Scator Holomorphic Functions

Scators form a linear space equipped with a specific non-distributive product. In the elliptic case they can be interpreted as a kind of hypercomplex numbers.The standard definition of holomorphy (requiring the directional derivative to be direction-independent) leads to a generalization of the Cauchy-Riemann equation and to scator holomorphic functions. In this paper we found a complete set of solutions to the generalized Cauchy-Riemann system in the (1+n)-dimensional elliptic scator space. For any n⩾2 the scator holomorphic functions consist of three classes: components exponential functions, linear functions (of a specific form) and some exceptional solutions parameterized by arbitrary functions of one variable. The obtained family of solutions, although relatively narrow, is greater than analogous functions in the quaternionic or Clifford analysis.