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Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order ρ ˜

Year 2025, Volume: 29 Issue: 1, 35 - 43, 25.04.2025

Ömer Kişi , Rümeysa Akbıyık

https://doi.org/10.19113/sdufenbed.1625365

Abstract

In the context of sequence spaces and summability theory, significant progress has been made in the theoretical foundations of classical concepts such as convergence and boundedness. However, the exploration of innovative convergence methods continues to emerge as a critical area of research. A notable gap in the literature is the integration of Bessel functions into these theoretical frameworks. This study aims to fill this gap by introducing the concepts of Bessel I_2-Δ_θ^n-asymptotically statistical equivalence of order ρ ˜ and strong Bessel I_2-Δ_θ^n-asymptotically equivalence of order ρ̃ for double sequences, examining the comprehensive interrelations of these concepts. The inclusion of Bessel functions in these theories not only extends the existing theoretical frameworks within mathematical analysis but also provides a solid foundation for further research and the solution of applied problems in the field. These innovative concepts contribute to a deeper understanding of sequence behavior related to Bessel functions and enhance their applicability in mathematical applications.

Keywords

Asymptotical equivalence Bessel statistical convergence Lacunary sequence

References

  • [1] Fast, H. 1951. Sur la Convergence Statistique. Colloquium Mathematicum, 2, 241-244.
  • [2] Fridy, J. J., Orhan, C. 1993. Lacunary Statistical Convergence. Pacific Journal of Mathematics, 160(1), 43-51.
  • [3] Çakan, C., Altay, B., Çoşkun, H. 2010. Double Lacunary Density and Lacunary Statistical Convergence of Double Sequences. Studia Scientiarum Mathematicarum Hungarica, 47(1), 35-45.
  • [4] Freedman, A. R., Sember, J. J., Raphael, M. 1978. Some Cesàro-type Summability Spaces. Proceedings of the London Mathematical Society, 3(3), 508-520.
  • [5] Kostyrko, P., Salat, T., Wilczynski, W. 2000-2001. T-Convergence. Real Analysis Exchange, 26(2), 669-686.
  • [6] Das, P., Kostyrko, P., Wilczyński, W., Malik, P. 2008. T and T^*-Convergence of Double Sequences. Mathematica Slovaca, 58(5), 605-620.
  • [7] Tripathy, B. C., Hazarika, B., Choudhary, B. 2012. Lacunary T-Convergent Sequences. Kyungpook Mathematical Journal, 52(4), 473-482.
  • [8] Das, P., Savaş, E., Ghosal, S. K. 2011. On Generalizations of Certain Summability Methods Using Ideals. Applied Mathematics Letters, 24, 1509-1514.
  • [9] Belen, C., Yıldırım, M. 2012. On Generalized Statistical Convergence of Double Sequences via Ideals. Annali dell'Universita di Ferrara, 58(1), 11-20.
  • [10] Kumar, S., Kumar, V., Bhatia, S. S. 2013. On Ideal Version of Lacunary Statistical Convergence of Double Sequences. General Mathematics Notes, 17(1), 32-44.
  • [11] Gürdal, M., Şahiner, A., Açık, I. 2009. Approximation Theory in 2-Banach Spaces. Nonlinear Analysis, 71(5-6), 1654-1661.
  • [12] Şahiner, A., Gürdal M., Yiğit, T. 2011. Ideal Convergence Characterization of the Completion of Linear n-Normed Spaces. Computers & Mathematics with Applications, 61(3), 683-689.
  • [13] Gürdal, M., Şahiner, A. 2012. Statistical Approximation with a Sequence of 2-Banach Spaces. Mathematical and Computer Modelling, 55(3-4), 471-479.
  • [14] Yamancı, U., Gürdal, M. 2013. On Lacunary Ideal Convergence in Random n-Normed Space. Journal of Mathematics , 2013(1); 868457, 1-8.
  • [15] Gürdal, M., Huban, M.B. 2014. On T-Convergence of Double Sequences in the Topology Induced by Random 2-Norms. Matematički Vesnik, 66(1), 73-83.
