On Some New Normed Narayana Sequence Spaces

Abstract

In this paper, we first establish the regular matrix $N$ using Narayana numbers. Then, we create new normed sequence spaces $Z(N)$ using the matrix $ N$ and demonstrate that these spaces are linearly isomorphic to $Z$ where $Z\in\{c_0, c, \ell_p, \ell_\infty\}$. Additionally, we provide inclusion relations for the spaces $c_0(N)$, $c(N)$, $\ell_p(N)$, and $\ell_\infty(N)$. Furthermore, we construct the Schauder bases of the $c_0(N)$, $c(N)$, and $\ell_p(N)$. Finally, we compute the $\alpha$-, $\beta$-, and $\gamma$-duals of these spaces and characterize the classes $(Z(N),X)$ for the certain choice of the sequence space $X$.

Keywords

• Narayana numbers
• Narayana matrices
• sequence spaces
• matrix transformations

References

1. J.-P. Allouche, T. Johnson, Narayana's cows and delayed morphisms, Journées d'Informatique Musicale (1996) 6 pages.
2. A. N. Singh, On the use of series in Hindu mathematics, Osiris 1 (1936) 606-628.
3. T. Koshy, Fibonacci and Lucas numbers with applications, 2nd Edition, John Wiley & Sons, New Jersey, 2019.
4. N. Sloane, The on-line encyclopedia of integer sequences, Mathematicae et Informaticae 41 (2013) 219-234.
5. G. Bilgici, The generalized order-$k$ Narayana's cows numbers, Mathematica Slovaca 66 (4) (2016) 795-802.
6. T. V. Didkivska, M. St'opochkina, Properties of Fibonacci-Narayana numbers, In the World of Mathematics 9 (1) (2003) 29-36.
7. C. Flaut, V. Shpakivskyi, On generalized Fibonacci quaternions and Fibonacci-Narayana quaternions, Advances in Applied Clifford Algebras 23 (3) (2013) 673-688.
8. T. Goy, On identities with multinomial coefficients for Fibonacci-Narayana sequence, Annales Mathematicae et Informaticae 49 (2018) 75-84.

9. J. L. Ramírez, V. F. Sirvent, A note on the k-Narayana sequence, Annales Mathematicae et Informatica 45 (2015) 91-105.
10. R. Zatorsky, T. Goy, Parapermanents of triangular matrices and some general theorems on number sequences, Journal of Integer Sequences 19 (2) (2016) Article 16.2.2 23 pages.
11. Y. Soykan, On generalized Narayana numbers, International Journal of Advances in Applied Mathematics and Mechanics 7 (3) (2020) 43-56.
12. P.-N. Ng, P.-Y. Lee, Cesáro sequences spaces of non-absolute type, Commentationes Mathematicae (Prace Matematyczne) 20 (2) (1978) 429-433.
13. C.-S. Wang, On Nörlund sequence spaces, Tamkang Journal of Mathematics 9 (1) (1978) 269-274.
14. B. Altay, F. Başar, M. Mursaleen, On the Euler sequence spaces which include the spaces $\ell_p$ and $\ell_\infty$, Information Sciences 176 (10) (2006) 1450-1462.
15. M. Şengonül, F. Başar, Some new Cesáro sequence spaces of non-absolute type which include, Soochow Journal of Mathematics 31 (1) (2005) 107-119.
16. M. Kirişçi, F. Başar, Some new sequence spaces derived by the domain of generalized difference matrix, Computers & Mathematics with Applications 60 (5) (2010) 1299-1309.
17. S. Erdem, S. Demiriz, A study on strongly almost convergent and strongly almost null binomial double sequence spaces, Fundamental Journal of Mathematics and Applications 4 (4) (2021) 271-279.
18. F. Başar, Summability theory and its applications, 2nd Edition, Chapman and Hall/CRC, New York, 2022.
19. J. Boos, F. P. Cass, Classical and modern methods in summability, Oxford University Press, Oxford, 2000.
20. E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, Journal of Inequalities and Applications 2013 (1) (2013) Article Number 38 15 pages.
21. M. Candan, E. E. Kara, A study of topological and geometrical characteristics of new Banach sequence spaces, Gulf Journal of Mathematics 3 (4) (2015) 67-84.
22. E. E. Kara, M. Ilkhan, Some properties of generalized Fibonacci sequence spaces, Linear and Multilinear algebra 64 (11) (2016) 2208-2223.
23. M. Karakaş, A. M. Karakaş, New Banach sequence spaces that is defined by the aid of Lucas numbers, Iğdır University Journal of the Institute of Science and Technology 7 (4) (2017) 103-111.
24. M. Karakaş, A. M. Karakaş, A study on Lucas difference sequence spaces $\ell_p(\hat{E}(r, s))$ and $\ell_\infty(\hat{E}(r, s))$, Maejo International Journal of Science and Technology 12 (1) (2018) 70-78.
25. T. Yaying, B. Hazarika, S. A. Mohiuddine, On difference sequence spaces of fractional-order involving Padovan numbers, Asian-European Journal of Mathematics 14 (06) (2021) 2150095 22 pages.
26. M. İ. Kara, E. E. Kara, Matrix transformations and compact operators on Catalan sequence spaces, Journal of Mathematical Analysis and Applications 498 (1) (2021) 124925 17 pages.
27. M. Karakaş, On the sequence spaces involving Bell numbers, Linear and Multilinear Algebra 71 (14) (2023) 2298-2309.
28. M. C. Dağli, A new almost convergent sequence space defined by Schröder matrix, Linear and Multilinear Algebra 71 (11) (2023) 1863-1874.
29. A. Wilansky, Summability through functional analysis, Elsevier, Amsterdam, 2000.
30. A. M. Jarrah, E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat 17 (2003) 59-78.
31. M. Stieglitz, H. Tietz, Matrix transformationen von Folgenräumen Eine Ergebnisübersicht, Mathematische Zeitschrift 154 (1977) 1-16.