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Results Math 72 (2017), 1747–1758

c

2017 Springer International Publishing AG

1422-6383/17/041747-12

published online August 8, 2017

DOI 10.1007/s00025-017-0732-2 Results in Mathematics

Fueter’s Theorem for Monogenic Functions

in Biaxial Symmetric Domains

Dixan Pe˜na Pe˜na, Irene Sabadini, and Franciscus Sommen

Abstract. Fueter’s theorem discloses a remarkable connection existing be-

tween holomorphic functions and monogenic functions in Rm+1 when m

is odd. It states that Δk+m−1

2

m+1 u(x0,|x|)+ x

|x|v(x0,|x|)Pk(x)is mono-

genic if u+iv is holomorphic and Pk(x) is a homogeneous monogenic

polynomial in Rm. Eelbode et al. (AIP Conf Proc 1479:340–343, 2012)

proved that this statement is still valid if the monogenicity condition on

Pk(x) is dropped. To obtain this result, the authors used representation

theory methods but their result also follows from a direct calculus we

established in our paper Pe˜na Pe˜na and Sommen (J Math Anal Appl

365:29–35, 2010). In this paper we generalize the result from Eelbode

et al. (2012) to the case of monogenic functions in biaxially symmetric

domains. In order to achieve this goal we ﬁrst generalize Pe˜na Pe˜na and

Sommen (2010) to the biaxial case and then derive the main result from

that.

Mathematics Subject Classiﬁcation. 30G35, 31A05.

Keywords. Cliﬀord monogenic functions, Fueter’s theorem, Fischer

decomposition.

1. Introduction

Let Rmbe the real Cliﬀord algebra generated by the standard basis {e1,...,e

m}

of the Euclidean space Rm(see [2,17]). The multiplication in this associative al-

gebra is determined by the relations: e2

j=−1, ejek+ekej=0,1≤j=k≤m.

Any Cliﬀord number a∈Rmmay thus be written as

a=

A

aAeA,a

A∈R,

where the basis elements eA=ej1...e

jkare deﬁned for every subset A=

{j1,...,j

k}of {1,...,m}with j1<···<j

k(for A=∅one puts e∅= 1).

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