Some Newly Defined Sequence Spaces Using Regular Matrix of Fibonacci Numbers

Year 2014, Volume: 14 Issue: 1, 1 - 3, 01.04.2014

Shyamal Debnath Subrata Saha

Abstract

The main purpose of this paper is to introduce the new sequence spaces (F), c(F) and (F) based on the newly defined regular matrix F of Fibonacci numbers. We study some basic topological and algebraic properties of these spaces. Also we investigate the relations related to these spaces. Let w be the space of all real sequences. Any vector subspace of w is called a sequence space. We shall write c, and for the sequence spaces of all convergent, null and bounded sequences. Let X, Y be two sequence spaces and A = ( ) be an infinite matrix of real numbers , where n, k N.Then, A defines a matrix mapping ( Debnath and Debnath, communicated; Malkowsky and Rakocevic, 2007) from X into Y and we denote it by A : X  Y, if for every sequence x= ( )  X, the sequence Ax = { (x)+ , the A-transform of x, is in Y; where (x) = ∑ , (n N)

Keywords

Newly, Defined, Sequence

Some Newly Defined Sequence Spaces Using Regular Matrix of Fibonacci Numbers

Abstract

The main purpose of this paper is to introduce the new sequence spaces (F), c(F) and (F) based on the newly defined regular matrix F of Fibonacci numbers. We study some basic topological and algebraic properties of these spaces. Also we investigate the relations related to these spaces. Let w be the space of all real sequences. Any vector subspace of w is called a sequence space. We shall write c, and for the sequence spaces of all convergent, null and bounded sequences. Let X, Y be two sequence spaces and A = ( ) be an infinite matrix of real numbers , where n, k N.Then, A defines a matrix mapping ( Debnath and Debnath, communicated; Malkowsky and Rakocevic, 2007) from X into Y and we denote it by A : X  Y, if for every sequence x= ( )  X, the sequence Ax = { (x)+ , the A-transform of x, is in Y; where (x) = ∑  , (n N)

Keywords

Fibonacci Number, Regular Matrix, Sequence Space

References

• Altay B., Basar F. and Mursaleen M., 2006. On the Euler sequence spaces which include the spaces and
• , Informations Science, 176, 1450-1462.
• Basar F., 2011. Summability Theory and Its Applications, Bentham Science Publishers, Istanbul.
• Debnath S. and Debnath J., On I-statistically convergent sequence spaces defined by sequences of Orlicz functions (Communicated). matrix transformation
• Kara E. E. and Basarir M., 2012. An application of Fibonacci numbers into infinite Toeplitz matrices, CJMS. 1(1), 43-47.
• Kalman D. and Mena R., June 2003. The Fibonacci numbers-Exposed, Mathematics Magazine. 76(3).
• Koshy T., 2001. Fibonacci and Lucas Numbers with Applications, Wiley.
• Mursaleen M. and Noman A. K., 2010. On the space of -convergent and bounded sequences, Thai J. Math. 8(2), 311-329.
• Malkowsky E. and Rakocevic V., 2007. On matrix domains of triangles, Appl. Math.Comput., 189(2), 1146-1163
• Tripathy B. C. and Sen M., 2002. On a new class of sequences related to the space , Tamkang J. Math. 33(2), 167-171.
• Vajda S., 1989. Fibonacci and Lucas Numbers, and Golden Section: Theory and Applications, Chichester: Ellis Horword.
• Wilansky A., 1984. Summability through functional analysis, North-Holland mathematics Studies 85, Elsevier Science Publishers, Amsterdam: New York: Oxford.