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K. L. Verma, Generalized general-term formulas for $k$-th order linear recurrence relations, Eur. J. Math. Appl. 6 (2026), Article ID 12.

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Volume 6 (2026), Article ID 12

https://doi.org/10.28919/ejma.2026.6.12

Published: 02/10/2026

Abstract:

This paper investigates the $k$-th order generalized linear recurrence relation $V_n = \sum_{j=1}^{k} p_j V_{n-j}$, where $p_k \neq 0$ and the coefficients $p_j$ and initial values $V_j = a_j$ are arbitrary integers. By constructing a generalized generating function, we derive a closed-form explicit formula that establishes a unifying framework, generalizing several existing results in the literature. We demonstrate that specializing the order $k$, the initial conditions, and the recurrence coefficients naturally reduces our formulation to classical sequences, such as the Fibonacci, Tribonacci, and Tetranacci sequences. Furthermore, by framing these higher-order sequences through a $k \times k$ state-transition matrix $M$, we extend this theoretical groundwork to demonstrate its utility in advanced applied domains, including cryptography, cyber security, and network vulnerability modeling.

How to Cite:

K. L. Verma, Generalized general-term formulas for $k$-th order linear recurrence relations, Eur. J. Math. Appl. 6 (2026), Article ID 12. https://doi.org/10.28919/ejma.2026.6.12