In this paper, we first introduce a new concept of delayed matrix \(q,\omega\) -exponential function, which helps us to construct the exact expression of the solutions for the linear homogeneous and nonhomogeneous time-delay Hahn difference systems (TDHDS). Then, the existence and uniqueness of solution for the nonlinear TDHDS is demonstrated via the fixed point method. Furthermore, \(q,\omega\) -Gronwall’s inequality and the Picard operator are used to establish the sufficient conditions for Ulam-Hyers stability (UHS) and Ulam-Hyers-Rassias stability (UHRS) on a finite time interval. Finally, our theoretical results are illustrated by several examples.
