Generalized explicit expressions for second-order recurrences

This article investigates the flexibility of closed-form generalized solutions for complex second-order linear recurrence relations. We derive and examine foundational results, including the generating function and a generalized Binet-type formula, in their most comprehensive forms. These tools are then used to establish new identities. Furthermore, we extend our results to generalize classical identities—such as those of Cassini, Catalan, Vajda, and D’Ocagne— for the first n terms of such recurrences.