Closed-form Solutions of Third-Order Generalized Leonardo Sequences with Polynomial Input
Yüksel Soykan *
Department of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University, 67100, Zonguldak, Turkey.
*Author to whom correspondence should be addressed.
Abstract
The objective of this study is to derive closed-form solutions to third-order nonhomogeneous linear recurrence relations, referred to as generalized Leonardo-type sequences, where the input function p(n) is a polynomial. The study considers the cases in which 1 appears as a root of the characteristic equation with multiplicity r = 0,1,2,3, and for each value of r explicit solutions are obtained for polynomial inputs p(n) of degree s = 0,1,2,3. The resulting formulas express the solution as the sum of homogeneous and particular components, with the coefficients determined through iterative relations. This unified framework provides a complete description of generalized Leonardo-type sequences in the nonhomogeneous setting with polynomial inputs, extending the classical theory of recurrence relations. The results extend classical recurrence theory by clarifying resonance phenomena and multiplicity corrections, while offering resonance-aware formulas that can be adapted to problems in mathematics, computer science, engineering, and physics. Beyond their theoretical contribution, the explicit examples provide pedagogical value by allowing students to engage directly with nonhomogeneous recurrences without excessive computation. Thus, the study demonstrates both the novelty and interdisciplinary impact of generalized Leonardo-type sequences in the nonhomogeneous setting.
Keywords: Leonardo numbers, Leonardo polynomials, nonhomogeneous linear recurrence relations, mogeneous recurrence relations, closed-form solutions