<p>This paper explores new approaches to analyzing Ulam stability within a class of implicit functional equations, utilizing techniques based on the inverse Z-transform. We begin by establishing the theoretical framework for Ulam stability, emphasizing the conditions under which it can be applied to implicit functional equations. By leveraging the inverse Z-transform, we provide a structured methodology for identifying stability bounds and refining stability conditions. We outline the applicability of these techniques in providing stability estimates and in extending Ulam stability concepts to more complex functional forms. These advancements underscore the versatility of inverse Z-transform methods in handling functional equations, with implications for various applications in both pure and applied mathematics. This research contributes a novel analytical perspective, facilitating a deeper understanding of functional equation stability and expanding the scope of functional stability analysis.</p>