New structure of Fibonacci numbers using concept of \(\Delta\)--operator

Volume 26, Issue 2, pp 101--112 http://dx.doi.org/10.22436/jmcs.026.02.01

Publication Date: November 05, 2021 Submission Date: September 09, 2021 Revision Date: October 13, 2021 Accteptance Date: October 16, 2021

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Authors

D. Fathima - Basic Science Department, College of Science and Theoretical Studies, Saudi Electronic University-Jeddah Female, Kingdom of Saudi Arabia. M. M. AlBaidani - Department of Mathematics, College of Science and Humanities Studies, rince Sattam Bin Abdulaziz University, Al Kharj 11942, Kingdom of Saudi Arabia. A. H. Ganie - Basic Science Department, College of Science and Theoretical Studies, Saudi Electronic University-Abha Male, Kingdom of Saudi Arabia. A. Akhter - Department of Applied Science and Humanities, SSM College of Engineering Parihaspora--Pattan, Jammu and Kashmir, India.

Abstract

The theory of sequence spaces is the fundamental of summability and applications to various sequences like Fibonacci sequences were deeply studied. In [A. H. Ganie, In: Matrix Theory-Applications and Theorems, \(\textbf{2018}\) (2018), 75--86], the author has analyzed the Fibonacci sequences and studied its various properties. By utilizing this concept, the notion of this paper is to introduce new scenario of spaces using Fibonacci numbers. By using Kizmaz operator, we shall introduce the difference sequence spaces \(c^{J}_{0}(\widetilde{\mho_g})\), \(c^{J}(\widetilde{\mho_g})\) and \(\ell^{J}_{\infty}(\widetilde{\mho_g})\) by involving Fibonacci sequence and the idea of ideal convergence. We will prove certain basic inclusion relations and study these for some topological properties.

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Keywords

• Fibonacci numbers
• ideal convergence
• BK-spaces

MSC

•  40A05
•  46A45
•  40C05

References

• [1] V. Baláž, T. Visnyai, $I$-Convergence of Arithmetical Functions, In: Number Theory and Its Applications, InTechOpen, 2020 (2020), 125--134
  • View Article
  • Google Scholar


• [2] M. Başarır, F. Başar, E. E. Kara, On the spaces of Fibonacci difference absolutely $p$-summable, null and convergent sequences, Sarajevo J. Math., 12 (2016), 167--182
  • View Article
  • MathSciNet
  • Google Scholar


• [3] M. Candan, E. E. Kara, A study on topological and geometrical characteristics of new Banach sequence spaces, Gulf J. Math., 3 (2015), 67--84
  • View Article
  • MathSciNet
  • Google Scholar


• [4] M. Candan, K. Kayaduman, Almost convergent sequence space derived by generalized Fibonacci matrix and Fibonacci core, J. Adv. Math. Comput. Sci., 7 (2015), 150--167
  • View Article
  • Google Scholar


• [5] A. Das, B. Hazarika, Some new Fibonacci difference spaces of non-absolute type and compact operators, Linear Multilinear Algebra, 65 (2017), 2551--2573
  • View Article
  • MathSciNet
  • Google Scholar


• [6] A. Das, B. Hazarika, Matrix transformation of Fibonacci band matrix on generalized $bv$-space and its dual spaces, Bol. Soc. Parana. Mat. (3), 36 (2018), 41--52
  • View Article
  • MathSciNet
  • Google Scholar


• [7] A. Das, B. Hazarika, Some properties of generalized Fibonacci difference bounded and $p$-absolutely convergent sequences, Bol. Soc. Parana. Mat. (3), 36 (2018), 37--50
  • View Article
  • MathSciNet
  • Google Scholar


• [8] P. Das, P. Kostyrko, W. Wilczynski, P. Malik, I and $I^{\ast}$-convergence of double sequences, Math. Slovaca, 58 (2008), 605--620
  • View Article
  • Google Scholar


• [9] H. Fast, Sur la convergence statistique, (French) Colloq. Math., 2 (1951), 241--244
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] D. Fathema, A. H. Ganie, On some new scenario of $\Delta$-spaces, J. Nonlinear Sci. Appl., 14 (2021), 163--167
  • View Article
  • MathSciNet


• [11] A. H. Ganie, Nature of Phylllotaxy and topology of H-matrix, In: Matrix Theory-Applications and Theorems, 2018 (2018), 75--86
  • View Article
  • Google Scholar


• [12] A. H. Ganie, Some new approach of spaces of non-integral order, J. Nonlinear Sci. Appl., 14 (2021), 89--96
  • View Article
  • MathSciNet
  • Google Scholar


• [13] A. H. Ganie, New spaces over modulus function, Bol. da Soc. Parana. de Mat., (accepted)
  • View Article
  • Google Scholar


• [14] A. H. Ganie, M. M. AlBaidani, Matrix Structure of Jacobsthal numbers, J. Funct. Space, 2021 (2021), 5 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [15] A. H. Ganie, A. Antesar, Certain sequence spaces using $\Delta$-operator, Adv. Stud. Contemp. Math. (Kyungshang), 30 (2020), 17--27
  • View Article


• [16] A. H. Ganie, D. Fathima, Almost convergence property of generalized Riesz spaces, J. Appl. Math. Comput., 4 (2020), 249--253
  • View Article
  • Google Scholar


• [17] A. H. Ganie, S. A. Lone, Generalised Cesaro difference sequence space of non- absolute type, J. Sci. Data Anal., 20 (2020), 147--153
  • View Article
  • Google Scholar