  • [16] Yamancı, U., Gürdal, M. 2014. T-Statistical Convergence in 2-Normed Space. Arab Journal of Mathematical Sciences, 20(1), 41-47.
  • [17] Gürdal, M., Yamancı, U. 2015. Statistical Convergence and Some Questions of Operator Theory. Dynamical Systems in Applications, 24(3), 305-311.
  • [18] Savaş, E., Gürdal M. 2015. T-Statistical Convergence in Probabilistic Normed Spaces. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 77(4), 195-204.
  • [19] Yamancı, U., Gürdal, M. 2016. Statistical Convergence and Operators on Fock Space. New York Journal of Mathematics, 22, 199-207.
  • [20] Nabiev, A.A., Savaş, E., Gürdal, M. 2019. Statistically Localized Sequences in Metric Spaces. Journal of Applied Analysis and Computation, 9(2), 739-746.
  • [21] Nabiev, A.A., Savaş, E., Gürdal, M. 2020. T-Localized Sequences in Metric Spaces. Facta Universitatis, Series: Mathematics and Informatics, 35(2), 459-469.
  • [22] Huban M.B, Gürdal M. 2021. Wijsman Lacunary Invariant Statistical Convergence for Triple Sequences via Orlicz Function. Journal of Classical Analysis, 17(2), 119-128.
  • [23] Yapalı, R., Gürdal, U. 2021. Pringsheim and Statistical Convergence for Double Sequences on L−Fuzzy Normed Space. AIMS Mathematics, 6(12), 13726-13733.
  • [24] Yapalı, R., Çoşkun, H., Gürdal, U. 2023. Statistical Convergence on L−Fuzzy Normed Space. Filomat, 37(7), 2077-2085.
  • [25] Yapalı, R., Korkmaz, E., Çınar, M., Çoskun, H. 2024. Lacunary Statistical Convergence on L-Fuzzy Normed Space. Journal of Intelligent & Fuzzy Systems, 46(1), 1985-1993.
  • [26] Çetin, S., Kişi, Ö., Gürdal, M. 2025. Exploration of Novel Convergence Concepts for Sequences in Octonion-Valued Metric Spaces. Advances in Mathematical Sciences and Applications, 34(1), 271-300.
  • [27] Marouf, M. 1993. Asymptotic Equivalence and Summability. International Journal of Mathematics and Mathematical Sciences, 16(2), 755–762.
  • [28] Patterson, R. F. 2003. On Asymptotically Statistically Equivalent Sequences. Demonstratio Mathematica, 36, 149-153.
  • [29] Patterson, R. F., Savaş, E. 2006. On Asymptotically Lacunary Statistically Equivalent Sequences. Thai Journal of Mathematics, 4, 267-272.
  • [30] Savaş, E. 2013. On T-Asymptotically Lacunary Statistical Equivalent Sequences. Advances in Difference Equations, 2013, Article ID 111. doi:10.1186/1687–1847–2013–111.
  • [31] Et, M., Çolak, R. 1995. On Generalized Difference Sequence Spaces. Soochow Journal of Mathematics, 21(4), 377-386.
  • [32] Et, M., Altınok, H., Altın, Y. 2004. On Some Generalized Sequence Spaces. Applied Mathematics and Computation, 154(1), 167-173.
  • [33] Et, M., Alotaibi, A., Mohiuddine, S. A. 2014. On (∆^m,T)-Statistical Convergence of order α. The Scientific World Journal, 2014, Article ID 535419, 1-5.
  • [34] Et, M., Şengül Kandemir, H., Çınar, M. 2022. On Asymptotically Lacunary Statistical Equivalent of order α ˜ of Difference Double Sequences. Mathematical Methods in the Applied Sciences, 45(18), 12023–12029.
  • [35] Repsold, J. A. 1919. Friedrich Wilhelm Bessel. Astronomische Nachrichten, 210(11), 161-216.
  • [36] Ibrahim, I. S., Yousif, M. A., Mohammed, P. O., Baleanu, D., Zeeshan, A., Abdelwahed, M. 2024. Bessel Statistical Convergence: New Concepts and Applications in Sequence Theory, PLoS ONE, 19(11), e0313273.
  • [37] Tripathy, B .C., Sarma, B. 2006. Statistically Convergent Double Sequence Spaces Defined by Orlicz Functions. Soochow Journal of Mathematics, 32(2), 211-221.
  • [38] Tripathy, B .C., Sarma, B. 2008. Statistically Convergent Difference Double Sequence Spaces. Acta Mathematica Sinica, English Series, 24(5), 737-742.

