On the Generalized Complex 2nd Order Recurrences

Abstract

This paper investigates a generalization of complex second-order recurrence relations and develops a unified framework for their analysis. Using the Z-transform technique, we derive the corresponding generating functions and Binet-type formulas. Distinct expressions for generating functions are further obtained for the oddand even-indexed subsequences, revealing additional structural properties. An explicit summation formula is also established, along with several fundamental identities for the resulting complex sequences. In addition, we present tridiagonal and 2×2 matrix representations of these generalized complex numbers, providing an algebraic perspective that connects recurrence relations with linear algebra. A key contribution of this work is that many well-known results for classical and generalized Fibonacci numbers appear as special cases within our broader framework, highlighting both the novelty and unifying power of the approach.