  In the literature on slice analysis in the hypercomplex setting, there are
two main approaches to define slice regular functions in one variable: one
consists in requiring that the restriction to any complex plane is holomorphic
(with the same complex structure of the complex plane), the second one makes
use of stem and slice functions. So far, in the setting of several hypercomplex
variables, only the second approach has been considered, i.e. the one based on
stem functions. In this paper, we use instead the first definition on the
so-called $n$-dimensional quadratic cone of octonions. These two approaches
yield the same class of slice regular functions on axially symmetric
slice-domains, however, they are different on other types of domains. We call
this new class of functions weak slice regular. We show that there exist weak
slice regular functions which are not slice regular in the second approach.
Moreover, we study various properties of these functions, including a Taylor
expansion.
