Fractional Polyanalyticity

Abstract

We investigate the concept of fractionally polyanalytic functions in the complex plane: namely, complex functions whose fractional d-bar derivatives are equal to zero. So far, fractional d-bar derivatives were defined in a Riemann–Liouville sense, which leads to a characterisation of fractionally polyanalytic functions involving fractional powers multiplied by polynomials. We also propose, for the first time, a Caputo version of the theory of fractional d-bar derivatives, in which fractional polyanalyticity is equivalent to classical polyanalyticity, due to the nature of Caputo-type operators. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.