Fueter’s Theorem for Monogenic Functions in Biaxial Symmetric Domains

Abstract

Fueter’s theorem discloses a remarkable connection existing between holomorphic functions and monogenic functions in $\mathbb R^{m+1}$ when m is odd. It states that $\Delta _{m+1}^{k+\frac{m-1}{2}}\bigl [\bigl (u(x_0,\vert \underline{x}\vert )+\frac{\underline{x}}{\vert \underline{x}\vert }\,v(x_0,\vert \underline{x}\vert )\bigr )P_k(\underline{x})\bigr ]$ is monogenic if $u+iv$ is holomorphic and $P_k(\underline{x})$ is a homogeneous monogenic polynomial in $\mathbb R^m$. Eelbode et al. (AIP Conf Proc 1479:340–343, 2012) proved that this statement is still valid if the monogenicity condition on $P_k(\underline{x})$ 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 Peña Peña 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 Peña Peña and Sommen (2010) to the biaxial case and then derive the main result from that.