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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 53 Issue: 4

Certain topological properties for almost convergent Catalan-Motzkin sequence space

Muhammet Cihat Dağlı(AKDENIZ UNIVERSITY)0000-0003-2859-902X
Pages
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Publication Date
2024-08-27
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In addition, by defining Catalan-Motzkin core of a complex-valued sequence, certain inclusion relations are derived for it.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:The purpose of this paper is to introduce a new almost convergent sequence space by means of a matrix involving Catalan and Motzkin numbers.
📊
Key Findings & Data:It is demonstrated that this novel sequence space and the space of all almost convergent sequences are linearly isomorphic and the $\beta$-dual of this new space is deduced.
💡
Conclusion & Impact:In addition, by defining Catalan-Motzkin core of a complex-valued sequence, certain inclusion relations are derived for it.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Catalan numbers · almost convergence · sequence spaces

Abstract

The purpose of this paper is to introduce a new almost convergent sequence space by means of a matrix involving Catalan and Motzkin numbers. It is demonstrated that this novel sequence space and the space of all almost convergent sequences are linearly isomorphic and the $\beta$-dual of this new space is deduced. In addition, by defining Catalan-Motzkin core of a complex-valued sequence, certain inclusion relations are derived for it.

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Cite This Article

Dağlı, M. C. (2024). Certain topological properties for almost convergent Catalan-Motzkin sequence space. Hacettepe Journal of Mathematics and Statistics, Volume: 53 Issue: 4. https://doi.org/10.15672/hujms.1298097

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References (50)

