  The theory of slice regular functions of a quaternionic variable on the unit
ball of the quaternions was introduced by Gentili and Struppa in 2006 and
nowadays it is a well established function theory, especially in view of its
applications to operator theory. In this paper, we introduce the notion of
fractional slice regular functions of a quaternionic variable defined as
null-solutions of a fractional Cauchy-Riemann operators. We present a
fractional Cauchy-Riemann operator in the sense of Riemann-Liouville and then
in the sense of Caputo, with orders associated to an element of $(0,1)\times
\mathbb R \times (0,1)\times \mathbb R$ for some axially symmetric slice
domains which are new in the literature. We prove a version of the
representation theorem, of the splitting lemma and we discuss a series
expansion.
