1 Introduction and preliminaries

Let w denote the space of all sequences. By \(\ell_{p}\), \(\ell_{\infty }\), c and \(c_{0}\), we denote the spaces of p-absolutely summable, bounded, convergent and null sequences, respectively, where \(1\leq p<\infty\). Let Z be a sequence space, then Kizmaz [1] introduced the following difference sequence spaces:

$$Z(\Delta)=\bigl\{ (x_{k})\in w:(\Delta x_{k})\in Z\bigr\} $$

for \(Z=\ell_{\infty},c,c_{0}\), where \(\Delta x_{k}=x_{k}-x_{k+1}\) for each \(k\in\mathbb{N}=\{1,2,3,\ldots\}\), the set of positive integers. Since then, many authors have studied further generalization of the difference sequence spaces [2–6]. Moreover, Altay and Polat [7], Başarir and Kara [8–12], Kara [13], Kara and İlkhan [14], Polat and Başar [15], and many others have studied new sequence spaces from a matrix point of view that represent difference operators.

For an infinite matrix \(A=(a_{n,k})\) and \(x=(x_{k})\in w\), the A-transform of x is defined by \(Ax=\{(Ax)_{n}\}\) and is supposed to be convergent for all \(n\in\mathbb{N}\), where \((Ax)_{n}=\sum_{k=0}^{\infty}a_{n,k}x_{k}\). For two sequence spaces X, Y and an infinite matrix \(A=(a_{n,k})\), the sequence space \(X_{A}\) is defined by \(X_{A}=\{x=(x_{k})\in w:Ax \in X\}\), which is called the domain of matrix A in the space X. By \((X: Y)\), we denote the class of all matrices such that \(X \subseteq Y_{A}\).

The Euler means \(E^{r}\) of order r is defined by the matrix \(E^{r}=(e_{n,k}^{r})\), where \(0< r<1\) and

$$e_{n,k}^{r}= \textstyle\begin{cases} \left({\scriptsize\begin{matrix}{} n\cr k \end{matrix}} \right)(1-r)^{n-k}r^{k}& \text{if $0\leq k\leq n$},\\ 0& \text{if $k>n$}. \end{cases} $$

The Euler sequence spaces \(e^{r}_{p}\) and \(e^{r}_{\infty}\) were defined by Altay, Başar and Mursaleen [16] as follows:

$$e^{r}_{p}=\left \{x=(x_{k})\in w: \sum _{n}\left \vert \sum _{k=0}^{n} \left ( \begin{matrix} n\\ k \end{matrix} \right ) (1-r)^{n-k}r^{k}x_{k} \right \vert ^{p}< \infty \right \} $$

and

$$e^{r}_{\infty}=\left \{x=(x_{k})\in w: \sup _{n\in\mathbb{N}}\left \vert \sum_{k=0}^{n} \left ( \begin{matrix} n\\ k \end{matrix} \right ) (1-r)^{n-k}r^{k}x_{k} \right \vert < \infty \right \}. $$

Altay and Polat [7] defined further generalization of the Euler sequence spaces \(e^{r}_{0}(\nabla)\), \(e^{r}_{c}(\nabla)\) and \(e^{r}_{\infty}(\nabla)\) by

$$\begin{aligned} &e^{r}_{0}(\nabla)= \bigl\{ x=(x_{k})\in w: ( \nabla x_{k})\in e^{r}_{0} \bigr\} , \\ &e^{r}_{c}(\nabla)= \bigl\{ x=(x_{k})\in w: ( \nabla x_{k})\in e^{r}_{c} \bigr\} \end{aligned}$$

and

$$\begin{aligned} e^{r}_{\infty}(\nabla)= \bigl\{ x=(x_{k})\in w: (\nabla x_{k})\in e^{r}_{\infty} \bigr\} , \end{aligned}$$

where \(\nabla x_{k}=x_{k}-x_{k-1}\) for each \(k\in\mathbb{N}\). Here any term with negative subscript is equal to naught. Many authors have used especially the Euler matrix for defining new sequence spaces, for instance, Kara and Başarir [17], Karakaya and Polat [18] and Polat and Başar [15].