• [18] A. H. Ganie, N. A. Sheikh, Infinite matrices and almost convergence, Filomat, 29 (2015), 1183--1188
  • View Article
  • MathSciNet
  • MATH


• [19] D. N. Georgiou, S. D. Iliadis, A. C. Megaritis, G. A. Prinos, Ideal-convergence classes, Topology Appl., 222 (2017), 217--226
  • View Article
  • MathSciNet
  • Google Scholar


• [20] F. Gökçek, M. A. Sarigöl, Some matrix and compact operators of the absolute Fibonacci spaces, Kragujevac J. Math., 44 (2020), 273--286
  • View Article
  • MathSciNet
  • Google Scholar


• [21] T. Jalal, Some ideal convergent multiplier sequence spaces using de la Vallee Poussin mean and Zweier operator, Proyecciones, 39 (2020), 91--105
  • View Article
  • MathSciNet
  • Google Scholar


• [22] T. Jalal, S. A. Gupkari, A. H. Ganie, Infinite matrices and $\sigma$-convergent sequences, Southeast Asian Bull. Math., 36 (2012), 825--830
  • View Article
  • Google Scholar
  • MATH


• [23] E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, J. Inequal. Appl., 2013 (2013), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [24] E. E. Kara, M. Ilkhan, Lacunary $I$-convergent and lacunary $I$-bounded sequence spaces defined by an Orlicz function, Electron. J. Math. Anal. Appl., 4 (2016), 150--159
  • View Article
  • MathSciNet
  • Google Scholar


• [25] M. Kiricsci, A. Karaisa, Fibonacci numbers, statistical convergence and applications, arXiv, 2016 (2016), 16 pages
  • View Article
  • Google Scholar


• [26] H. Kizmaz, On certain sequences spaces, Canad. Math. Bull., 24 (1981), 169--176
  • View Article
  • Google Scholar


• [27] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York (2001)
  • View Article
  • MathSciNet
  • Google Scholar


• [28] P. Kostyrko, M. Macaj, T. Salat, Statistical convergence and $I$-convergence, Real Analysis Exchange, New York (1999)
  • Google Scholar


• [29] E. Malkowsky, V. Rakocevic, Summability Methods and Applications, Mathematical Institute of the Serbian Academy of Sciences and Arts, Serbia (2020)
  • View Article
  • Google Scholar


• [30] S. A. Mohiuddine, B. Hazarika, Some Classes of Ideal Convergent Sequences and Generalized Difference Matrix Operator, Filomat, 31 (2017), 1827--1834
  • View Article
  • MathSciNet
  • Google Scholar


• [31] M. Mursaleen, A. H. Ganie, N. A. Sheikh, New type of difference sequenence spaces and matrix transformation, Filomat, 28 (2014), 1381--1392
  • View Article
  • MathSciNet
  • Google Scholar


• [32] M. Mursaleen, S. K. Sharma, Spaces of ideal convergent sequences, Sci. World J., 2014 (2014), 6 pages
  • View Article
  • Google Scholar


• [33] T. Šalát, B. C. Tripathy, M. Ziman, On some properties of I-convergence, Tatra Mt. Math. Publ., 28 (2004), 279--286
  • View Article
  • MathSciNet
  • Google Scholar


• [34] T. Šalát, B. C. Tripathy, M. Ziman, On $I$-convergence field, Ital. J. Pure Appl. Math., 17 (2005), 45--54
  • View Article
  • MathSciNet
  • MATH


• [35] N. A. Sheikh, A. H. Ganie, A new paranormed sequence space and some matrix transformation, Acta Math. Acad. Paed. Nyir., 28 (2012), 47--58
  • View Article
  • Google Scholar


• [36] N. A. Sheikh, A. H. Ganie, Some matrix transformations and almost convergence, Kathmandu Uni. J. Sci. Eng. Tech., 8 (2012), 89--92
  • View Article
  • Google Scholar


• [37] N. A. Sheikh, A. H. Ganie, A new type of sequence space of non-absolute type and matrix transformation, WSEAS Trans. J. Math., 8 (2013), 852--859
  • View Article
  • Google Scholar


• [38] N. A. Sheikh, T. Jalal, A. H. Ganie, New type of sequence spaces of non-absolute type and some matrix transformations, Acta Math. Acade. Paedag. Nyireg., 29 (2013), 51--66
  • View Article
  • Google Scholar


• [39] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73--74
  • Google Scholar


• [40] T. A. Tarray, P. A. Naik, R. A. Najar, Matrix representation of an all-inclusive Fibonacci sequence, Asian J. Math. Stat., 11 (2018), 18--26
  • View Article
  • Google Scholar


• [41] B. C. Tripathy, B. Hazarika, $I$-convergent sequence spaces associated with multiplier sequences, Math. Inequal. Appl., 11 (2008), 543--548
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [42] B. C. Tripathy, B. Hazarika, Some $I$-convergent sequence spaces defined by Orlicz functions, Acta Math. Appl. Sin. Engl. Ser., 27 (2011), 149--154
  • View Article
  • MathSciNet
  • Google Scholar


• [43] O. Tuğ, On some Generalized Norlund Ideal Convergent Sequence Spaces, ZANCO J. Pure Appl. Sci., 28 (2016), 97--103
  • View Article
  • Google Scholar


• [44] F. Yaşar, K. Kayaduman, On the domain of the Fibonacci difference matrix, Mathematics, 7 (2019), 10 pages
  • View Article
  • Google Scholar