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Dergiler / 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
Sayfa
1–17
Yayın Tarihi
2026-06-20
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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 Maddede Bu Makale

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Amaç & Hipotez:In this paper, several new identities are given for the $k$-Fibonacci difference sequence.
📊
Temel Bulgular & Veri:This is accomplished by solving a class of non-homogeneous, linear recurrence relations.
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Sonuç & Katkı: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.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Binet’s formula · $k$-Fibonacci sequence · \(k\)-Lucas sequence

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.

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.

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Bu Makaleye Atıf Yap

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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Kaynakça (13)

  1. Bolat, C., Köse, H., (2010). On the properties of k-Fibonacci numbers. Int. J. Contemp. Math. Sciences, v. 5, n. 22, p. 1097-1105.
  2. Catarino, P. (2014). On some identities for k-Fibonacci sequence. Int. J. Contemp. Math. Sci., 9, 37–42.
  3. Falc´on, Sergio. (2011). On the k-Lucas Numbers. Chaos, Solitons and Fractals, v. 06, n. 21, p. 1039- 1050.
  4. Falcon, Sergio (2013). On the generating matrices of the k-Fibonacci numbers. Proyecciones, 32, 347–357.
  5. Falcon, Sergio (2024). On the Extended (k, t)-Fibonacci Numbers. J. Adv. Math. Comput. Sci., 39, 81–89.
  6. Falcon, S. (2016). The k–Fibonacci difference sequences. Chaos, Solitons & Fractals, 87, 153-157.
  7. Falcon, S.; Plaza, A. (2007). The k-Fibonacci sequence and the Pascal 2-triangle. Chaos, Solitons and Fractals, v. 32, n. 1, p. 38-49.
  8. Falc´on, S.; Plaza, ´A. (2007). On the Fibonacci k-numbers. Chaos, Solitons and Fractals, v. 32, n. 5, p. 1615-1624.
  9. Koshy, T. (2017). Fibonacci and Lucas Numbers with Applications; John Wiley & Sons: Hoboken, NJ, USA, Volume 1.
  10. Mesquita, ´E. G. C.; Costa, E. A.; Catarino, P. M. M. C. On (2k, k)-Fibonacci sequence: a new approach to k-Fibonacci sequence. [submitted]
  11. Vajda, S. (2008). Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications; Courier Corporation: Chelmsford, MA, USA.
  12. Wani, A. A.; Badshah, V. H.; Rathore, G. P. S.; Catarino, P. (2020). Generalized Fibonacci and k-Pell matrix sequences. Punjab Univ. J. Math., 51, 17–28.
  13. Wani, A. A.; Badshah, V. H.; Rathore, G. P. S.; Catarino, P. (2020). Generalized Fibonacci and k-Pell matrix sequences. Punjab Univ. J. Math., 51, 17–28.