ρ ˜ Katlı Bessel I_2-Δ_θ^n-Asimptotik İstatistiksel Denk Çift Dizilerin Bazı Özellikleri

Year 2025, Volume: 29 Issue: 1, 35 - 43, 25.04.2025

Ömer Kişi , Rümeysa Akbıyık

https://doi.org/10.19113/sdufenbed.1625365

Abstract

Dizi uzayları ve toplanabilirlik teorisi çerçevesinde, klasik yakınsama ve sınırlılık kavramlarının teorik temelleri üzerine önemli ilerlemeler kaydedilmiş olmakla birlikte, yenilikçi yakınsama yöntemlerinin araştırılması hala kritik bir alan olarak öne çıkmaktadır. Literatürde, özellikle Bessel fonksiyonlarının bu teorik yapılarla entegrasyonu yönünde belirgin bir eksiklik bulunmaktadır. Bu çalışma, çift diziler için ρ̃ katlı Bessel I_2-Δ_θ^n-asimptotik istatistiksel denklik ve güçlü Bessel I_2-Δ_θ^n-asimptotik denklik kavramlarını tanıtarak, bu boşluğu doldurmayı ve söz konusu kavramlar arasındaki kapsama ilişkilerini kapsamlı bir biçimde incelemeyi amaçlamaktadır. Bessel fonksiyonlarının bu bağlamda teorilere dahil edilmesi, yalnızca matematiksel analiz alanındaki mevcut teorik çerçeveleri genişletmekle kalmayıp, aynı zamanda alanın ileri düzeydeki araştırmalarına ve uygulamalı problemlerin çözümüne yönelik sağlam bir temel teşkil etmektedir. Bu yenilikçi kavramlar, Bessel fonksiyonlarıyla ilişkili dizi davranışlarının daha derin bir şekilde anlaşılmasına katkıda bulunmakta ve bu anlayışın matematiksel uygulamalarda kullanılabilirliğini artırmaktadır.

