Hypermonogenic functions of two vector variables

Abstract

In this paper we introduce the modified Dirac operators $\mathcal{M}_{\m{x}}^\kappa f:= \,\partial_{\m{x}}f-\frac{\kappa}{x_m}e_m\cdot f$ and $\mathcal{M}_{\m{y}}^\tau f:= \,\partial_{\m{y}}f-\frac{\tau}{y_m}e_m \cdot f$, where $f:\Omega\subset \mathbb{R}_+^{m}\times \mathbb{R}_+^{m}\to\Cl_{0,m}$ is differentiable function, and $\Cl_{0,m}$ is the Clifford algebra generated by the basis vectors of $\mathbb{R}^m$. We look for solutions $f(\m{x},\m{y}) =f(\U{x},x_m,\U{y},y_m)$ of the system $\mathcal{M}_{\m{x}}^\kappa f(\m{x,y})= \mathcal{M}_{\m{y}}^\tau f(\m{x,y})=0$, where the first and third variables are invariant under rotations. These functions are called $(\kappa,\tau)$-hypermonogenic functions. We discuss about axially symmetric functions with respect to the symmetric group $SO(m-1)$. Some examples of axially symmetric $(\kappa,\tau)$-hypermonogenic functions generated by homogeneous functions and hypergeometric functions are presented.