In this paper, we introduce the concepts of generalized \(\mathrm{Z}\) -Riemann-Liouville fractional integral and generalized \(\mathrm{Z}\) -Caputo fractional derivative of order \((\alpha,\beta)\) with respect to a weight function for \(\mathcal{Z}^+\) -valued functions and some related properties along with specific illustrative examples. The \(\mathrm{Z}\) -Laplace transforms for generalized \(\mathrm{Z}\) -fractional operators are established and applied to determine the general solution of some classes of \(\mathrm{Z}\) -fractional linear differential systems. In addition, in order to find \(\mathcal{Z}^+\) -solution of \(\mathrm{Z}\) -fractional differential systems, an important result on the Newton-Leibniz-type formula is also presented. Moreover, we consider an initial value problem to \(\mathrm{Z}\) -fractional differential system under granular differentiability and then, prove the existence and uniqueness of \(\mathcal{Z}^+\) -integral solution of this problem. Finally, some qualitative properties of the obtained \(\mathcal{Z}^+\) -integral solution such as continuous dependence on data or stability are also shown.