Recently Bişgin [19, 20] defined another type of generalization of the Euler sequence spaces and introduced the binomial sequence spaces \(b^{r,s}_{0}\), \(b^{r,s}_{c}\), \(b^{r,s}_{\infty }\) and \(b^{r,s}_{p}\). Let \(r,s\in\mathbb{R}\) and \(r+s\neq0\). Then the binomial matrix \(B^{r,s}=(b_{n,k}^{r,s})\) is defined by

$$b_{n,k}^{r,s}= \textstyle\begin{cases} \frac{1}{(s+r)^{n}}\left({\scriptsize\begin{matrix}{} n\cr k\end{matrix}} \right)s^{n-k}r^{k}& \text{if $0\leq k\leq n$},\\ 0& \text{if $k>n$}, \end{cases} $$

for all \(k,n\in\mathbb{N}\). For \(sr>0\) we have

  1. (i)

    \(\Vert B^{r,s} \Vert <\infty\),

  2. (ii)

    \(\lim_{n\rightarrow\infty}b_{n,k}^{r,s}=0\) for each \(k\in \mathbb{N}\),

  3. (iii)

    \(\lim_{n\rightarrow\infty}\sum_{k}b_{n,k}^{r,s}=1\).

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 [20] defined the following spaces of binomial sequences:

$$b^{r,s}_{p}=\left \{x=(x_{k})\in w: \sum _{n}\left \vert \frac{1}{(s+r)^{n}}\sum _{k=0}^{n}\left ( \begin{matrix} n\\ k \end{matrix} \right )s^{n-k}r^{k}x_{k} \right \vert ^{p}< \infty \right \} $$

and

$$b^{r,s}_{\infty}=\left \{x=(x_{k})\in w: \sup _{n\in\mathbb{N}}\left \vert \frac {1}{(s+r)^{n}}\sum _{k=0}^{n}\left ( \begin{matrix} n\\ k \end{matrix} \right )s^{n-k}r^{k}x_{k} \right \vert < \infty \right \}. $$

The main purpose of the present paper is to study the normed difference spaces \(b^{r,s}_{p}(\nabla)\) and \(b^{r,s}_{\infty}(\nabla)\) of the binomial sequence whose \(B^{r,s}(\nabla)\)-transforms are in the spaces \(\ell_{p}\) and \(\ell_{\infty}\), respectively. These new sequence spaces are the generalization of the sequence spaces defined in [7] and [20]. Also, we compute the bases and α-, β- and γ-duals of these sequence spaces.

2 The binomial difference sequence spaces

In this section, we introduce the spaces \(b^{r,s}_{p}(\nabla)\) and \(b^{r,s}_{\infty}(\nabla)\) and prove that these sequence spaces are linearly isomorphic to the spaces \(\ell_{p}\) and \(\ell_{\infty}\), respectively.

We first define the binomial difference sequence spaces \(b^{r,s}_{p}(\nabla)\) and \(b^{r,s}_{\infty}(\nabla)\) by

$$b^{r,s}_{p}(\nabla)= \bigl\{ x=(x_{k})\in w: (\nabla x_{k})\in b^{r,s}_{p} \bigr\} $$

and

$$b^{r,s}_{\infty}(\nabla)= \bigl\{ x=(x_{k})\in w: (\nabla x_{k})\in b^{r,s}_{\infty} \bigr\} . $$

Let us define the sequence \(y=(y_{n})\) as the \(B^{r,s}(\nabla )\)-transform of a sequence \(x=(x_{k})\), that is,

$$ y_{n}= \bigl[B^{r,s}(\nabla x_{k}) \bigr]_{n}=\frac{1}{(s+r)^{n}}\sum_{k=0}^{n} \left ( \begin{matrix} n\\ k \end{matrix} \right )s^{n-k}r^{k}(\nabla x_{k}) $$
(2.1)

for each \(n\in\mathbb{N}\). Then the binomial difference sequence spaces \(b^{r,s}_{p}(\nabla)\) or \(b^{r,s}_{\infty}(\nabla)\) can be redefined by all sequences whose \(B^{r,s}(\nabla)\)-transforms are in the space \(\ell_{p}\) or \(\ell_{\infty}\).

