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Journals / Journal of Universal Mathematics / 2026 / Cilt: 9 Sayı: 1

ON SOME IDENTITIES FOR k-FIBONACCI DIFFERENCE SEQUENCE

Eudes Antonio Costa(Federal University of Tocantins)0000-0001-6684-9961
Elis G. Mesquita(UFT)0000-0003-2385-4108
Paula Maria Machado Cruz Catarino(UTAD)0000-0001-6917-5093
Pages
1–17
Publication Date
2026-06-20
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In addition, we present some results concerning the limit of the ratio involving terms of the $k$-Fibonacci difference sequence for both positive and negative indexes, along with various identities for partial sums.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, several new identities are given for the $k$-Fibonacci difference sequence.
📊
Key Findings & Data:This is accomplished by solving a class of non-homogeneous, linear recurrence relations.
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Conclusion & Impact:In addition, we present some results concerning the limit of the ratio involving terms of the $k$-Fibonacci difference sequence for both positive and negative indexes, along with various identities for partial sums.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Binet’s formula · $k$-Fibonacci sequence · \(k\)-Lucas sequence

Abstract

In this paper, several new identities are given for the $k$-Fibonacci difference sequence. This is accomplished by solving a class of non-homogeneous, linear recurrence relations. We address a sequence of the $k$-Fibonacci type, called the $k$-Fibonacci difference sequence. This sequence is obtained by applying the finite difference operator to the $k$-Fibonacci sequence a finite number of times. By taking advantage of the strong dependence this sequence has on its initial terms, as well as on the $k$-Fibonacci and $k$-Lucas sequences, we were able to derive a wide range of properties, including classical identities such as those of Tagiuri-Vajda, Catalan, and Cassini identities, among others. Moreover, we obtained its extension to negative indexes with explicit expressions. Several types of generating functions were also derived, including exponential and Poisson-type generating functions. In addition, we present some results concerning the limit of the ratio involving terms of the $k$-Fibonacci difference sequence for both positive and negative indexes, along with various identities for partial sums.

Özet

In this paper, several new identities are given for the $k$-Fibonacci difference sequence. This is accomplished by solving a class of non-homogeneous, linear recurrence relations. We address a sequence of the $k$-Fibonacci type, called the $k$-Fibonacci difference sequence. This sequence is obtained by applying the finite difference operator to the $k$-Fibonacci sequence a finite number of times. By taking advantage of the strong dependence this sequence has on its initial terms, as well as on the $k$-Fibonacci and $k$-Lucas sequences, we were able to derive a wide range of properties, including classical identities such as those of Tagiuri-Vajda, Catalan, and Cassini identities, among others. Moreover, we obtained its extension to negative indexes with explicit expressions. Several types of generating functions were also derived, including exponential and Poisson-type generating functions. In addition, we present some results concerning the limit of the ratio involving terms of the $k$-Fibonacci difference sequence for both positive and negative indexes, along with various identities for partial sums.

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Cite This Article

Costa, E. A., Mesquita, E. G., Catarino, P. M. M. C. (2026). ON SOME IDENTITIES FOR k-FIBONACCI DIFFERENCE SEQUENCE. Journal of Universal Mathematics, Cilt: 9 Sayı: 1, 1–17. https://doi.org/10.33773/jum.1753190

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References (13)

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