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In this paper, we introduce the binomial sequence spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b^{a,b}_{0}(B^{(m)})$\end{document}b0a,b(B(m)), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b^{a,b}_{c}(B^{(m)})$\end{document}bca,b(B(m)) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b^{a,b}_{\infty}(B^{(m)})$\end{document}b∞a,b(B(m)) by combining the binomial transformation and difference operator. We prove the BK-property and some inclusion relations. Furthermore, we obtain Schauder bases and compute the α-, β- and γ-duals of these sequence spaces. Finally, we characterize matrix transformations on the sequence space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{c}^{a,b}(B^{(m)})$\end{document}bca,b(B(m)).


Introduction and preliminaries
Let w denote the space of all sequences. By p , ∞ , c and c  , we denote the spaces of absolutely p-summable, bounded, convergent and null sequences, respectively, where  ≤ p < ∞. A Banach sequence space Z is called a BK -space [] provided each of the maps p n : Z → C defined by p n (x) = x n is continuous for all n ∈ N, which is of great importance in the characterization of matrix transformations between sequence spaces. One can prove that the sequence spaces ∞ , c and c  are BK -spaces with their usual sup-norm.
Let Z be a sequence space, then Kizmaz [] introduced the following difference sequence spaces: For an infinite matrix A = (a n,k ) and x = (x k ) ∈ w, the A-transform of x is defined by Ax = {(Ax) n } and is supposed to be convergent for all n ∈ N, where (Ax) n = ∞ k= a n,k x k .
For two sequence spaces X and Y and an infinite matrix A = (a n,k ), the sequence space X A is defined by which is called the domain of matrix A in the space X. By (X : Y ), we denote the class of all matrices such that X ⊆ Y A . The Euler means E r of order r is defined by the matrix E r = (e r n,k ), where  < r <  and Altay and Polat [] defined further generalization of the Euler sequence spaces e r  (∇), e r c (∇) and e r ∞ (∇) by Here any term with negative subscript is equal to naught.
Polat and Başar [] employed the matrix domain technique of the triangle limitation method for obtaining the following sequence spaces: for all k, n, m ∈ N. Also, Başarir and Kayikçi [] defined the matrix B (m) = (b (m) n,k ) by which is reduced to the matrix ∇ (m) in the case r = , s = -. Kara for all k, n ∈ N. For ab >  we have n,k = . Thus, the binomial matrix B a,b is regular for ab > . Unless stated otherwise, we assume that ab > . If we take a + b = , we obtain the Euler matrix E r , so the binomial matrix generalizes the Euler matrix. Bişgin defined the following binomial sequence spaces: The purpose of the present paper is to study the binomial difference spaces b a,b  (B (m) ), b a,b c (B (m) ) and b a,b ∞ (B (m) ) whose B a,b (B (m) )-transforms are in the spaces c  , c and ∞ , respectively. These new sequence spaces are the generalization of the sequence spaces defined in [, ] and []. Also, we give some inclusion relations and compute the bases and α-, β-and γ -duals of these sequence spaces.

The binomial difference sequence spaces
In this section, we introduce the spaces b a,b ) and prove the BK -property and inclusion relations.
We first define the binomial difference sequence spaces b a,b By using the notion of (.), the sequence spaces b a,b It is obvious that the sequence spaces b a,b  (B (m) ), b a,b c (B (m) ) and b a,b ∞ (B (m) ) may be reduced to some sequence spaces in the special cases of a, b, s, r and m ∈ N. For instance, if we take a + b = , then we obtain the spaces e r  (B (m) ), e r c (B (m) ) and e r ∞ (B (m) ), defined by Kara and Başarir []. If we take a + b = , r =  and s = -, then we obtain the spaces e r  (∇ (m) ), e r c (∇ (m) ) and e r ∞ (∇ (m) ), defined by Polat and Başar []. Especially, taking r =  and s = -, we obtain the new binomial difference sequence spaces b a,b ∞ (∇ (m) ). Let us define the sequence y = (y n ) as the B a,b (B (m) )-transform of a sequence x = (x k ), that is, can be redefined by all sequences whose B a,b (B (m) )-transforms are in the spaces c  , c and ∞ . Let y = (y n ) ∈ c  and define the sequence x = (x k ) by for each k ∈ N. Then we have The following theorems give some inclusion relations for this class of sequence spaces. We have the well known inclusion c  ⊆ c ⊆ ∞ , then the corresponding extended versions also preserve this inclusion. Proof Similarly, we only prove the inclusion e a  (B (m) ) ⊆ b a,b  (B (m) ). If a + b = , we have E a = B a,b . So e a  (B (m) ) ⊆ b a,b  (B (m) ) holds. Let  < a <  and b = . We define a sequence x = (x k ) by x k = (- a ) k for each k ∈ N. It is clear that ). This shows that the inclusion e a  (B (m) ) ⊆ b a,b  (B (m) ) strictly holds.

The Schauder basis and α-, βand γ -duals
For a normed space (X, · ), a sequence {x k : x k ∈ X} k∈N is called a Schauder basis [] if for every x ∈ X, there is a unique scalar sequence (λ k ) such that x -n k= λ k x k →  as n → ∞. We shall construct Schauder bases for the sequence spaces b a,b  (B (m) ) and b a,b c (B (m) ). We define the sequence Theorem . The sequence (g (k) (a, b)) k∈N is a Schauder basis for the binomial sequence space b a,b  (B (m) ) and every x = (x i ) ∈ b a,b  (B (m) ) has a unique representation by (a, b)) = e k ∈ c  , where e k is the sequence with  in the kth place and zeros elsewhere for each k ∈ N. This implies that g (k) (a, b) ∈ b a,b  (B (m) ) for each k ∈ N.
For x ∈ b a,b  (B (m) ) and n ∈ N, we put a, b).
By the linearity of B a,b (B (m) ), we have For every ε > , there is a positive integer n  such that To show the uniqueness of this representation, we assume that a, b).
which is a contradiction with the assumption that This shows the uniqueness of this representation.
Theorem . Let g = (, , , , . . .) and lim k→∞ λ k (a, b) = l. The set {g, g () (a, b), g () (a, b), . . . , g (k) (a, b), . . .} is a Schauder basis for the space b a,b c (B (m) ) and every x ∈ b a,b c (B (m) ) has a unique representation by a, b). (.) Proof Obviously, B a,b (B (m) g k i (a, b)) = e k ∈ c and g ∈ b a,b c (B (m) ). For x ∈ b a,b c (B (m) ), we put y = xlg and we have y ∈ b a,b  (B (m) ). Hence, we deduce that y has a unique representation by (.), which implies that x has a unique representation by (.). Thus, we complete the proof.

Corollary . The sequence spaces b a,b
 (B (m) ) and b a,b c (B (m) ) are separable.

Köthe and Toeplitz []
first computed the dual whose elements can be represented as sequences and defined the α-dual (or Köthe-Toeplitz dual). Next, we compute the α-,βand γ -duals of the sequence spaces b a,b  (B (m) ), b a,b c (B (m) ) and b a,b ∞ (B (m) ). For the sequence spaces X and Y , define multiplier space M(X, Y ) by Then the α-, β-and γ -duals of a sequence space X are defined by respectively. Let us give the following properties: Proof Let u = (u k ) ∈ w and x = (x k ) be defined by (.), then we have Therefore, we deduce that ux Theorem . We have the following relations: Proof Let u = (u k ) ∈ w and x = (x k ) be defined by (.), then we consider the following equation: domain of the Euler matrix. In order to give full knowledge to the reader on related topics with applications and a possible line of further investigation, the e-book [] is added to the list of references.