Theorem 2.1

The sequence spaces \(b^{r,s}_{p}(\nabla)\) and \(b^{r,s}_{\infty}(\nabla )\) are complete linear metric spaces with the norm defined by

$$f_{b^{r,s}_{p}(\nabla)}(x)= \Vert y \Vert _{p}= \Biggl(\sum _{n=1}^{\infty} \vert y_{n} \vert ^{p} \Biggr)^{\frac{1}{p}} $$

and

$$f_{b^{r,s}_{\infty}(\nabla)}(x)= \Vert y \Vert _{\infty}=\sup_{n\in\mathbb{N}} \vert y_{n} \vert , $$

where \(1\leq p<\infty\) and the sequence \(y=(y_{n})\) is defined by the \(B^{r,s}(\nabla)\)-transform of x.

Proof

The proof of the linearity is a routine verification. It is obvious that \(f_{b^{r,s}_{p}}(\alpha x)= \vert \alpha \vert f_{b^{r,s}_{p}}(x)\) and \(f_{b^{r,s}_{p}}(x)=0\) if and only if \(x=\theta\) for all \(x\in b^{r,s}_{p}(\nabla)\), where θ is the zero element in \(b^{r,s}_{p}\) and \(\alpha\in\mathbb{R}\). We consider \(x,z \in b^{r,s}_{p}(\nabla)\), then we have

$$\begin{aligned} f_{b^{r,s}_{p}(\nabla)}(x+z)&= \biggl(\sum_{n} \bigl\vert \bigl(B^{r,s} \bigl[\nabla (x_{k}+z_{k}) \bigr] \bigr)_{n} \bigr\vert ^{p} \biggr)^{\frac{1}{p}} \\ &\leq \biggl(\sum_{n} \bigl\vert \bigl[B^{r,s}(\nabla x_{k}) \bigr]_{n} \bigr\vert ^{p} \biggr)^{\frac {1}{p}}+ \biggl(\sum_{n} \bigl\vert \bigl[B^{r,s}(\nabla z_{k}) \bigr]_{n} \bigr\vert ^{p} \biggr)^{\frac {1}{p}} = f_{b^{r,s}_{p}(\nabla)}(x)+f_{b^{r,s}_{p}(\nabla)}(z). \end{aligned}$$

Hence \(f_{b^{r,s}_{p}(\nabla)}\) is a norm on the space \(b^{r,s}_{p}(\nabla)\).

Let \((x_{m})\) be a Cauchy sequence in \(b^{r,s}_{p}(\nabla)\), where \(x_{m}=(x_{m_{k}})_{k=1}^{\infty}\in b^{r,s}_{p}(\nabla)\) for each \(m\in\mathbb{N}\). For every \(\varepsilon>0\), there is a positive integer \(m_{0}\) such that \(f_{b^{r,s}_{p}(\nabla)}( x_{m}-x_{l})<\varepsilon\) \(\text{for } m,l\geq m_{0}\). Then we get

$$\bigl\vert \bigl(B^{r,s} \bigl[\nabla(x_{m_{k}}-x_{l_{k}}) \bigr] \bigr)_{n} \bigr\vert \leq \biggl(\sum _{n} \bigl\vert \bigl(B^{r,s} \bigl[ \nabla(x_{m_{k}}-x_{l_{k}}) \bigr] \bigr)_{n} \bigr\vert ^{p} \biggr)^{\frac {1}{p}}< \varepsilon $$

