Hacer Sengul Kandemir, Mikail Et, Hüseyin Çakallı

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Study of difference sequences is a recent development in the summability theory. Sometimes a situation may arise that we have a sequence at hand and we are interested in sequences formed by its successive differences and in the structure of these new sequences. Studies on difference sequences were introduced in the 1980s and after that many mathematicians studied these kind of sequences and obtained some generalized difference sequence spaces. In this study, we generalize the concepts of weighted statistical convergence and weighted $\overline{\left[ N_{p}\right] }-$summability of real (or complex) numbers sequences to the concepts of $\Delta^{m}-$weighted statistical convergence of order $\alpha$ and weighted $\left[ \overline{N_{p}}^{\alpha}\right] \left(\Delta^{m},r\right)-$summability of order $\alpha$ by using generalized difference operator $\Delta^{m}$ and examine the relationships between $\Delta^{m}-$weighted statistical convergence of order $\alpha$ and weighted $\left[ \overline {N_{p}}^{\alpha}\right] \left( \Delta^{m},r\right) -$summability of order $\alpha.$ Our results are more general than the corresponding results in the existing literature.


statistical convergence, difference sequences, weighted summability

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Y. Altin, M. Et and M. Basarir, On some generalized dierence sequences of fuzzy numbers. Kuwait J. Sci. Engrg. 34 (2007), no. 1A, 1-14.

H. Altinok, M. Et and R. Colak, Some remarks on generalized sequence space of bounded variation of sequences of fuzzy numbers. Iran. J. Fuzzy Syst. 11(5) (2014),

-46, 109.

K. E. Akbas and M. Isik, On asymptotically lambda-statistical equivalent sequences of order alpha in probability, Filomat 34(13) (2020), 4359-4365.

N. D. Aral, H. Sengul Kandemir, I-lacunary statistical convergence of order of difference sequences of fractional order. Facta Univ. Ser. Math. Inform. 36 (2021), no.1, 43-55.

C. Belen and S. A. Mohiuddine: Generalized weighted statistical convergence and application. Applied Mathematics and Computation, 219(18) (2013), 9821--9826.

H. Cakallı, A new approach to statistically quasi Cauchy sequences, Maltepe Journal of Mathematics 1(1) (2019) 1-8.

H. Cakallı, C. G. Aras and A. Sonmez, Lacunary statistical ward continuity, AIP Conf. Proc. 1676, 020042 (2015).

H. Cakallı and H. Kaplan, A variation on lacunary statistical quasi Cauchy sequences, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 66(2) (2017) 71-79.

M. Candan and A. Gunes, Paranormed sequence space of non-absolute type founded using generalized difference matrix. Proc. Nat. Acad. Sci. India Sect. A 85(2) (2015), 269-276.

M. Candan, Domain of the double sequential band matrix in the classical sequence

spaces. J. Inequal. Appl. 2012, 2012:281.

R. Colak, (2010), Statistical convergence of order ; Modern Methods in Analysis and its Applications, Anamaya Publ. New Delhi, India,121-129.

R. Colak, Y. Altin and M. Mursaleen, On some sets of dierence sequences of fuzzy numbers, Soft Comput. 15 (2011), no. 4, 787-793.

J. S. Connor, (1988),The statistical and strong p-Cesaro convergence of sequences, Analysis 8, 47-63.

M. Et, R. Colak and Y. Altın, Strongly almost summable sequences of order alpha; Kuwait J. Sci. 41(2), (2014), 35-47.

M. Karakas, M. Et and V. Karakaya, Some geometric properties of a new difference sequence space involving lacunary sequences. Acta Math. Sci. Ser. B (Engl. Ed.) 33(6)

(2013), 1711-1720

M. Et and R. Colak, On generalized difference sequence spaces, Soochow J. Math. 21(4) (1995), 377-386.

H. Fast, Sur la convergence statistique, Colloq. Math., pp. (1951), 241-244.

J. A. Fridy, On statistical convergence, Analysis 5 (1985),301-313.

S. Ghosal, Weighted statistical convergence of order alpha and its applications, J. Egyptian Math. Soc. 24(1), (2016), 60-67.

M. Gungor and M. Et, r-strongly almost summable sequences dened by Orlicz functions. Indian J. Pure Appl. Math. 34(8) (2003),1141-1151.

M. Gungor and M. Et and Y. Altin, Strongly (V; ; q)-summable sequences defined

by Orlicz functions. Appl. Math. Comput. 157(2) (2004), 561-571.

M. Isk and K. E. Akbas, On lambda-statistical convergence of order alpha in probability, J. Inequal. Spec. Funct. 8(4), (2017), 57-64.

M. Isk and K. E. Akbas, On asymptotically lacunary statistical equivalent sequences of order alpha in probability, ITM Web of Conferences 13,(2017), 01024.

H. Kızmaz, H. On certain sequence spaces, Canad. Math. Bull. 24(2) (1981), 169-176.

V. Karakaya and T. A. Chishti, Weighted statistical convergence, Iran. J. Sci. Technol. Trans. A Sci. 33(33) (2009), 219-223.

M. Mursaleen, statistical convergence, Math. Slovaca 50(1) (2000), 111-115.

M. Mursaleen, V. Karakaya, M. Erturk and F. Gursoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput.

(18) (2012), 9132-9137.

T. Salat, On statistically convergent sequences of real numbers; Math. Slovaca 30 (1980), 139-150.

E. Savas and M. Et, On (m, I)-statistical convergence of order ; Period. Math.

Hungar. 71(2), (2015), 135-145.

I. J. Schoenberg,The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361-375.

H. Sengul and M. Et, On I-lacunary statistical convergence of order alpha of sequences of sets. Filomat 31(8) (2017), 2403-2412.

H. S. Kandemir, I-Deferred Statistical Convergence in Topological Groups, Maltepe Journal of Mathematics 1(2) (2019) 48-55.

H. S. Kandemir; M. Et and H. Cakallı; m-Weighted Statistical Convergence, International Conference of Mathematical Sciences, (ICMS 2020), Maltepe University, Istanbul, Turkey.

H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73-74.

S. Yıldız, Lacunary Statistical -Quasi Cauchy Sequences, Maltepe Journal of Mathematics 1(1) (2019) 9-17.

DOI: https://doi.org/10.22190/FUMI211004022S


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