Öz
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.
Anahtar Kelimeler
Catalan numbers Fibonacci numbers Generating functions Hankel transform $k$-Mersenne sequences $k$-Mersenne-Lucas sequences
Kaynakça
- [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.
Öz
Kaynakça
- [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.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Cebir ve Sayı Teorisi |
| Bölüm | Makaleler |
| Yazarlar | |
| Erken Görünüm Tarihi | 23 Şubat 2025 |
| Yayımlanma Tarihi | 25 Mart 2025 |
| Gönderilme Tarihi | 13 Ekim 2024 |
| Kabul Tarihi | 6 Şubat 2025 |
| Yayımlandığı Sayı | Yıl 2025 Cilt: 8 Sayı: 1 |
Kaynak Göster
Cited By
On New Sequences of $p$-Binomial and Catalan Transforms of the $k$-Mersenne Numbers and Associated Generating Functions
Universal Journal of Mathematics and Applications
https://doi.org/10.32323/ujma.1641001
Universal Journal of Mathematics and Applications
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