Keywords

Asimptotik denklik Bessel istatistiksel yakınsaklık Lacunary dizi

References

  • [1] Fast, H. 1951. Sur la Convergence Statistique. Colloquium Mathematicum, 2, 241-244.
  • [2] Fridy, J. J., Orhan, C. 1993. Lacunary Statistical Convergence. Pacific Journal of Mathematics, 160(1), 43-51.
  • [3] Çakan, C., Altay, B., Çoşkun, H. 2010. Double Lacunary Density and Lacunary Statistical Convergence of Double Sequences. Studia Scientiarum Mathematicarum Hungarica, 47(1), 35-45.
  • [4] Freedman, A. R., Sember, J. J., Raphael, M. 1978. Some Cesàro-type Summability Spaces. Proceedings of the London Mathematical Society, 3(3), 508-520.
  • [5] Kostyrko, P., Salat, T., Wilczynski, W. 2000-2001. T-Convergence. Real Analysis Exchange, 26(2), 669-686.
  • [6] Das, P., Kostyrko, P., Wilczyński, W., Malik, P. 2008. T and T^*-Convergence of Double Sequences. Mathematica Slovaca, 58(5), 605-620.
  • [7] Tripathy, B. C., Hazarika, B., Choudhary, B. 2012. Lacunary T-Convergent Sequences. Kyungpook Mathematical Journal, 52(4), 473-482.
  • [8] Das, P., Savaş, E., Ghosal, S. K. 2011. On Generalizations of Certain Summability Methods Using Ideals. Applied Mathematics Letters, 24, 1509-1514.
  • [9] Belen, C., Yıldırım, M. 2012. On Generalized Statistical Convergence of Double Sequences via Ideals. Annali dell'Universita di Ferrara, 58(1), 11-20.
  • [10] Kumar, S., Kumar, V., Bhatia, S. S. 2013. On Ideal Version of Lacunary Statistical Convergence of Double Sequences. General Mathematics Notes, 17(1), 32-44.
  • [11] Gürdal, M., Şahiner, A., Açık, I. 2009. Approximation Theory in 2-Banach Spaces. Nonlinear Analysis, 71(5-6), 1654-1661.
  • [12] Şahiner, A., Gürdal M., Yiğit, T. 2011. Ideal Convergence Characterization of the Completion of Linear n-Normed Spaces. Computers & Mathematics with Applications, 61(3), 683-689.
  • [13] Gürdal, M., Şahiner, A. 2012. Statistical Approximation with a Sequence of 2-Banach Spaces. Mathematical and Computer Modelling, 55(3-4), 471-479.
  • [14] Yamancı, U., Gürdal, M. 2013. On Lacunary Ideal Convergence in Random n-Normed Space. Journal of Mathematics , 2013(1); 868457, 1-8.
  • [15] Gürdal, M., Huban, M.B. 2014. On T-Convergence of Double Sequences in the Topology Induced by Random 2-Norms. Matematički Vesnik, 66(1), 73-83.
  • [16] Yamancı, U., Gürdal, M. 2014. T-Statistical Convergence in 2-Normed Space. Arab Journal of Mathematical Sciences, 20(1), 41-47.
  • [17] Gürdal, M., Yamancı, U. 2015. Statistical Convergence and Some Questions of Operator Theory. Dynamical Systems in Applications, 24(3), 305-311.
  • [18] Savaş, E., Gürdal M. 2015. T-Statistical Convergence in Probabilistic Normed Spaces. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 77(4), 195-204.
  • [19] Yamancı, U., Gürdal, M. 2016. Statistical Convergence and Operators on Fock Space. New York Journal of Mathematics, 22, 199-207.
  • [20] Nabiev, A.A., Savaş, E., Gürdal, M. 2019. Statistically Localized Sequences in Metric Spaces. Journal of Applied Analysis and Computation, 9(2), 739-746.
  • [21] Nabiev, A.A., Savaş, E., Gürdal, M. 2020. T-Localized Sequences in Metric Spaces. Facta Universitatis, Series: Mathematics and Informatics, 35(2), 459-469.
  • [22] Huban M.B, Gürdal M. 2021. Wijsman Lacunary Invariant Statistical Convergence for Triple Sequences via Orlicz Function. Journal of Classical Analysis, 17(2), 119-128.
  • [23] Yapalı, R., Gürdal, U. 2021. Pringsheim and Statistical Convergence for Double Sequences on L−Fuzzy Normed Space. AIMS Mathematics, 6(12), 13726-13733.
  • [24] Yapalı, R., Çoşkun, H., Gürdal, U. 2023. Statistical Convergence on L−Fuzzy Normed Space. Filomat, 37(7), 2077-2085.
  • [25] Yapalı, R., Korkmaz, E., Çınar, M., Çoskun, H. 2024. Lacunary Statistical Convergence on L-Fuzzy Normed Space. Journal of Intelligent & Fuzzy Systems, 46(1), 1985-1993.
  • [26] Çetin, S., Kişi, Ö., Gürdal, M. 2025. Exploration of Novel Convergence Concepts for Sequences in Octonion-Valued Metric Spaces. Advances in Mathematical Sciences and Applications, 34(1), 271-300.
  • [27] Marouf, M. 1993. Asymptotic Equivalence and Summability. International Journal of Mathematics and Mathematical Sciences, 16(2), 755–762.
  • [28] Patterson, R. F. 2003. On Asymptotically Statistically Equivalent Sequences. Demonstratio Mathematica, 36, 149-153.
  • [29] Patterson, R. F., Savaş, E. 2006. On Asymptotically Lacunary Statistically Equivalent Sequences. Thai Journal of Mathematics, 4, 267-272.
  • [30] Savaş, E. 2013. On T-Asymptotically Lacunary Statistical Equivalent Sequences. Advances in Difference Equations, 2013, Article ID 111. doi:10.1186/1687–1847–2013–111.
  • [31] Et, M., Çolak, R. 1995. On Generalized Difference Sequence Spaces. Soochow Journal of Mathematics, 21(4), 377-386.
  • [32] Et, M., Altınok, H., Altın, Y. 2004. On Some Generalized Sequence Spaces. Applied Mathematics and Computation, 154(1), 167-173.
  • [33] Et, M., Alotaibi, A., Mohiuddine, S. A. 2014. On (∆^m,T)-Statistical Convergence of order α. The Scientific World Journal, 2014, Article ID 535419, 1-5.
  • [34] Et, M., Şengül Kandemir, H., Çınar, M. 2022. On Asymptotically Lacunary Statistical Equivalent of order α ˜ of Difference Double Sequences. Mathematical Methods in the Applied Sciences, 45(18), 12023–12029.
  • [35] Repsold, J. A. 1919. Friedrich Wilhelm Bessel. Astronomische Nachrichten, 210(11), 161-216.
  • [36] Ibrahim, I. S., Yousif, M. A., Mohammed, P. O., Baleanu, D., Zeeshan, A., Abdelwahed, M. 2024. Bessel Statistical Convergence: New Concepts and Applications in Sequence Theory, PLoS ONE, 19(11), e0313273.
  • [37] Tripathy, B .C., Sarma, B. 2006. Statistically Convergent Double Sequence Spaces Defined by Orlicz Functions. Soochow Journal of Mathematics, 32(2), 211-221.
  • [38] Tripathy, B .C., Sarma, B. 2008. Statistically Convergent Difference Double Sequence Spaces. Acta Mathematica Sinica, English Series, 24(5), 737-742.
There are 38 citations in total.

