آمار
| تعداد نشریات | 6 |
| تعداد شماره‌ها | 131 |
| تعداد مقالات | 1,634 |
| تعداد مشاهده مقاله | 1,915,268 |
| تعداد دریافت فایل اصل مقاله | 1,769,840 |
New generalizations and identities of Mersenne-Lucas numbers and polynomials with structural constraints | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 27 اردیبهشت 1405 اصل مقاله (443.44 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.30674.2571 | ||
| نویسندگان | ||
| Kalika Prasad1؛ Mritunjay Kumar Singh2؛ Munesh Kumari* 1 | ||
| 1Department of Mathematics, Government Engineering College Bhojpur, Bihar, India | ||
| 2Government Polytechnic, Nawada, Bihar, 805122, India | ||
| چکیده | ||
| This paper introduces and investigates two new sequences, $\{R_{n}^{(k)}\}$ and $\{R_{n}^{(k)}(x)\}$, which provide a distinct generalization of the Mersenne--Lucas numbers and polynomials, respectively, where the index $n$ is expressed in the form $n = sk + r$, with $0 \le r < k$. We derive several identities for these sequences in relation to the classical Mersenne and Mersenne--Lucas numbers and polynomials. Furthermore, we examine their algebraic properties and establish connections with existing sequences and polynomial families. In addition, we obtain closed-form expressions, Cassini-type identities, partial sums, recurrence relations, and various combinatorial identities associated with these sequences. | ||
| کلیدواژه‌ها | ||
| Mersenne-Lucas numbers؛ Mersenne polynomials؛ Mersenne-Lucas polynomials؛ recurrence relation | ||
| مراجع | ||
|
[1] P. Catarino, H. Campos, and P. Vasco, On the Mersenne sequence, Ann. Math. Inform. 46 (2016), 37–53.
[2] P. Catarino, P. Vasco, H. Campos, A.P. Aires, and A. Borges, New families of Jacobsthal and Jacobsthal-Lucas numbers, Algebra Discrete Math. 20 (2015), no. 1, 40–54.
[3] M. Chelgham and A. Boussayoud, On the k-Mersenne–Lucas numbers, Notes Number Theory Discrete Math. 27 (2021), no. 1, 7–13. https://doi.org/10.7546/nntdm.2021.27.1.7-13
[4] A. Dasdemir and G. Bilgici, Gaussian Mersenne numbers and generalized Mersenne quaternions, Notes Number Theory Discrete Math. 25 (2019), no. 3, 87–96. https://doi.org/10.7546/nntdm.2019.25.3.87-96
[5] M. El-Mikkawy and T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010), no. 12, 4456–4461. https://doi.org/10.1016/j.amc.2009.12.069
[6] R. Frontczak and T. Goy, Mersenne-Horadam identities using generating functions, Carpathian Math. Publ. 12 (2020), no. 1, 34–45. https://doi.org/10.15330/cmp.12.1.34-45
[7] R. Granger and A. Moss, Generalised Mersenne numbers revisited, Math. Comp. 82 (2013), no. 284, 2389–2420.
[8] A. Horadam, A synthesis of certain polynomial sequences, Proceedings of ‘The Sixth International Research Conference on Fibonacci Numbers and Their Applications’ (Pullman, Washington, USA), Applications of Fibonacci Numbers, vol. 6, Washington State University, Springer, June 2022, pp. 215–229.
[9] M. Kumari, Mersenne like sequences, Fibonacci numbers, associated matrices and their application, Ph.D. thesis, INFLIBNET / Shodhganga, 2025.
[10] M. Kumari, K. Prasad, and R. Mohanta, Algebra of quaternions and octonions involving higher order Mersenne numbers, Proc. Indian Nat. Sci. Acad. 91 (2025), 333–342.
[11] M. Kumari, K. Prasad, and J. Tanti, On the generalization of Mersenne and Gaussian Mersenne polynomials, J. Anal. 32 (2024), 931–947.
[12] M. Kumari, J. Tanti, and K. Prasad, On some new families of $k$-Mersenne and generalized $k$-Gaussian Mersenne numbers and their polynomials, Contrib. Discrete Math. 18 (2023), no. 2, 244–260. https://doi.org/10.55016/ojs/cdm.v18i2.74415
[13] E. Özkan and M. Taştan, On Gauss Fibonacci polynomials, on Gauss Lucas polynomials and their applications, Comm. Algebra 48 (2020), no. 3, 952–960. https://doi.org/10.1080/00927872.2019.1670193
[14] E. Özkan and M. Uysal, $d$-Gaussian Fibonacci, $d$-Gaussian Lucas polynomials, and their matrix representations, Ukr. Math. J. 75, no. 4, 562–585. https://doi.org/10.1007/s11253-023-02217-3
[15] E. Özkan, B. Şen, H. Akkuş, and M. Uysal, A study on the $k$-Mersenne and $k$-Mersenne-Lucas sequences, Univ. J. Math. Appl. 8 (2025), no. 1, 1–7.
[16] S. Palimar and B.R. Shankar, Mersenne primes in real Quadratic fields, J. Integer Seq. 15 (2012), no. 2, 12.5.6.
[17] K. Prasad and M. Kumari, Some new properties of frank matrices with entries Mersenne numbers, Nat. Acad. Sci. Lett. 48 (2025), no. 6, 743–749. https://doi.org/10.1007/s40009-024-01538-6
[18] K. Prasad, M. Kumari, and J. Tanti, Generalized $k$-balancing and $k$-Lucas balancing numbers and associated polynomials, Kyungpook Math. J. 63 (2023), no. 4, 539–550. https://doi.org/10.5666/KMJ.2023.63.4.539
[19] K. Prasad, M. Kumari, and J. Tanti, On norms, spread, characteristic polynomial and determinant of hankel and Toeplitz matrices with Mersenne sequence, Commun. Comb. Optim. 11 (2026), no. 3, 941–958. https://doi.org/10.22049/cco.2024.30037.2287
[20] K. Prasad, R. Mohanty, M. Kumari, and H. Mahato, Some new families of generalized k-Leonardo and Gaussian Leonardo numbers, Commun. Comb. Optim. 9 (2024), no. 3, 539–553. https://doi.org/10.22049/cco.2023.28236.1485
[21] R. M. Robinson, Mersenne and Fermat numbers, Proc. Amer. Math. Soc. 5 (1954), no. 5, 842–846. https://doi.org/10.2307/2031878
[22] N. Saba and A. Boussayoud, On the bivariate Mersenne Lucas polynomials and their properties, Chaos, Solitons and Fractals 146 (2021), 110899. https://doi.org/10.1016/j.chaos.2021.110899
[23] Y. Soykan, A study on generalized Mersenne numbers, Journal of Progressive Research in Mathematics 18 (2021), no. 3, 90–108.
[24] Y. Soykan, On generalized p-Mersenne numbers, Earthline J. Math. Sci. 8 (2022), no. 1, 83–120. https://doi.org/10.34198/ejms.8122.83120
[25] D. Taşcı and E. Sevgi, Some properties between Mersenne, Jacobsthal and Jacobsthal-Lucas hybrid numbers, Chaos, Solitons and Fractals 146 (2021), 110862. https://doi.org/10.1016/j.chaos.2021.110862 | ||
|
آمار تعداد مشاهده مقاله: 165 تعداد دریافت فایل اصل مقاله: 90 |
||