On the solutions of a class of iterated generalized Bers–Vekua equations in Clifford analysis

Abstract

We consider functions with values in the Clifford algebra

Solutions to a class of polynomially generalized Bers–Vekua equations using Clifford analysis

“…is a solution to the equation D k w(x) = 0, expression (8) in Corollary 3.1 gives the decomposition for the solution to D k w(x) = 0 as in [10]. Further, when a(x) ≡ 0, decomposition (8) in Corollary 3.1 corresponds exactly to that in [2].…”

“…. , n j − 1) are given similarly to (2). That is, [10] and to the polynomially generalized Cauchy-Riemann operator in [16,9,13].…”

“…. , l − 1) of (24) can be explicitly given (see [2]) similarly to those of (11) in [1]. If it exists, the Bauer-type differential operator is defined similarly to 7and (8) of Section 3 in [1], which is implied in [2,10], when considering the generalized Bers-Vekua equation.…”

“…, l − 1) of (24) can be explicitly given (see [2]) similarly to those of (11) in [1]. If it exists, the Bauer-type differential operator is defined similarly to 7and (8) of Section 3 in [1], which is implied in [2,10], when considering the generalized Bers-Vekua equation. A sufficient condition for the existence of a Bauer-type differential operator was provided in [2]; however, designing other sufficient conditions for the existence of a Bauer-type differential operator is still work in progress.…”

“…In [5]- [9], [13]- [12] Delanghe, Brackx and others studied the polynomially generalized Cauchy-Riemann equations in R n+1 , and obtained the solutions to the polynomially generalized Cauchy-Riemann equations and to their Riemann boundary value problems, by means of integral formulas and Fischer-type decomposition theorems for null solutions to the polynomially generalized Cauchy-Riemann equations, respectively. Recently, Berglez (see [1,2]) discussed a class of iterated generalized Bers-Vekua equations in Clifford analysis, which is a generalization of a special Vekua-type equation in the complex plane (see e.g. [3,15]) to higher dimensions, and obtained their solutions under some conditions.…”