Details

Primary Language English
Subjects Pure Mathematics (Other)
Journal Section Articles
Authors

Ömer Kişi 0000-0001-6844-3092

Rümeysa Akbıyık 0009-0004-8751-0325

Publication Date April 25, 2025
Submission Date January 22, 2025
Acceptance Date February 8, 2025
Published in Issue Year 2025 Volume: 29 Issue: 1

Cite

APA Kişi, Ö., & Akbıyık, R. (2025). Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order ρ ˜. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 29(1), 35-43. https://doi.org/10.19113/sdufenbed.1625365
AMA Kişi Ö, Akbıyık R. Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order ρ ˜. J. Nat. Appl. Sci. April 2025;29(1):35-43. doi:10.19113/sdufenbed.1625365
Chicago Kişi, Ömer, and Rümeysa Akbıyık. “Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order ρ ˜”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 29, no. 1 (April 2025): 35-43. https://doi.org/10.19113/sdufenbed.1625365.
EndNote Kişi Ö, Akbıyık R (April 1, 2025) Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order ρ ˜. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 29 1 35–43.
IEEE Ö. Kişi and R. Akbıyık, “Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order ρ ˜”, J. Nat. Appl. Sci., vol. 29, no. 1, pp. 35–43, 2025, doi: 10.19113/sdufenbed.1625365.
ISNAD Kişi, Ömer - Akbıyık, Rümeysa. “Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order ρ ˜”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 29/1 (April 2025), 35-43. https://doi.org/10.19113/sdufenbed.1625365.
JAMA Kişi Ö, Akbıyık R. Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order ρ ˜. J. Nat. Appl. Sci. 2025;29:35–43.
MLA Kişi, Ömer and Rümeysa Akbıyık. “Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order ρ ˜”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 29, no. 1, 2025, pp. 35-43, doi:10.19113/sdufenbed.1625365.
Vancouver Kişi Ö, Akbıyık R. Some Properties of Bessel I_2-Δ_θ^n-Asymptotically Statistical Equivalent Double Sequence of Order ρ ˜. J. Nat. Appl. Sci. 2025;29(1):35-43.
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