presenter: Hilde De Ridder, Ghent University, Belgium

Joint work with F. Sommen

In 1935, R. Fueter described in his paper [1] a technique to obtain monogenic quaternionic functions, starting from a holomorphic function in the upper half of the complex plane. This classical result was later generalized to $\mathbb{R}_{0,m}$ for $m$ odd by Sce [2], for $m$ even by [3] and by Sommen in [4]: if $m$ is an odd positive integer and $P_k(\underline{x})$ is a homogeneous monogenic polynomial of degree $k$ in $\mathbb{R}^m$, then $\Delta_x^{k+\frac{m-1}{2}} \left[ \left( u(x_0, r) + \underline{\omega} \, v(x_0,r)\right) P_k(\underline{x}) \right] $ is also monogenic in $\widetilde{\Omega}$.

In this presentation, we consider a discretization of Sommen's result to the setting of (hermitean) discrete Clifford analysis in even dimension: starting from a discrete monogenic function in $\mathbb{Z}^2$, we will construct monogenic functions on $\mathbb{Z}^m$.

[1] R. Fueter, Die Funktionentheorie der Differentialgleichungen $\Delta u = 0$ und $\Delta \Delta u = 0$ mit vier Variablen, Comm. Math. Helv. 7 (1935), pp. 307-330.

[2] M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 23, 1957, pp. 220-225.

[3] K. I. Kou and T. Qian and F. Sommen, Generalizations of Fueter's theorem, Methods Appl. Anal. 9 no. 2, 2002, pp. 273-289.

[4] F. Sommen, On a generalization of Fueter's theorem, Z.Anal. Anwendungen 19 no.4, 2000, pp. 899-902.