Covariance property for a quaternionic fractional Cauchy-Riemann operator under affine transformation

RESUMO

In this paper, we continue the development of the foundations of a quaternionic function theory that is associated to a fractional proportional Cauchy-Riemann type operator acting on quaternionic functions. We introduce a new class of fractional hyperholomorphic functions adapted to affine transformations of the domain, and study the corresponding quaternionic right module structure. In particular, we establish versions of the Stokes, Borel-Pompeiu and Cauchy formulas for this function theory. A covariance property of the fractional operator under affine transformations is proved, showing that the composition with an affine map, weighted by a suitable quaternionic factor, preserves the class of fractional hyperholomorphic functions. The result is shown to include the case considered previously by the authors, and extends their covariance property to the setting of affine transformations.