Qualitative analysis of Caputo fractional delayed difference system: a novel delayed discrete fractional sine and cosine-type function

In this paper, we provide an explicit solution for the homogeneous fractional delay oscillation difference equation with an order \(2{\delta}\) ranging from 1 to 2. This solution is achieved through the construction of discrete sine and cosine-type delayed matrix functions. Subsequently, we employ the discrete Laplace transform technique, a powerful method for handling nonhomogeneous terms, to investigate the solution of the corresponding nonhomogeneous equation. The study then delves into the Ulam-Hyers-type stabilities of the homogeneous equation, leveraging the representation of the solution. To validate the stability theory, we illustrate a numerical example. Finally, we extend our analysis by presenting an exact solution for the nonhomogeneous fractional difference equation with \(1<2{\delta}<2\), utilizing the discrete two-parameter delayed sine and cosine-type function.