Transmutation operators and complete systems of solutions for the radial Bicomplex Vekua equation

Abstract

The construction of a pair of transmutation operators for the radial main Vekua equation with a Bicomplex-valued coefficient is presented. The pair of operators transform the Bicomplex analytic functions into the solutions of the main Vekua equation. The analytical properties of the operators in the space of classical solutions and the pseudoanalytic Bergman space are established. With the aid of such operators, a complete system of solutions for the main Vekua equation, called radial formal powers, is obtained. The completeness of the radial formal powers is proven with respect to the uniform convergence on compact subsets and in the Bicomplex pseudoanalytic Bergman space with the $L_2$-norm. For the pseudoanalytic Bergman space on a disk, we show that the radial formal powers are an orthogonal basis, and a representation for the Bergman kernel in terms of such basis is obtained.