The Solution of a System of Higher-Order Difference Equations in Terms of Balancing Numbers

Ahmed Ghezal, Imane Zemmouri

Abstract

In this paper, we are interested in the closed-form solution of the following system of nonlinear difference equations of higher order,
un+1 = 1/34-vn-m , vn+1 = 1/34-un-m, n, m āˆˆ N0,
and the initial values u-j and v-j , jāˆˆ{0, 1, ..., m} are real numbers do not equal 34. We show that the solutions of this system are associated with Balancing numbers. As consequence, these solutions are also associated with Pell numbers, Pell-Lucas numbers, and Lucas-Balancing numbers. It is shown that the global stability of positive solutions of this system holds. Our results are illustrated via numerical examples.

Full Text:

PDF
Pan-Amer. J. Math. 2 (2023), 9
DOI: https://doi.org/10.28919/cpr-pajm/2-9
This article was published on August 2, 2023 by Mathyze, under a Creative Commons Attribution 4.0 International License.