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 number. The requirement that the scator partial derivatives are direction-independent leads to a generalization of the Cauchy–Riemann equation and to scator holomorphic functions. In this paper we find a complete set of C2-solutions to the generalized Cauchy–Riemann system in the (1+n)-dimensional elliptic scator space. For any n⩾2 this set consists of three classes: components exponential functions (already known), a new class of affine linear functions, and some exceptional solutions parameterized by arbitrary functions of one variable. We show, however, that the last class of solutions is not scator holomorphic and the generalized Cauchy–Riemann system should be supplemented with additional constraints to avoid such spurious solutions. The obtained family of scator holomorphic functions, although relatively narrow, is greater than that of analogous functions in quaternionic or Clifford analysis.