Difference Sequence Spaces Derived by using Pascal Transform

The essential goal of this manuscript is to investigate some novel sequence spaces of $p_{\infty }\left( \Delta \right) $, $p_{c}\left( \Delta \right) $ and $p_{0}\left( \Delta \right) $ which are comprised by all sequence spaces whose differences are in Pascal sequence spaces $p_{\infty }$, $p_{c}$ and $p_{0}$, respectively. Furthermore, we determine both $\gamma $-, $\beta $-, $\alpha $- duals of newly defined difference sequence spaces of $p_{\infty }\left( \Delta \right) $, $% p_{c}\left( \Delta \right) $ and $p_{0}\left( \Delta \right) $. We also obtain bases of the newly defined difference sequence spaces of $p_{c}\left( \Delta \right) $ and $p_{0}\left( \Delta \right) $. Finally, necessary and sufficient conditions on an infinite matrix belonging to the classes $(p_{c}\left( \Delta \right) :l_{\infty })$ and $(p_{c}\left( \Delta \right) :c)$ are characterized.

Keywords:

Pascal difference sequence spaces, Difference operator matrix mappings, Pascal difference sequence spaces,

PDF

___

[1] B. Choudhary and S. Nanda, Functional Analysis with Applications, Wiley, New Delhi (1989).
[2] W.H. Ruckle, Sequence spaces Pitman Publishing, Toronto (1981).
[3] B. Altay and H. Polat, On some new Euler difference sequence spaces, Southeast Asian Bull. Math. 30 (2006), 209-220.
[4] R. Brawer, Potenzen der Pascal matrix und eine Identitat der Kombinatorik, Elem. der Math. 45 (1990), 107-110.
[5] A. Edelman and G. Strang, Pascal Matrices, The Mathematical Association of America, Monthly 111 (2004), 189-197.
[6] C. Lay David, Linear Algebra and Its Applications: 4th Ed. Boston: Pearson, Addison-Wesley, (2012).
[7] G. H. Lawden, Pascal matrices, Mathematical Gazette, 56 (398) (1972), 325--327.
[8] H. Polat, Some New Pascal Sequence Spaces, Fundamental Journal of Mathematics and Applications 1 (2018), 61-68.
[9] H. Kızmaz, On certain sequence space, Canad. Math. Bull. 24(2) (1981), 169 176.
[10] R. Çolak and M. Et, On some generalized difference sequence spaces and related matrix transformations, Hokkaido Math J, 26(3), (1997), 483-492.
[11] B. Altay and F. Başar, The fine spectrum and the matrix domain of the difference operator on the sequence space l_{p}; (0
[12] V. Karakaya and H. Polat, Some New Paranormed Sequence Spaces de ned by Euler and Difference Operators, Acta Sci. Math(Szeged), 76(2010), 87-100.
[13] H. Polat and F. Başar, Some Euler spaces of difference sequences of order m, Acta Math. Sci. Ser. B Engl. Ed. 27B(2) (2007), 254- 266.
[14] H.Polat, V.Karakaya and N. Şimşek, Difference Sequence Spaces Derived by Generalized Weighted Mean, Applied Mathematics Letters 24 (2011), 608-614.
[15] M. Et and M. Başarır, On some genaralized difference sequence spaces, Period. Math. Hung. 35 (3) (1997), 169- 175.
[16] I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, Cambridge (1988).
[17] D. J.H. Garling, The α-, β- and γ-duality of Sequence Spaces, Proc. Comb. Phil. Soc. 63 (1967), 963-981.
[18] M. Stieglitz and H. Tietz, Matrixtransformationen von Folgenraumen Eine Ergebnisubersict, Math. Z., 154(1977), 1-16.