Riemann Boundary Value Problems on the Sphere in Clifford Analysis

Abstract

We present and study a type of Riemann boundary value problems (for short RBVPs) for polynomially monogenic functions, i.e. null solutions to polynomially generalized Cauchy-Riemann equations, over the sphere of \({\mathbb{R}^{n+1}}\). Making use of Fischer type decomposition and the Clifford calculus for polynomially monogenic functions, we obtain explicit expressions of solutions of this kind of boundary value problems over the sphere of \({\mathbb{R}^{n+1}}\). As special cases the solutions of the corresponding boundary value problems for classical polyanalytic functions and metaanalytic functions are derived respectively.