Maltepe Journal of Mathematics 2667-7660 Hüseyin ÇAKALLI 10.47087/mjm.1741148 Applied Mathematics (Other) Uygulamalı Matematik (Diğer) The Triple Difference Operator of Binomial Poisson Matrix of Fractional Priya C. Bishop Heber College https://orcid.org/0000-0002-5895-673X Subramanian Nagarajan SASTRA Esi Ayhan TURGUT OZAL UNIVERSITY https://orcid.org/0000-0001-6798-1868 Özdemir Mustafa Kemal İNÖNÜ ÜNİVERSİTESİ 04 28 2026 8 1 18 29 07 24 2025 01 19 2026 Copyright © 2019, Maltepe Journal of Mathematics 2019 Maltepe Journal of Mathematics

In this article we introduce binomial Poisson matrix difference sequence spaces of fractional order $\alpha$, $b_{\Gamma^{3}}^{rs}$ and $b_{\Lambda ^{3}}^{rs}$ by employing fractional difference operator $\Delta^{\alpha}$ defined by \[ \Delta^{\alpha} x_{mnk} = \sum_{u=0}^{\infty} \sum_{v=0}^{\infty} \sum_{w=0}^{\infty} (-1)^{u+v+w} \frac{ \Gamma(\alpha+1)\Gamma(\beta+1)\Gamma(\gamma+1) }{ u!\,v!\,w!\, \Gamma(\alpha-u+1) \Gamma(\beta-v+1) \Gamma(\gamma-w+1) } x_{m-u,n-v,k-w}. \] We give some topological properties, obtain the Schauder basis, and discuss various duals. Finally, we present a graphical interpretation of the operator $B^{rs}(\Delta^{\alpha})$.

 Binomial poisson difference sequence space difference operator $\Delta^{\alpha}$ Schauder basis D. Barlak, Statistical convergence of order β for (λ; μ) double sequences of fuzzy numbers, J. Intell. Fuzzy Syst. 39 (2020), 6949–6954. A. Esi and N. Subramanian, On triple sequence spaces of Bernstein operator of χ3 rough λ-statistical convergence in probability of random variables defined by Musielak–Orlicz func- tion p-metric, Electron. J. Math. Anal. Appl. 6 (2018), no. 1, 198–203. A. Esi, N. Subramanian and M. K. Ozdemir, Chlodowsky type (λ,q)-Bernstein–Stancu operator of rough fuzzy Borel summability of triple sequences, Int. J. Open Probl. Comput. Sci. Math. 15 (2022), no. 1, 1–19. B. Hazarika, N. Subramanian and A. Esi, On rough weighted ideal convergence of triple sequence of Bernstein polynomials, Proc. Jangjeon Math. Soc. 21 (2018), no. 3, 497–506. C. Priya, N. Saivaraju and N. Subramanian, The ideal convergent sequence spaces over np-metric spaces defined by sequence of modulus, Far East J. Math. Sci. (FJMS) 92 (2014), no. 2, 173–203. C. Priya, N. Saivaraju and N. Subramanian, The Fibonacci numbers of χ2 over p-metric spaces defined by sequence of modulus, Far East J. Math. Sci. (FJMS) 93 (2014), no. 1, 1–21. C. Priya, N. Saivaraju and N. Subramanian, The Ces`aro lacunary ideal double sequence χ2 of φ-statistical convergence defined by a Musielak–Orlicz function, Appl. Math. Inf. Sci. 10 (2016), no. 4, 1585–1591. N. Subramanian and A. Esi, Wijsman rough convergence of triple sequences, Matematychni Studii 48 (2017), no. 2, 171–179. N. Subramanian, A. Esi and V. A. Khan, The rough intuitionistic fuzzy Zweier lacunary ideal convergence of triple sequence spaces, J. Math. Stat. 14 (2018), no. 1, 72–78. N. Subramanian, A. Esi and M. K. Ozdemir, Rough statistical convergence on triple sequence of Bernstein operator of random variables in probability, Songklanakarin J. Sci. Technol. 41 (2019), no. 3, 567–579. N. Subramanian, C. Priya and N. Saivaraju, Randomness of lacunary statistical convergence of χ2 over p-metric spaces defined by sequence of modulus, Far East J. Math. Sci. (FJMS) 94 (2014), no. 1, 89–111. N. Subramanian, C. Priya and N. Saivaraju, The χ2 sequence space over p-metric spaces defined by Musielak modulus, Songklanakarin J. Sci. Technol. 36 (2014), no. 5, 591–598. N. Subramanian, C. Priya and N. Saivaraju, The R χ2I of real numbers over Musielak p- metric space, Southeast Asian Bull. Math. 39 (2015), 133–148. N. Subramanian, N. Saivaraju and C. Priya, The ideal of χ2 of fuzzy real numbers over fuzzy p-metric spaces defined by Musielak, J. Math. Anal. 6 (2015), no. 1, 1–12.