Let w denote the space of all sequences. By ℓ_p, ℓ_∞, c and c₀, we denote the spaces of p-absolutely summable, bounded, convergent and null sequences respectively. Let Z be a sequence space, then Kizmaz[12] introduced the following difference sequence spaces

for Z ∈ {ℓ_∞, c, c₀}, where Δx_k = x_k − x_k+1 for each k ∈ ℕ = {1, 2, 3...}–the set of positive integers. Since then, many authors have studied further generalization of the difference sequence spaces [5, 9, 15, 17]. Moreover, Altay and Polat [3], Polat and Başar [14] and many others have studied new sequence spaces from matrix point of view that represent difference operators.

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=0∞an,kxk. For two sequence spaces X and Y and an infinite matrix A = (a_{n, k}), the sequence space X_A is defined by X_A = {x = (x_k) ∈ w : Ax ∈ X}, 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 Er=(en,kr), where 0 < r < 1 and

The Euler sequence spaces e0r,ecr and e∞r were defined by Altay and Başar [1] and Altay, Başar and Mursaleen [2] as follows

and

Altay and Polat [3] defined further generalization of the Euler sequence spaces e0r(∇),ecr(∇) and e∞r(∇) by

for Z∈{e0r,ecr,e∞r}, where ∇x_k = x_k − x_k−1 for each k ∈ ℕ. Here any term with negative subscript is equal to naught. Moreover, many authors have used especially the Euler matrix for defining new sequence spaces. For instance, Kara and Başarir [10], Karakaya and Polat [11] and Polat and Başar [14].

Recently Bişgin [6, 7] defined another type of generalization of the Euler sequence spaces and introduced the binomial sequence spaces b0r,s,bcr,s and b∞r,s. Let r, s ∈ ℝ and r + s ≠ 0. Then the binomial matrix Br,s=(bn,kr,s) is defined by

for all k, n ∈ ℕ. For sr > 0 we have

• ||B^{r, s} ||< ∞,

• limn→∞ bn,kr,s=0 for each k ∈ ℕ,

• limn→∞ ∑kbn,kr,s=1.

Thus, the binomial matrix B^{r, s} is regular for sr > 0. Unless stated otherwise, we assume that sr > 0. If we take s + r = 1, we obtain the Euler matrix E^r. So, the binomial matrix generalizes the Euler matrix. Bişgin defined the following spaces of binomial sequences

and

The main purpose of the present paper is to study the difference spaces b0r,s(∇),bcr,s(∇) and b∞r,s(∇) of the binomial sequence whose B^{r, s}(∇)-transforms are in the spaces c₀, c and ℓ_∞, respectively. These new sequence spaces are the generalization of the sequence spaces defined in [3, 6, 7]. Also, we compute the bases and the α-, β- and γ-duals of these sequence spaces.

In this section, we introduce the spaces b0r,s(∇),bcr,s(∇) and b∞r,s(∇) and prove that these sequence spaces are linearly isomorphic to the spaces c₀, c and ℓ_∞, respectively.

We first define the binomial difference sequence spaces b0r,s(∇),bcr,s(∇) and b∞r,s(∇) by

and

Let us define the sequence y = (y_n) as the B^{r, s}(∇)-transform of a sequence x = (x_k), that is

for each n ∈ ℕ. Then, the binomial difference sequence spaces b0r,s(∇),bcr,s(∇) and b∞r,s(∇) can be redefined by all sequences whose B^{r, s}(∇)-transforms are in the space c₀, c and ℓ_∞. Let X be the one of the spaces b0r,s(∇),bcr,s(∇) and b∞r,s(∇). It is obvious that these sequence spaces are linear spaces normed by

Theorem 2.1

The sequence space X is a complete linear metric space with the norm defined by theequation (2.2).

Theorem 2.2

The sequence spacesb0r,s(∇),bcr,s(∇)andb∞r,s(∇)are linearly isomorphic to the spaces c₀, c and ℓ_∞, respectively.

Theorem 2.3

The inclusionsc0(∇)⊆e0r(∇)⊆b0r,s(∇),c(∇)⊆ecr(∇)⊆bcr,s(∇)andℓ∞(∇)⊆e∞r(∇)⊆b∞r,s(∇)strictly hold.

For a normed space (X, || · ||), a sequence {x_k : x_k ∈ X}_k∈ℕ is called a Schauder basis [8] if for every x ∈ X, there is a unique scalar sequence (λ_k) such that ‖x-∑k=0nλkxk‖→0, as n→∞. Next, we shall give a Schauder basis for the sequence spaces b0r,s(∇) and bcr,s(∇).

We define the sequence g(k)(r,s)={gi(k)(r,s)}i∈ℕ by

for each k ∈ ℕ.

Theorem 3.1

The sequence (g^(k)(r, s))_k∈ℕis a Schauder basis for the binomial sequence spaceb0r,s(∇)and everyx=(xi)∈b0r,s(∇)has a unique representation by

where λ_k(r, s) = [B^{r, s}(∇x_i)]_k for each k ∈ ℕ.

Theorem 3.2

Let g = (1, 2, 3, 4, …) and lim_k→∞ λ_k(r, s) = l. The set {g, g⁽⁰⁾(r, s), g⁽¹⁾(r, s), …, g^(k)(r, s), …} is a Schauder basis for the spacebcr,s(∇)and everyx∈bcr,s(∇)has a unique representation by

Corollary 3.1

The sequence spacesb0(r,s)(∇)andbc(r,s)(∇)are separable.

For the duality theory, the study of sequence spaces is more useful when we investigate them equipped with linear topologies. Köthe and Toeplitz [13] first computed the duals whose elements can be represented as sequences and defined the α-dual (or Köthe-Toeplitz dual). Next, we compute the α-, β- and γ-duals of the binomial sequence spaces b0r,s(∇),bcr,s(∇) and b∞r,s(∇).

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:

where Γ is the collection of all finite subsets of ℕ.

Lemma 3.1.([16])

Let A = (a_{n, k}) be an infinite matrix. Then the following statements hold:

• A ∈ (c₀ : ℓ₁) = (c : ℓ₁) = (ℓ_∞ : ℓ₁) if and only if (3.3) holds.

• A ∈ (c₀ : c) if and only if (3.4) and (3.5) hold.

• A ∈ (c : c) if and only if (3.4), (3.5) and (3.6) hold.

• A ∈ (ℓ_∞ : c) if and only if (3.5) and (3.7) hold.

• A ∈ (c₀ : ℓ_∞) = (c : ℓ_∞) = (ℓ_∞ : ℓ_∞) if and only if (3.4) holds.

Theorem 3.3

The α-dual of the spacesb0r,s(∇),bcr,s(∇)andb∞r,s(∇)is the set

Theorem 3.4

We have the following relations:

• [b0r,s(∇)]β=U2r,s∩U3r,s,

• [bcr,s(∇)]β=U2r,s∩U3r,s∩U5r,s,

• [b∞r,s(∇)]β=U3r,s∩U4r,s.

Theorem 3.5

The γ-dual of the spacesb0r,s(∇),bcr,s(∇)andb∞r,s(∇)is the setU2r,s.

By considering the definitions of the binomial matrix Br,s=(bn,kr,s) and difference operator, we introduce the sequence spaces b0r,s(∇),bcr,s(∇) and b∞r,s(∇). These spaces are the natural continuation of [3, 6, 7]. Our results are the generalization of the matrix domain of the Euler matrix of order r. In order to give full knowledge to the reader on related topics with applications and a possible line of further investigation, the e-book[4] is added to the list of references.