Some Pascal Spaces of Difference Sequences Spaces of Order m

Yıl 2019, Cilt: 2 Sayı: 1, 97 - 103, 30.10.2019

Saadettin Aydın , Harun Polat

Öz

The main purpose of this article is to introduce new sequence spaces $p_{\infty }\left( \Delta ^{(m)}\right) $, $p_{c}\left( \Delta ^{(m)}\right) $ and $p_{0}\left( \Delta ^{(m)}\right) $ which are consisted by sequences whose $m^{th}$ order differences are in the Pascal sequence spaces $p_{\infty }$, $p_{c}$ and $p_{0}$, respectively. Furthermore, the bases of the new difference sequence spaces $p_{c}\left( \Delta ^{(m)}\right) $ and $p_{0}\left( \Delta ^{(m)}\right) ,$ and the $% \alpha $-, $\beta $-$\ $and $\gamma $-duals of the new difference sequence spaces $p_{\infty }\left( \Delta ^{(m)}\right) $, $p_{c}\left( \Delta ^{(m)}\right) $ and $p_{0}\left( \Delta ^{(m)}\right) \ $have been determined. Finally, necessary and sufficient conditions on an infinite matrix belonging to the classes $(p_{c}\left( \Delta ^{(m)}\right) :l_{\infty })$ and $(p_{c}\left( \Delta ^{(m)}\right) :c)$ are obtained.

Anahtar Kelimeler

Pascal difference sequence spaces, difference operator of order m

Kaynakça

• [1] B. Choudhary, S. Nanda, Functional Analysis with Applications, Wiley, New Delhi, 1989.
• [2] W.H. Ruckle, Sequence spaces, Pitman Publishing, Toronto, 1981.
• [3] H. Kızmaz, On certain sequence space, Canad. Math. Bull., 24 (2) (1981), 169-176.
• [4] R. Brawer, Potenzen der Pascal matrix und eine Identitat der Kombinatorik, Elem. der Math., 45 (1990), 107-110.
• [5] A. Edelman, 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] M. Et, R. Çolak, On some genaralized difference sequence spaces, Soochow J. Math., 21 (4) (1995), 377-386.
• [10] M. Mursaleen, Generalized spaces of difference sequences. J. Math. Anal. Appl., 203 (1996), 738-745.
• [11] E. Malkowsky, S. D. Parashar, Matrix transformations in spaces of bounded and convergent difference sequences of order m, Analysis, 17 (1997), 87-97.
• [12] R. Çolak, M. Et, On some generalized difference sequence spaces and related matrix transformations, Hokkaido Math. J., 26 (3) (1997), 483-492.
• [13] M. Et, M. Ba¸sarır, On some genaralized difference sequence spaces, Period. Math. Hung., 35 (3) (1997), 169-175.
• [14] B. Altay, H. Polat, On some new Euler difference sequence spaces, Southeast Asian Bull. Math., 30 (2006), 209-220.
• [15] B. Altay, F. Ba¸sar, The fine spectrum and the matrix domain of the difference operator on the sequence space lp; (0 < p < 1), Commun. Math. Anal., 2 (2) (2007), 1-11.
• [16] E. Malkowsky, M. Mursaleen, The Dual Spaces of Sets of Difference Sequences of Order m and Matrix Transformations, Acta Mathematica Sinica, 23 (3), (2007), 521-532.
• [17] H. Polat, F. Ba¸sar, Some Euler spaces of difference sequences of order m, Acta Math. Sci. Ser. B Engl. Ed., 27B (2) (2007), 254-266.
• [18] V. Karakaya, H. Polat, Some New Paranormed Sequence Spaces defined by Euler and Difference Operators, Acta Sci. Math(Szeged), 76 (2010), 87-100.
• [19] M. Mursaleen, A. K. Noman, On some new difference sequence spaces of non-absolute type, Math. Comput. Modelling, 52, 3-4 (2010), 603-617.
• [20] H. Polat, V.Karakaya, N. ¸Sim¸sek, Difference Sequence Spaces Derived by Generalized Weighted Mean, Applied Mathematics Letters, 24 (2011), 608-614.
• [21] M. Stieglitz, H. Tietz, Matrix transformationen von Folgenraumen Eine Ergebnisübersict, Math. Z., 154 (1977), 1-16.