A Study on the k-Mersenne and k-Mersenne-Lucas Sequences

Engin Özkan; Bayram Şen; Hakan Akkuş; Mine Uysal

doi:10.32323/ujma.1566270

EN

A Study on the k-Mersenne and k-Mersenne-Lucas Sequences

Abstract

In this study, we examine an application of $k-$Mersenne and $k-$Mersenne-Lucas sequences. We present the Catalan transforms of these sequences and give the properties of these Catalan transforms. Catalan transforms of $k-$Mersenne and $k-$Mersenne-Lucas sequences have terms according to different values of $k$, and some of them are associated with the sequences in OEIS. We obtain the generating functions of the Catalan transforms of $k-$Mersenne and $k-$Mersenne-Lucas sequences. Moreover, we apply the Hankel transform to the Catalan transforms of these sequences. Finally, the determinants of the created matrices are calculated by applying the Hankel transform to the terms of the Catalan transforms of these sequences.

Keywords

Catalan numbers, Fibonacci numbers, Generating functions, Hankel transform, $k$-Mersenne sequences, $k$-Mersenne-Lucas sequences

References

[1] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, 2019.
[2] S. Falcon, A. Plaza, On the Fibonacci k-Numbers, Chaos Solutions Fractals, 32(5) (2007), 1615-1624.
[3] S. Falcon, Catalan transform of the k-Fibonacci sequence, Communications of the Korean Mathematical Society, 28(4) (2013), 827–832.
[4] Y. Soykan, On generalized p-Mersenne numbers, Earthline Journal of Mathematical Sciences, 8(1) (2022), 83-120.
[5] R. Frontczak, T. P. Goy, Mersenne-Horadam identities using generating functions, Carpathian Math. Publ., 12(1) (2020), 34-45.
[6] M. Uysal, M. Kumari, B. Kuloğlu, K. Prasad, E. Özkan, On the hyperbolic k-Mersenne and k-Mersenne-Lucas octonions, Kragujevac J. Math.,49(5) (2025), 765-779.
[7] D. Tasci, On Gaussian Mersenne numbers, J. Sci. Arts, 21(4) (2021), 1021-1028.
[8] P. Barry, A Catalan transform and related transformations on integer sequences, J. Integer Seq. J., 8(4) (2005), 1-24.
[9] M. Chelgham, A. Boussayoud, On the k-Mersenne-Lucas numbers, Notes Number Theory Discrete Math., 27(1) (2021), 7-13.
[10] M. Kumari, K, Prasad, J, Tanti, On the generalization of Mersenne and Gaussian Mersenne polynomials, J. Anal., 32(3) (2024), 931-947.

[11] Y. Soykan, A study on generalized Mersenne numbers, Journal of Progressive Research in Mathematics, 18(3) (2021), 90-112.
[12] S. Demiriz, S. Erdem, Mersenne matrix operator and its application in p-summable sequence space, Commun. Adv. Math. Sci., 7(1)(2024), 42-55.
[13] E. Özkan, M. Uysal, Mersenne-Lucas hybrid numbers, Math. Montisnigri, 52(2021), 17-29.
[14] M. Kumari, K. Prasad, R. Mohanta, Algebra of quaternions and octonions involving higher order Mersenne numbers, Proc. Indian Nat. Sci. Acad., (2024), 1-10.
[15] M. Kumari, J. Tanti, K. Prasad, On some new families of k-Mersenne and generalized k-Gaussian Mersenne numbers and their polynomials, (2021), arXiv preprint arXiv:2111.09592.
[16] P. Catarino, H. Campos, H., P. Vasco, On the Mersenne sequence, Ann. Math. Inform., 46 (2016), 37-53.
[17] N. Saba, A. Boussayoud, K. V. V. Kanuri, Mersenne Lucas numbers and complete homogeneous symmetric functions, J. Math. Comput. Sci., 24(2) (2021), 127- 139.
[18] A. Patra, M. K. Kaabar, Catalan transform of k-Balancing sequences, Int. J. Math. Math. Sci., 2021(1) 2021, 9987314.
[19] S. Kapoor, P. Kumar, Catalan Transformation of (s; t) Padovan Sequences, Asian Journal of Pure and Applied Mathematics, 5(1) (2023), 170-178.
[20] P. J. Larcombe, P. D. Wilson, On the generating function of the Catalan sequence: A historical perspective, Congr. Numer., 149 (2001), 97-108.
[21] J. W. Layman, The Hankel transform and some of its properties, J. Integer Seq., 4(1) 2001, 1-11.
[22] P. M. Rajkovi´c, M. D. Petkovi´c, P. Barry, The Hankel transform of the sum of consecutive generalized Catalan numbers, Integral Transforms Spec. Funct., 18(4) (2007), 285-296.
[23] K. Parmar, V. R. Gorty, Quaternion Hankel Transform and its Generalization, Sahand Commun. Math. Anal., 21(1) (2024), 67-81.
[24] E. Özkan, M. Uysal, B. Kulogğu, Catalan transform of the incomplete Jacobsthal numbers and incomplete generalized Jacobsthal polynomials, Asian-Eur. J. Math., 15(6) (2022), 2250119.

Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

Engin Özkan ^*
0000-0002-4188-7248
Türkiye

Bayram Şen
0009-0009-5570-4981
Türkiye

Hakan Akkuş
0000-0001-9716-9424
Türkiye

Mine Uysal
0000-0002-2362-3097
Türkiye

Early Pub Date

February 23, 2025

Publication Date

March 25, 2025

Submission Date

October 13, 2024

Acceptance Date

February 6, 2025

Published in Issue

Year 2025 Volume: 8 Number: 1

DOI

https://doi.org/10.32323/ujma.1566270

IZ

https://izlik.org/JA78KT82HC

APA

Özkan, E., Şen, B., Akkuş, H., & Uysal, M. (2025). A Study on the k-Mersenne and k-Mersenne-Lucas Sequences. Universal Journal of Mathematics and Applications, 8(1), 1-7. https://doi.org/10.32323/ujma.1566270

AMA

1.Özkan E, Şen B, Akkuş H, Uysal M. A Study on the k-Mersenne and k-Mersenne-Lucas Sequences. Univ. J. Math. Appl. 2025;8(1):1-7. doi:10.32323/ujma.1566270

Chicago

Özkan, Engin, Bayram Şen, Hakan Akkuş, and Mine Uysal. 2025. “A Study on the K-Mersenne and K-Mersenne-Lucas Sequences”. Universal Journal of Mathematics and Applications 8 (1): 1-7. https://doi.org/10.32323/ujma.1566270.

EndNote

Özkan E, Şen B, Akkuş H, Uysal M (March 1, 2025) A Study on the k-Mersenne and k-Mersenne-Lucas Sequences. Universal Journal of Mathematics and Applications 8 1 1–7.

IEEE

[1]E. Özkan, B. Şen, H. Akkuş, and M. Uysal, “A Study on the k-Mersenne and k-Mersenne-Lucas Sequences”, Univ. J. Math. Appl., vol. 8, no. 1, pp. 1–7, Mar. 2025, doi: 10.32323/ujma.1566270.

ISNAD

Özkan, Engin - Şen, Bayram - Akkuş, Hakan - Uysal, Mine. “A Study on the K-Mersenne and K-Mersenne-Lucas Sequences”. Universal Journal of Mathematics and Applications 8/1 (March 1, 2025): 1-7. https://doi.org/10.32323/ujma.1566270.

JAMA

1.Özkan E, Şen B, Akkuş H, Uysal M. A Study on the k-Mersenne and k-Mersenne-Lucas Sequences. Univ. J. Math. Appl. 2025;8:1–7.

MLA

Özkan, Engin, et al. “A Study on the K-Mersenne and K-Mersenne-Lucas Sequences”. Universal Journal of Mathematics and Applications, vol. 8, no. 1, Mar. 2025, pp. 1-7, doi:10.32323/ujma.1566270.

Vancouver

1.Engin Özkan, Bayram Şen, Hakan Akkuş, Mine Uysal. A Study on the k-Mersenne and k-Mersenne-Lucas Sequences. Univ. J. Math. Appl. 2025 Mar. 1;8(1):1-7. doi:10.32323/ujma.1566270