Abstract. Using the Clifford algebra we consider a generalization of the Cauchy-Riemann system. These equations are related to some important equations such as the Maxwell equations, the Dirac equation and the Moisil-Theodorescu system.

With the help of first-order and higher-order partial differential operators, transmutations of holomorphic functions, their relatives and of solutions of certain related differential equations to the equations considered here are given. For the first time explicit representations for a class of generalized Clifford holomorphic functions obeying a generalized Bers-Vekua equation are derived and relations between the solutions of the characterizing differential equations with different parameters are presented. Similar representations and relations are given for the solutions of a differential equation of second order also.

Received: August 3, 2011

AMS Subject Classification: 30G20, 30G35, 35C05

Key Words and Phrases: generalized Cauchy-Riemann system, transmutations, differential operators, Clifford analysis

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