# zbMATH — the first resource for mathematics

On the classes of fractional order difference sequence spaces and their matrix transformations. (English) Zbl 1328.46002
Summary: The main purpose of the present article is to introduce the classes of generalized fractional order difference sequence spaces $$\ell_\infty(\operatorname{\Gamma}, \operatorname{\Delta}^{\widetilde{\alpha}}, p), c_0(\operatorname{\Gamma}, \operatorname{\Delta}^{\widetilde{\alpha}}, p)$$ and $$c(\operatorname{\Gamma}, \operatorname{\Delta}^{\widetilde{\alpha}}, p)$$ by defining the fractional difference operator $$\operatorname{\Delta}^{\widetilde{\alpha}} x_k = \sum_{i = 0}^\infty(- 1)^i \frac{\operatorname{\Gamma}(\widetilde{\alpha} + 1)}{i! \operatorname{\Gamma}(\widetilde{\alpha} - i + 1)} x_{k + i}$$, where $$\widetilde{\alpha}$$ is a positive proper fraction and $$k \in \mathbb{N} = \{1, 2, 3 \ldots . \}$$. Results concerning the linearity and various topological properties of these spaces are established and also the alpha-, beta-, gamma- and $$N$$-duals of these spaces are obtained. The matrix transformations from these classes into Maddox spaces are also characterized. Throughout the article we use the notation $$\operatorname{\Gamma}(n)$$ as the Gamma function of $$n$$, defined by an improper integral $$\operatorname{\Gamma}(n) = \int_0^\infty e^{- t} t^{n - 1} {dt}$$, where $$n \notin \{0, - 1, - 2, \ldots \}$$ and $$\operatorname{\Gamma}(n + 1) = n \operatorname{\Gamma}(n)$$.

##### MSC:
 46A45 Sequence spaces (including Köthe sequence spaces) 40A05 Convergence and divergence of series and sequences
Full Text:
##### References:
 [1] Kızmaz, H., On certain sequence spaces, Can. Math. Bull., 24, 2, 169-176, (1981) · Zbl 0454.46010 [2] Et, M.; Çolak, R., On some generalized difference sequence spaces, Soochow J. Math., 21, 377-386, (1995) · Zbl 0841.46006 [3] 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 [4] Bektas, C. A.; Et, M., The dual spaces of the sets of difference sequences of order m, J. Inequal. Pure Appl. Math., 7, 3, (2006), (Art. 101) · Zbl 1132.46004 [5] Et, M., On some topological properties of generalized difference sequence spaces, Int. J. Math. Sci., 24, 785-791, (2000) · Zbl 0966.40002 [6] Et, M.; Nuray, F., $$\operatorname{\Delta}^m$$-statistical convergence, Indian J. Pure Appl. Math., 32, 961-969, (2001) · Zbl 1028.46033 [7] Ahmad, Z. U.; Mursaleen, M., Köthe-Toeplitz duals of some new sequence spaces, Publ. Inst. Math. (Belgrade), 42, 57-61, (1987) · Zbl 0647.46006 [8] Et, M.; Basarir, M., On some new generalized difference sequence spaces, Period. Math. Hung., 35, 3, 169-175, (1997) · Zbl 0922.40003 [9] Altay, B.; Başar, F.; Mursaleen, M., On the Euler sequence spaces which includes the spaces $$\ell_p$$ and $$\ell_\infty$$, Inform. Sci., 176, 10, 1450-1462, (2006) · Zbl 1101.46015 [10] Altay, B.; Başar, F., Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, J. Math. Anal. Appl., 336, 1, 632-645, (2007) · Zbl 1152.46003 [11] Dutta, S.; Baliarsingh, P., On certain new difference sequence spaces generated by infinite matrices, Thai J. Math., 11, 1, 75-86, (2013) · Zbl 1287.46005 [12] 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 [13] Srivastava, P. D.; Kumar, S., On the fine spectrum of the generalized difference operator $$\operatorname{\Delta}_\nu$$ over the sequence space $$c_0$$, Commun. Math. Anal., 6, 1, 8-21, (2009) · Zbl 1173.47022 [14] Srivastava, P. D.; Kumar, S., Fine spectrum of the generalized difference operator $$\operatorname{\Delta}_\nu$$ on sequence space $$\ell_1$$, Thai J. Math., 8, 2, 221-233, (2010) · Zbl 1236.47029 [15] 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 [16] Altay, B.; Başar, F., Generalization of sequence space $$\ell_p$$ derived by weighted mean, J. Math. Anal. Appl., 330, 1, 174-185, (2007) · Zbl 1116.46003 [17] Başar, F.; Altay, B., On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J., 55, 1, 136-147, (2003) · Zbl 1040.46022 [18] Başar, F., Summability theory and its applications, (2012), Bentham Science Publishers Ístanbul, (e-books, Monographs) [19] Basarir, M.; Et, M., On some new generalized difference sequence spaces, Period. Math. Hung., 35, 3, 169-175, (1997) · Zbl 0922.40003 [20] Malkowsky, E., Absolute and ordinary Köthe-Toeplitz duals of certain sequence spaces, Publ. Inst. Math. (Belgrade), 46, 60, 97-104, (1989) [21] Malkowsky, E.; Parashar, S. D., Matrix transformations in spaces of bounded and convergent sequences of order m, Analysis, 17, 87-97, (1997) · Zbl 0872.40002 [22] Malkowsky, E.; Mursaleen, M.; Suantai, S., The dual spaces of sets of difference sequences of order m and matrix transformations, Acta Math. Sin. (Engl. Ser.), 23, 3, 521-532, (2007) · Zbl 1123.46007 [23] Mursaleen, M., Generalized spaces of difference sequences, J. Math. Anal. Appl., 203, 738-745, (1996) · Zbl 0873.46014 [24] Mursaleen, M.; Noman, A. K., On some new difference sequence spaces of non-absolute type, Math. Comput. Modell., 52, 603-617, (2010) · Zbl 1201.40003 [25] Kamthan, P. K.; Gupta, M., Sequence spaces and series, (1981), Marcel Dekker Incl. New York, Basel · Zbl 0447.46002 [26] Maddox, I. J., Elements of functional analysis, (1978), Cambridge Univ Press · Zbl 0193.08601 [27] Dutta, S.; Baliarsingh, P., On the spectrum of 2nd order generalized difference operator $$\operatorname{\Delta}^2$$ over the sequence space $$c_0$$, Bol. Soc. Paran. Mat., 31, 2, 235-244, (2013) · Zbl 1413.47013 [28] Baliarsingh, P., Some new difference sequence spaces of fractional order and their dual spaces, Appl. Math. Comput., 219, 18, 9737-9742, (2013) · Zbl 1300.46004 [29] Lascarides, C. G.; Maddox, I. J., Matrix transformations between some classes of sequences, Proc. Cambridge Philos. Soc., 68, 99-104, (1970) · Zbl 0193.41102 [30] Dutta, S.; Baliarsingh, P., On the class of new difference sequence spaces, J. Indian Math. Soc., 80, 3-4, 203-211, (2013) · Zbl 1300.46005 [31] Grosse-Erdmann, K. G., Matrix transformations between the sequence spaces of maddox, J. Math. Anal. Appl., 180, 223-238, (1993), (Second edition) · Zbl 0791.47029
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.