Abstract
In this paper, we study q-statistical convergence for double sequences. The definitions of q-analog of statistical Cauchy and statistical pre-Cauchy for double sequences are given. The necessary and sufficient condition for a double sequence to have different statistical limits is also obtained. We show that a q-statistical convergent sequence is q-statistical Cauchy and vice-versa.
Similar content being viewed by others
References
H. Aktuğlu, Ş Bekar, \(q\)-Cesàro matrix and \(q\)-statistical convergence. Jour. Comput. Appl. Math. 235(16), 4717–4723 (2011)
A.A.H. Al-Abied, M. Ayman Mursaleen, M. Mursaleen, Szász type operators involving Charlier polynomials and approximation properties. Filomat 35(15), 5149–5159 (2021)
M. Ayman Mursaleen, S. Serra-Capizzano, Statistical convergence via \(q\)-calculus and a KorovkinÂs type approximation theorem. Axioms 11(2), 70 (2022)
Q.-B. Cai, A. Kilicman, M. Ayman Mursaleen, Approximation properties and q-statistical convergence of Stancu type generalized Baskakov–Szász operators. J. Funct. Spaces 2022, 2286500 (2022)
M.-Y. Chen, Md. Nasiruzzaman, M. Ayman Mursaleen, N. Rao, A.A. Kiliçman, On shape parameter a based approximation properties and \(q\)-statistical convergence of Baskakov–Gamma operators. J. Math. 2022, 4190732 (2022)
M. Çinar, M. Et, \(q\)-double Cesàro matrices and \(q\)-statistical convergence of double sequences. Nat. Acad. Sci. Lett. 43(1), 73–76 (2020)
J.S. Connor, J. Fridy, J. Kline, Statistically pre-Cauchy sequences. Analysis 14(4), 311–318 (1994)
J.S. Connor, The statistical and strong \(p\)-Cesàro convergence of sequences. Analysis 8(1–2), 47–63 (1988)
H. Fast, Sur la convergence statistique. Colloq. Math. 2(3–4), 241–244 (1951)
A. Freedman, J. Sember, Densities and summability. Pac. J. Math. 95(2), 293–305 (1981)
J.A. Fridy, On stastistical convergence. Analysis 5, 301–313 (1985)
V. Kac, P. Cheung, Quantum Calculus. Universitext (Springer, New York, 2002)
F. Móricz, Statistical convergence of multiple sequences. Arch. Math. 81(1), 82–89 (2003)
M. Mursaleen, O.H.H. Edely, Statistical convergence of double sequences. J. Math. Anal. Appl. 288, 223–231 (2003)
M. Mursaleen, S. Tabassum, R. Fatma, On q-statistical summability method and its properties. Iran. J. Sci. Technol. Trans. Sci. 46(2), 455–460 (2022)
M. Nasiruzzaman, A. Kiliçman, M. Ayman-Mursaleen, Construction of q-Baskakov operators by wavelets and approximation properties. Iran. J. Sci. Technol. Trans. Sci. 46(5), 1495–1503 (2022)
T. Šalát, On statistically convergent sequences of real numbers. Math. Slovaca 30(2), 139–150 (1980)
I.J. Schoenberg, The integrability of certain functions and related summability methods. Am. Math. Mon. 66(5), 361–375 (1959)
H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2(1), 73–74 (1951)
O. Toeplitz, Über allgemeine lineare Mittelbildungen. Prace Matematyczno-Fizyczne 22(1), 113–119 (1911)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Mursaleen, M., Tabassum, S. & Fatma, R. On the q-statistical convergence of double sequences. Period Math Hung 88, 324–334 (2024). https://doi.org/10.1007/s10998-023-00556-z
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-023-00556-z
Keywords
- Sequence spaces
- Double sequences
- Statistical convergence
- q-statistical Cauchy
- q-statistical pre-Cauchy
- Cesàro matrix
- q-Cesàro matrix