On two-parameter generalization of Oresme and Oresme-Lucas sequences

Hasan Gökbaş; Francisco R. V. Alves; Douglas C. Santos; Eudes A. Costa; Hasan Gökbaş; Francisco R. V. Alves; Douglas C. Santos; Eudes A. Costa

doi:10.3934/era.2026218

Electronic Research Archive

2026, Volume 34, Issue 7: 4931-4946. doi: 10.3934/era.2026218

Previous Article Next Article

Research article

On two-parameter generalization of Oresme and Oresme-Lucas sequences

1.
Department of Mathematics, Science and Arts Faculty, Bitlis Eren University, Bitlis, Türkiye
2.
Department of Mathematics, Federal Institute of Educations, Science and Technology of State of Ceará, Fortaleza, Brazil
3.
Education Department of the State of Bahia, Barreiras, Brazil
4.
Department of Mathematics, Federal University of Tocantins, Arraias, Brazil

Received: 01 April 2026 Revised: 24 April 2026 Accepted: 08 May 2026 Published: 09 June 2026

In this paper, we introduced and studied the $ (t, s) $-Oresme and $ (t, s) $-Oresme-Lucas sequences, which were generalizations of the classical Oresme and Oresme-Lucas numbers. We presented the recurrence relations, generating functions, and Binet formulas for these sequences. Furthermore, we derived various algebraic identities, including those of Tagiuri, d'Ocagne, Catalan, Cassini, Ruggles, and Honsberger. A matrix representation for these sequences was also established, and we defined the associated $ (t, s) $-Oresme matrix sequence, exploring its fundamental properties and relations. Additionally, we provided the summation formulas for the first $ n $ terms of these sequences. This work extended the existing theory of Oresme numbers and offered a unified framework for their further investigation.
- Fibonacci sequence,
- Oresme matrix,
- Oresme sequence,
- Oresme-Lucas sequence
Citation: Hasan Gökbaş, Francisco R. V. Alves, Douglas C. Santos, Eudes A. Costa. On two-parameter generalization of Oresme and Oresme-Lucas sequences[J]. Electronic Research Archive, 2026, 34(7): 4931-4946. doi: 10.3934/era.2026218

Related Papers:

Abstract

In this paper, we introduced and studied the $ (t, s) $-Oresme and $ (t, s) $-Oresme-Lucas sequences, which were generalizations of the classical Oresme and Oresme-Lucas numbers. We presented the recurrence relations, generating functions, and Binet formulas for these sequences. Furthermore, we derived various algebraic identities, including those of Tagiuri, d'Ocagne, Catalan, Cassini, Ruggles, and Honsberger. A matrix representation for these sequences was also established, and we defined the associated $ (t, s) $-Oresme matrix sequence, exploring its fundamental properties and relations. Additionally, we provided the summation formulas for the first $ n $ terms of these sequences. This work extended the existing theory of Oresme numbers and offered a unified framework for their further investigation.

