Let ∂ x denote the Dirac operator in R m . In this paper, we present a refinement of the biharmonic functions and at the same time an extension of the monogenic functions by considering the equation ∂ x f ∂ x = 0. The solutions of this "sandwich" equation, which we call inframonogenic functions, are used to obtain a new Fischer decomposition for homogeneous polynomials in R m . RESUMENDenotemos por ∂ x el operador de Dirac en R m . En este artículo, nosotros presentamos un refinamiento de las funciones biarmónicas y al mismo tiempo una extensión de las funciones monogénicas considerando la ecuación ∂ x f ∂ x = 0. Las soluciones de esta ecuación tipo "sándwich", las cuales llamaremos inframonogénicas, son utilizadas para obtener una nueva descomposición de Fischer para polinomios homogéneos en R m . 190Helmuth R. Malonek, Dixan Peña Peña and Frank Sommen CUBO 12, 2 (2010)

In this paper we introduce the quaternionic Witt basis in Hm = H ⊗ R Rm, m = 4n. We then define a notion of quaternionic hermitian vector derivative which leads to hermitian monogenic functions. We study the resolutions associated to quaternionic hermitian systems in the 4 and 8 dimensional cases. We finally prove Martinelli-Bochner type formulae. Mathematics Subject Classification (2000). 30G35.Keywords. Witt basis, hermitian vector derivative, resolutions, MartinelliBochner integral formulae. The quaternionic Witt basisLet us consider the algebra of quaternions H whose elements will be denoted by q = x 0 + ix 1 + jx 2 + kx 3 with i 2 = j 2 = k 2 = −1 and ij = −ji = k. In this paper we will combine quaternions with the (real) Clifford algebra R m , with m = 4n; the generators of the algebra will be denoted by e 1 , . . . , e m with e r e s = −e s e r , e 2 r = −1. As it is well known, the algebra of quaternions may be identified with a Clifford algebra in two ways:(i) H ∼ = R 2 where we make the identifications i → e 1 , j → e 2 and k → e 1 e 2 ; (ii) H ∼ = R + 3 (where R + 3 denoted the even part of R 3 ) where we make the identifications i → e 2 e 3 , j → e 3 e 1 and k → e 1 e 2 .It is well known that on the Clifford algebra R m one can consider the following automorphisms:(i) the conjugation:ē r = −e r , and for any a, b ∈ R m , ab =bā; (ii) the main involutionẽ r = −e r , and for any a, b ∈ R m , ab =ãb; (iii) the reversion, defined by a * =ã.In the sequel, the conjugation and the main involution will be the most important automorphism for us. The two isomorphisms described above obviously allow to

In the even dimensional case the Dirac equation may be reduced to the so-called isotonic Dirac system, in which different Dirac operators appear from both sides in half the dimension. This system is then used to derive the classical Martinelli-Bochner formula for several complex variables. Mathematics Subject Classification (2000). 30G35, 32A25, 58A10.

SUMMARYIn this paper we present a generalization of the Martinelli-Bochner formula in several complex variables which is derived for a Hermitian Dirac operator.

In this paper, we study some new special functions that arise naturally within the framework of Hermitian Clifford analysis, which concerns the study of Dirac‐like systems in several complex variables. In particular, we focus on Hermite polynomials, Bessel functions, and generalized powers. We also derive a Vekua system for solutions of Hermitian systems in axially symmetric domains. Copyright © 2012 John Wiley & Sons, Ltd.