  1. B. Altay, F. Başar and E. Malkowsky, Matrix transformations on some sequence spaces related to strong Cesàro summability and boundedness, Appl. Math. Comput. 211, 255–264, 2009.
  2. C. Aydın and F. Başar, On the new sequence spaces which include the spaces $c_{0}$ and $c$, Hokkaido Math. J. 33 (2), 383–398, 2004.
  3. C. Aydın and F. Başar, Some new difference sequence spaces, Appl. Math. Comput. 157 (3), 677–693, 2004.
  4. F. Başar, $f$−conservative matrix sequences, Tamkang J. Math. 22, 205–212 1991.
  5. F. Başar, Summability Theory and Its Applications, 2nd ed., CRC Press/Taylor & Francis Group, Boca Raton · London · New York, 2022.
  6. F. Başar and B. Altay, On the space of sequences of p−bounded variation and related matrix mappings, (English, Ukrainian summary) Ukrain. Mat. Zh. 55 (1) (2003), 108–118; reprinted in Ukrainian Math. J. 55 (1), 136–147, 2003.
  7. F. Başar and R. Çolak, Almost-conservative matrix transformations, Turkish J. Math. 13, 91–100, 1989.
  8. F. Başar and M. Kirişçi, Almost convergence and generalized difference matrix, Comput. Math. App. 61 (3), 602–611, 2011.
  9. F. Başar, E. Malkowsky and B. Altay, Matrix transformations on the matrix domains of triangles in the spaces of strongly C1−summable and bounded sequences, Publ. Math. Debrecen 73 (1-2), 193–213, 2008.
  10. F. Başar and İ. Solak, Almost-coercive matrix transformations, Rend. Mat. Appl. 11–2, 249–256, 1991.
  11. M. Candan, Vector-valued FK-spaces defined by a modulus function and an infinite matrix, Thai J. Math. 12 (1), 155–165, 2014.
  12. M. Candan, Almost convergence and double sequential band matrix, Acta Math. Sci. 34 (2), 354–366 2014.
  13. J. Connor, J.A. Fridy and C. Orhan, Core equality results for sequences, J. Math. Anal. Appl. 321, 515–523, 2006.
  14. R.G. Cooke, Infinite matrices and sequence spaces, MacMillan, New York, 1950.
  15. C. Çakan and H. Coşkun, Some new inequalities related to the invariant means and uniformly bounded function sequences, Appl. Math. Lett. 20, 605–609, 2007.
  16. H. Çoşkun and C. Çakan, A class of statistical and $\sigma$−conservative matrices, Czechoslovak Math. J. 55, 791–801, 2005.
  17. H. Çoşkun, C. Çakan and M. Mursaleen, On the statistical and $\sigma$−cores, Stud. Math. 154, 29–35, 2003.
  18. M.C. Dağlı, A novel conservative matrix arising from Schröder numbers and its properties, Linear Multilinear Algebra 71 (8), 1338–1351, 2023.
  19. M.C. Dağlı, A new almost convergent sequence space defined by Schröder matrix, Linear Multilinear Algebra, 71 (11), 1863–1874, 2023.
  20. M.C. Dağlı, Matrix mappings and compact operators for Schröder sequence spaces, Turkish J. Math. 46 (6), 2304–2320, 2022.
  21. K. Demirci, A-statistical core of a sequence, Demonstr. Math. 33, 43–51, 2000.
  22. S. Demiriz, M. İlkhan and E.E. Kara, Almost convergence and Euler totient matrix, Ann. Funct. Anal. 11, 604–616, 2020.
  23. J.P. Duran, Infinite matrices and almost convergence, Math. Z. 128, 75–83, 1972.
  24. H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241–244, 1951.
  25. J.A. Fridy and C. Orhan, Statistical core theorems, J. Math. Anal. Appl. 208, 520– 527, 1997.
  26. M. İlkhan, Matrix domain of a regular matrix derived by Euler totient function in the spaces $c_{0}$ and $c$, Mediterr. J. Math. 17 (1), Article no: 27, 2020.
  27. M. İlkhan, A new conservative matrix derived by Catalan numbers and its matrix domain in the spaces $c$ and $c_{0}$, Linear Multilinear Algebra 68 (2), 417–434, 2020.
  28. M. İlkhan and E.E. Kara, A new Banach space defined by Euler totient matrix operator, Oper. Matrices 13 (2), 527–544, 2019.
  29. M.İlkhan, E.E.Kara and F. Usta, Compact operators on the Jordan totient sequence spaces, Math. Methods Appl. Sci. 44 (9), 7666–7675, 2021.
  30. M. İlkhan, N. Simsek and E.E. Kara, A new regular infinite matrix defined by Jordan totient function and its matrix domain in ℓp, Math. Methods Appl. Sci. 44 (9), 7622– 7633, 2020.
  31. S. Jasrotia, U.P. Singh and K. Raj, Some new observations on Catalan almost convergent sequence spaces and the Catalan core, Acta Sci. Math. (Szeged) 87 (1-2), 153–163, 2021.
  32. E.E. Kara and S. Demiriz, Some new paranormed difference sequence spaces derived by Fibonacci numbers, Miskolc Math. Notes 16 (2), 907–923, 2015.
  33. E.E. Kara and M. İlkhan, Some properties of generalized Fibonacci sequence spaces, Linear Multilinear Algebra 64 (11), 2208–2223, 2016.
  34. M.İ. Kara and E.E. Kara, Matrix transformations and compact operators on Catalan sequence spaces, J. Math. Anal. Appl. 498 (1), Article no: 124925, 2021.
  35. A. Karaisa and F. Özger, On almost convergence and difference sequence spaces of order m with core theorems, Gen. Math. Notes 26 (1), 102–125, 2015.
  36. M. Karakas, On the sequence spaces involving bell numbers, Linear Multilinear Algebra 71 (14), 2298–2309, 2023.
  37. M. Karakas and M.C. Dağlı, Some topologic and geometric properties of new Catalan sequence spaces, Adv. Oper. Theory, 8 (1), Article number: 14, 2023.
  38. G. Kılınç and M. Candan, Some generalized Fibonacci difference spaces defined by a sequence of modulus functions, Facta Univ. Ser. Math. Inf. 32, 95–116, 2017.
  39. M. Kirişçi and F. Başar, Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math. Appl. 60 (5), 1299–1309, 2010.
  40. M. Mursaleen and F. Başar, Sequence Spaces: Topics in Modern Summability Theory, CRC Press/Taylor & Francis Group, Series: Mathematics and Its Applications, Boca Raton · London · New York, 2020.
  41. F. Özger and F. Başar, Domain of the double sequential band matrix $B(\tilde{r},\tilde{s})$ on some Maddox’s spaces, Acta Math. Sci. 34, 394–408, 2014.
  42. A.A. Shcherbakov, Kernels of sequences of complex numbers and their regular transformations, Math. Notes 22, 948–953, 1977.
  43. J.A. Sıddıqi, Infinite matrices summing every almost periodic sequences, Pacific J. Math. 39, 235–251, 1971.
  44. S. Simons, Banach limits, infinite matrices and sublinear functionals, J. Math. Anal. Appl. 26, 640–655, 1969.
  45. R.P. Stanley, Catalan Numbers, Cambridge University Press, New York, 2015.
  46. H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2, 73–74, 1951.
  47. H. Steinhaus, Quality control by sampling, Colloq. Math. 2, 98–108, 1951.
  48. Q. Qamaruddin and S.A. Mohuiddine, Almost convergence and some matrix transformations, Filomat 21, 261–266, 2007.
  49. T. Yaying and B. Hazarika, On sequence spaces defined by the domain of a regular Tribonacci matrix, Math. Slovaca 70 (3), 697–706, 2020.
  50. T. Yaying and M.İ. Kara, On sequence spaces defined by the domain of tribonacci matrix in $c_{0}$ and $c$, Korean J. Math. 29 (1), 25–40, 2021.