## Some new difference sequence spaces of fractional order and their dual spaces.(English)Zbl 1300.46004

Summary: The main objective of the present paper is to introduce certain difference sequence spaces of fractional order and investigate their topological structures as well as some interesting results concerning the fractional difference operator $$\Delta^\alpha$$. In this paper, we define some new difference sequence spaces such as $$\ell_\infty(\Gamma, \Delta^\alpha, u)$$, $$c_0(\Gamma, \Delta^\alpha, u)$$ and $$c(\Gamma, \Delta^\alpha, u)$$ by introducing a fractional difference operator $$\Delta^\alpha$$ and, for a positive fraction $$\alpha$$, the difference operator $$\Delta^\alpha$$ is defined by $$\Delta^\alpha(\chi_k) = \sum_{i=0}^\infty(-1)^i \frac{\Gamma(\alpha + 1)}{i!\Gamma(\alpha - i + 1)}\chi_{k+i}$$. Also, we establish their $$\alpha$$-, $$\beta$$- and $$\gamma$$-duals.

### MSC:

 46A45 Sequence spaces (including Köthe sequence spaces)
Full Text:

### References:

 [1] Kızmaz, H., On certain sequence spaces, Canad. Math. Bull., 24, 2, 169-176, (1981) · Zbl 0454.46010 [2] Srivastava, P. D.; Nanda, S.; Dutta, S., On certain paranormed function spaces, Rend. Mat., 3, 3, 413-425, (1983) · Zbl 0563.46018 [3] Dutta, S.; Baliarsingh, P., On the fine spectra of the generalized rth difference operator $$\operatorname{\Delta}_\nu^r$$ on the sequence space $$\ell_1$$, Appl. Math. Comput., 219, 1776-1784, (2012) · Zbl 1311.47045 [4] Et, M.; Çolak, R., On some generalized difference sequence spaces, Soochow J. Math., 21, 4, 377-386, (1995) · Zbl 0841.46006 [5] Mursaleen, M., Generalized spaces of difference sequences, J. Math. Anal. Appl., 203, 3, 738-745, (1996) · Zbl 0873.46014 [6] Et, M.; Basarir, M., On some new generalized difference sequence spaces, Periodica Math. Hungar., 35, 3, 169-175, (1997) · Zbl 0922.40003 [7] Et, M., On some topological properties of generalized difference sequence spaces, Int. J. Math. Math. Sci., 24, 11, 785-791, (2000) · Zbl 0966.40002 [8] Et, M.; Altin, Y.; Altinok, H., On some generalized difference sequence spaces defined by a modulus function, Filomat, 17, 23-33, (2003) · Zbl 1050.40001 [9] Et, M.; Altinok, H.; Altin, Y., On some generalized sequence spaces, Appl. Math. Comput., 154, 1, 167-173, (2004) · Zbl 1056.46007 [10] Altay, B.; Başar, F., Some paranormed sequence spaces of non absolute type derived by weighted mean, J. Math. Anal. Appl., 319, 2, 494-508, (2006) · Zbl 1105.46005 [11] Subramanian, N., The difference sequence space defined on Orlicz functions, Int. J. Math. Anal., 2, 13-16, 721-729, (2008) · Zbl 1175.46005 [12] Tripathy, B. C.; Sarma, B., Some classes of difference paranormed sequence spaces defined by Orlicz functions, Thai. J. Math., 3, 2, 209-218, (2005) · Zbl 1154.46300 [13] Tripathy, B. C.; Altin, Y.; Et, M., Generalized difference sequence spaces defined by Orlicz functions, Math. Slovaca, 58, 3, 315-324, (2008) · Zbl 1174.40003 [14] Aydın, C.; Başar, F., Some new paranormed sequence spaces, Inform. Sci., 160, 1-4, 27-40, (2004) · Zbl 1049.46002 [15] Aydın, C.; Başar, F., Some new difference sequence spaces, Appl. Math. Comput., 157, 3, 677-693, (2004) · Zbl 1072.46007 [16] Bektas, C. A.; Et, M.; Çolak, R., Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl., 292, 423-432, (2004) · Zbl 1056.46004 [17] Malkowsky, E.; Mursaleen, M.; Suantai, S., The dual spaces of sets of difference sequence spaces of order m and martix transformations, Acta. Math. Sin. (Engl. Ser.), 23, 3, 521-532, (2007) · Zbl 1123.46007 [18] Tripathy, B. C., On some class of difference paranormed sequence spaces associated with multiplier sequences, Int. J. Math. Sci., 2, 1, 159-166, (2003) · Zbl 1064.40002 [19] Dutta, S.; Baliarsingh, P., On certain new difference sequence spaces generated by infinite matrices, Thai J. Math., 11, 1, (2013), (in press) · Zbl 1287.46005 [20] Çolak, R.; Et, M., On some generalized difference sequence spaces and related matrix transformations, Hokkaido Math. J., 26, 3, 483-492, (1997) · Zbl 0888.40002 [21] Mursaleen, M.; Noman, A. K., On some new difference sequence spaces of non-absolute type, Math. Comput. Model., 52, 603-617, (2010) · Zbl 1201.40003 [22] Stieglitz, M.; Tietz, H., Matrix transformationen von folgenraumen eine ergebnisubersict, Math. Z., 154, 1-16, (1977) · Zbl 0331.40005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.