References

[1]	S. Falcon, A. Plaza, On the Fibonacci $k$-numbers, Chaos Solitions Fractals, 32 (2007), 1615–1624. https://doi.org/10.1016/j.chaos.2006.09.022 doi: 10.1016/j.chaos.2006.09.022
[2]	C. J. Harman, Complex Fibonacci numbers, Fibonacci Q., 19 (1981), 82–86. https://doi.org/10.1080/00150517.1981.12430133 doi: 10.1080/00150517.1981.12430133
[3]	A. F. Horadam, A generalized Fibonacci sequence, Am. Math. Mon., 68 (1961), 455–459. https://doi.org/10.1080/00029890.1961.11989696 doi: 10.1080/00029890.1961.11989696
[4]	A. F. Horadam, Oresme numbers, Fibonacci Q., 12 (1974), 267–270. https://doi.org/10.1080/00150517.1974.12430733 doi: 10.1080/00150517.1974.12430733
[5]	C. K. Cook, Some sums related to sums of Oresme numbers, in Applications of Fibonacci Numbers, (2004), 87–99. https://doi.org/10.1007/978-0-306-48517-6_10
[6]	Y. Soykan, A study on generalized $p$-Oresme numbers, Asian J. Adv. Res. Rep., 15 (2021), 1–25. https://doi.org/10.9734/ajarr/2021/v15i730410 doi: 10.9734/ajarr/2021/v15i730410
[7]	E. Özkan, H. Akkuş, A new approach to $k$-Oresme and $k$-Oresme-Lucas sequences, Symmetry, 16 (2024), 1407. https://doi.org/10.3390/sym16111407 doi: 10.3390/sym16111407
[8]	S. Erdem, S. Demiriz, Oresme numbers and associated BK-sequence spaces, Iran. J. Sci., 49 (2025), 725–732. https://doi.org/10.1007/s40995-024-01734-5 doi: 10.1007/s40995-024-01734-5
[9]	T. Goy, R. Zatorsky, On Oresme numbers and their connection with Fibonacci and Pell numbers, Fibonacci Q., 57 (2019), 238–245. https://doi.org/10.1080/00150517.2019.12427646 doi: 10.1080/00150517.2019.12427646
[10]	E. V. P. Spreafico, P. M. M. Catarino, On generalized Oresme numbers and the Fibonacci fundamental system, Rev. Sergipana Mat. Educ. Mat., 9 (2024), 68–80.
[11]	G. Cerda-Morales, Oresme polynomials and their derivatives, preprint, arXiv: 1904.01165.
[12]	M. Edson, O. Yayenie, A new generalization of Fibonacci sequence & extended Binet's formula, Integers, 9 (2009), 639–654. https://doi.org/10.1515/INTEG.2009.051 doi: 10.1515/INTEG.2009.051
[13]	M. R. Bacon, C. K. Cook, Some properties of Oresme numbers and convolutions with various second order sequences, Fibonacci Q., 62 (2024), 233–240. https://doi.org/10.1080/00150517.2024.12459539 doi: 10.1080/00150517.2024.12459539
[14]	O. Yayenie, New identities for generalized Fibonacci sequences and new generalization of Lucas sequences, Southeast Asian Bull. Math., 36 (2012).
[15]	F. Caldarola, G. d'Atri, M. Maiolo, G. Pirillo, New algebraic and geometric constructs arising from Fibonacci numbers, Soft Comput., 24 (2020), 17497–17508. https://doi.org/10.1007/s00500-020-05256-1 doi: 10.1007/s00500-020-05256-1
[16]	H. Özimamoğlu, $d$-Oresme polynomials and their matrix representations, J. Math. Ext., 18 (2024), 1–20. https://doi.org/10.30495/JME.2024.3026 doi: 10.30495/JME.2024.3026
[17]	S. Halıcı, E. Sayın, On Oresme numbers and their geometric interpretations, J. Inst. Sci. Technol., 14 (2024), 1291–1300. https://doi.org/10.21597/jist.1421298 doi: 10.21597/jist.1421298
[18]	H. Al-Zoubi, A. Al-Kateeb, Generalized balancing and balancing-Lucas numbers, Proyecciones, 43 (2024), 707–723. https://doi.org/10.22199/issn.0717-6279-5712 doi: 10.22199/issn.0717-6279-5712
[19]	E. A. Costa, E. V. P. Spreafico, P. M. M. C. Catarino, A note on the generalized bi-periodic Lucas-balancing numbers, Braz. Electron. J. Math., 6 (2025), 1–15. https://doi.org/10.14393/BEJOM-v6-2025-75273 doi: 10.14393/BEJOM-v6-2025-75273
[20]	M. Rubajczyk, A. Szynal-Liana, On bivariate balancing and Lucas-balancing hybrinomials, Symmetry, 17 (2025), 537. https://doi.org/10.3390/sym17040537 doi: 10.3390/sym17040537
[21]	T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2001.
[22]	H. W. Gould, A history of the Fibonacci $Q$-matrix and a higher-dimensional problem, Fibonacci Q., 19 (1981), 250–257. https://doi.org/10.1080/00150517.1981.12430088 doi: 10.1080/00150517.1981.12430088
[23]	F. R. V. Alves, Sequência de Oresme e algumas propriedades (matriciais) generalizadas, Rev. Eletronica Paulista Mat., 16 (2019), 28–52. https://doi.org/10.21167/cqdvol16201923169664frva2852 doi: 10.21167/cqdvol16201923169664frva2852

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)