A Study on Explicit Particular Solutions for First and Second Order Generalized Leonardo-Type Recurrences with Polynomial-Exponential Forcing

English (OpenAlex)

Recurrence relations offer a versatile framework for analyzing numerical sequences, with applications across both classical and modern branches of mathematics. In earlier work, explicit iterative formulas were established for polynomial–exponential type particular solutions of generalized Leonardo-type sequences. The present article builds on that foundation by presenting concrete examples for orders m = 1,2 where the input function takes the form c(n) = p(n)dn, with p(n) = \(\sum_{i=0}^s c_i n^i\) a polynomial in n. For such sequences, we construct particular solutions of the form \[W_n^{(C)}\ = nr \left(\sum_{i=0}^sA_i n^i\right) d^n,\] and illustrate the computation of the coefficients Ai using the established iterative scheme. These examples show how the multiplicity r of the root d in the characteristic equation shapes the structure of the particular solution, and they highlight resonance phenomena in non-homogeneous cases. By working through explicit instances, the paper provides a clear andaccessible demonstration of the general theory, strengthening the link between abstract recurrence relations and concrete symbolic computation. We present two representative examples that demonstrate how resonance and root multiplicities influence the construction of particular solutions in polynomial–exponential-driven recurrence relations. In the case of the generalized Fibonacci sequence, the input polynomial-exponential is non-resonant, allowing the particular solution to be obtained directly without the need for correction. By contrast, the generalized Mersenne sequence highlights the resonant situation, where the root 2 of the characteristic equation necessitates a multiplicity-aware adjustment in the solution process.