Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis

Abstract

In this paper, we mainly study polynomial generalized Vekua-type equation ${p(\mathcal{D})w=0}$ and polynomial generalized Bers–Vekua equation ${p(\mathcal{\underline{D}})w=0}$ defined in ${\Omega\subset\mathbb{R}^{n+1}}$ where ${\mathcal{D}}$ and ${\mathcal{\underline{D}}}$ mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including ${\left(\mathcal{D}-\lambda\right)^{k}w=0,\left(\mathcal {\underline{D}}-\lambda\right)^{k}w=0\left(k\in\mathbb{N}\right)}$ with complex parameter λ as special cases, which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in ${\Omega\subset\mathbb{R}^{n+1}}$. Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in ${\Omega\subset\mathbb{R}^{n+1}}$ under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equation ${p(\mathcal{\underline{D}})w=v}$ defined in ${\Omega\subset\mathbb{R}^{n+1}}$, and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation ${p(\mathcal{\underline{D}})w=v}$ defined in ${\Omega\subset\mathbb{R}^{n+1}}$.