  The quaternionic valued functions of a quaternionic variable, often referred
to as slice regular functions has been studied extensively due to the large
number of generali\-zed results of the theory of one complex variable, see
\cite{cgs,CSS,GSC,GS2,gssbook,gp,gpr,GS} and the references given there.
Recently, several global properties of these functions has been found of the
study of a differential operator, see \cite{GlobalOp,GP_2, G, GG1,GG2}.
Particularly, given a structural set $\psi$ the Borel-Pompieu formula induced
by the operator ${}^{\psi}G$ and its consequences in the slice regular function
theory were studied in \cite{GG1}. The aim of this paper is to present some
global and local properties of a kind of quaternionic generalized slice regular
functions. We shall see that the global properties are consequences of the
study of the perturbed global-type operator: \begin{align*} {}^{\psi}G_v [f] :=
{}^{\psi} G [f] -\frac{{\bf x}_{\psi}}{ 2} ({\bf x}_{\psi} v + v {\bf x}_{\psi}
) f , \end{align*} where $v$ is a quaternionic constant and $f$ is a
quaternionic-valued continuously differentiable function with domain in
$\mathbb H$ since our generalized slice regular function space coincides with
$\textrm{Ker} {}^{\psi_{\textrm{st}}}G_v$ associated to an axially symmetric
s-domain, where the $\psi_{\textrm{st}}$ is standard structural set. Among the
local properties studied in this work are the versions of Splitting Lemma and
Representation Theorem that show us a deep relationship between this
generalized slice regular function space with a complex generalized analytic
function space on each slice.
