Binomial difference sequence spaces of fractional order

In this paper, we introduce the sequence spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b^{r,s}_{0}( \nabla^{(\alpha)})$\end{document}b0r,s(∇(α)), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b^{r,s}_{c}(\nabla^{(\alpha)})$\end{document}bcr,s(∇(α)), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b^{r,s}_{\infty }(\nabla^{(\alpha)})$\end{document}b∞r,s(∇(α)). We investigate some functional properties, inclusion relations, and the α-, β-, γ-, and continuous duals of these sets.


Introduction
Let w, p , ∞ , c, and c 0 denote the spaces of all, p-absolutely summable, bounded, convergent, and null sequences x = (x k ) with complex terms x k , respectively, where 1 ≤ p < ∞ and k ∈ N = {0, 1, 2, . . .}. A sequence space X is called a BK -space if it is a Banach space with continuous coordinates p k : X → C defined by p k (x) = x k for x = (x k ) ∈ X and k ∈ N. The most important result of the theory of BK -spaces is that matrix mappings between BK -spaces are continuous [13]. The sequence spaces ∞ , c, and c 0 with their sup-norm are BK -spaces.
The concept of a difference sequence space was firstly introduced by Kizmaz [22] by defining the set Z( ) = {x = (x k ) : ( x k ) ∈ Z} for Z ∈ { ∞ , c, c 0 }, where x k = x kx k+1 for k ∈ N. The idea of a difference sequence was generalized by Et and Çolak [14][15][16] by defining the spaces For a positive proper fraction α, Baliarsingh and Dutta [4,5] defined the fractional difference operator α by x k+i for k ∈ N, where the Euler gamma function (p) of a real number p with p / ∈ {0, -1, -2, -3, . . .} can be expressed as an improper integral (p) = ∞ 0 e -t t p-1 dt. It is observed that (i) (p + 1) = p! for p ∈ N, (ii) (p + 1) = p (p) for p ∈ R \ {0, -1, -2, -3, . . .}. Some definitions of fractional derivatives have been generalized by using a set of new difference sequence spaces of fractional order [3]. Application of fractional derivatives becomes more apparent in diffusion processes, modeling mechanical systems, and many other fields.
Let X, Y be two sequence spaces, and let A = (a n,k ) be an infinite matrix with complex numbers a n,k , n, k ∈ N. Let the sequence space X A defined by k=0 a n,k x k , n ∈ N. Let (X : Y ) denote the class of all matrices such that X ⊆ Y A . The matrix domain approach has been employed by Başarir and Kara [6][7][8][9][10], Kara and İlkhan [19][20][21], Polat and Başar [26], Song and Meng [23][24][25]27], and many others to introduce new sequence spaces.

Difference sequence spaces of fractional order
In this chapter, we introduce the binomial difference sequence spaces b r,s 0 (∇ (α) ), b r,s c (∇ (α) ), and b r,s ∞ (∇ (α) ) of fractional order and investigate some functional properties and inclusion relations.

Theorem 2.2 The inclusion b r,s
Proof Proof follows from Lemma 2.3 of Et and Nuray [17].
Proof We only give the proof of the inclusion e r 0 (∇ (α) ) ⊆ b r,s 0 (∇ (α) ). The others can be proved similarly.
Proof We prove the theorem only for the space b r,s 0 (∇ (α) ). To prove b r,s 0 (∇ (α) ) ∼ = c 0 , we will show the existence of a linear bijection between the spaces b r,s 0 (∇ (α) ) and c 0 . Let us denote the transformation T : b r,s 0 (∇ (α) ) → c 0 by T(x) = B r,s (∇ (α) )(x k ). The linearity of T is clear, and x = 0 whenever T(x) = 0. Hence T is injective.
Let y = (y n ) ∈ c 0 and define the sequence x = (x k ) by for k ∈ N. Then we have which implies that x ∈ b r,s 0 (∇ (α) ). Therefore, we obtain that T is surjective and norm preserving. This completes the proof.
We shall construct the Schauder bases for the sequence spaces b r,s 0 (∇ (α) ) and b r,s c (∇ (α) ). Because the isomorphism T between b r,s 0 (∇ (α) ) and c 0 (or between b r,s c (∇ (α) ) and c) is onto, the inverse image of the basis of the space c 0 (or c) is the basis of the space b r,s 0 (∇ (α) ) (or b r,s c (∇ (α) )). For k ∈ N, define the sequence g (k) (r, s) = {g (k) i (r, s)} i∈N by Theorem 2.5 The sequence (g (k) (r, s)) k∈N is the Schauder basis for the space b r,s 0 (∇ (α) ), and every x in b r,s 0 (∇ (α) ) has a unique representation by

The α-, β-, γ -, and continuous duals
In this section, we determine the α-, β-, γ -, and continuous duals of the spaces b r,s 0 (∇ (α) ), b r,s c (∇ (α) ), and b r,s ∞ (∇ (α) ). For two sequence spaces X and Y , the set M(X, Y ) is defined by Let bs and cs denote the sequence spaces of all bounded and convergent series, respectively. In particular, cs), and X γ = M(X, bs) are called the α-, β-, and γ -duals of the sequence space X, respectively. The space of all bounded linear functionals on X denoted by X * is called the continuous dual of the space X. Let us give the following properties needed in Lemma 3.1: sup K∈ n k∈K a n,k < ∞,