In this paper, we introduce the notion of a cosine sequence \(\{C_\tau ^n\}_{n\in \mathbb {N}_0}\) generated by a closed linear operator A in a Banach space X. We provide a systematic study of the properties of \(\{C^n_\tau \}_{n\in \mathbb {N}_0}\) , showing that it satisfies a discrete d’Alembert’s functional equation. We also explore its connections with its generator A, the resolvent operator \(R_\tau := \tau ^{-2}(\tau ^{-2} - A)^{-1}\) , and its corresponding sine family \(\{S^n_\tau \}_{n\in \mathbb {N}_0}\) . Moreover, we show that the solution to the abstract discrete system of second order \(\begin{aligned} \nabla ^{2}_\tau u^n = Au^n + f^n, \quad n \ge 2, \end{aligned}\) subject to the initial conditions \(u^0 = x_0\) , \(u^1 = x_1\) , where \(f: \mathbb {N}_0 \rightarrow X\) is a given sequence, \(\tau > 0\) is a specified step size, and \(\nabla ^{2}_\tau u^n\) is the backward operator of order two, can be expressed as a discrete variation parameter formula in terms of \(C^n_\tau \) and its corresponding sine sequence.
