Fractional Powers of the Quaternionic d-Bar Derivative

Abstract

This work introduces fractional d-bar derivatives in the setting of quaternionic analysis, by giving meaning to fractional powers of the quaternionic d-bar derivative. The definition is motivated by starting from nth-order d-bar derivatives for n is an element of N, and further justified by various natural properties such as composition laws and its action on special functions such as Fueter polynomials.