for \(m,l\geq m_{0}\) and each \(k\in\mathbb{N}\). So \((B^{r,s}(\nabla x_{m_{k}}))_{m=1}^{\infty}\) is a Cauchy sequence in the set of real numbers \(\mathbb{R}\). Since \(\mathbb{R}\) is complete, we have \(\lim_{m\rightarrow\infty}B^{r,s}(\nabla x_{m_{k}})=B^{r,s}(\nabla x_{k})\) for each \(k\in\mathbb{N}\). We compute

$$\begin{aligned} \sum_{n=0}^{i} \bigl\vert \bigl(B^{r,s} \bigl[\nabla(x_{m_{k}}-x_{l_{k}}) \bigr] \bigr)_{n} \bigr\vert \leq f_{b^{r,s}_{p}(\nabla)}( x_{m}-x_{l})< \varepsilon \end{aligned}$$
(2.2)

for \(m>m_{0}\). We take i and l →∞, then the inequality (2.2) implies that

$$f_{b^{r,s}_{p}(\nabla)}( x_{m}-x)\rightarrow0. $$

We have

$$f_{b^{r,s}_{p}(\nabla)}(x)\leq f_{b^{r,s}_{p}(\nabla )}(x_{m}-x)+f_{b^{r,s}_{p}(\nabla)}(x_{m})< \infty, $$

that is, \(x\in b^{r,s}_{p}(\nabla)\). Thus, the space \(b^{r,s}_{p}(\nabla )\) is complete. For the space \(b^{r,s}_{\infty}(\nabla)\), the proof can be completed in a similar way. So, we omit the detail. □

Theorem 2.2

The sequence spaces \(b^{r,s}_{p}(\nabla)\) and \(b^{r,s}_{\infty}(\nabla )\) are linearly isomorphic to the spaces \(\ell_{p}\) and \(\ell_{\infty}\), respectively, where \(1\leq p< \infty\).

Proof

Similarly, we only prove the theorem for the space \(b^{r,s}_{p}(\nabla)\). To prove \(b^{r,s}_{p}(\nabla)\cong\ell _{p}\), we must show the existence of a linear bijection between the spaces \(b^{r,s}_{p}(\nabla)\) and \(\ell_{p}\).

Consider \(T:b^{r,s}_{p}(\nabla)\rightarrow\ell_{p}\) by \(T(x)=B^{r,s}(\nabla x_{k})\). The linearity of T is obvious and \(x=\theta\) whenever \(T(x)=\theta\). Therefore, T is injective.

Let \(y=(y_{n})\in\ell_{p} \) and define the sequence \(x=(x_{k})\) by

$$\begin{aligned} x_{k}=\sum_{i=0}^{k}(s+r)^{i} \sum_{j=i}^{k} \left ( \begin{matrix} j\\ i \end{matrix} \right )r^{-j}(-s)^{j-i}y_{i} \end{aligned}$$
(2.3)

for each \(k \in\mathbb{N}\). Then we have

$$\begin{aligned} f_{b^{r,s}_{p}(\nabla)}(x) =& \bigl\Vert \bigl[B^{r,s}(\nabla x_{k}) \bigr]_{n} \bigr\Vert _{p} \\ =&\left (\sum_{n=1}^{\infty} \left \vert \frac{1}{(s+r)^{n}}\sum_{k=0}^{n} \left ( \begin{matrix} n\\ k \end{matrix} \right )s^{n-k}r^{k}( \nabla x_{k}) \right \vert ^{p} \right )^{\frac{1}{p}} \\ =& \Biggl(\sum_{n=1}^{\infty} \vert y_{n} \vert ^{p} \Biggr)^{\frac{1}{p}} = \Vert y \Vert _{p}< \infty, \end{aligned}$$

which implies that \(x\in b^{r,s}_{p}(\nabla)\) and \(T(x)=y\). Consequently, T is surjective and is norm preserving. Thus, \(b^{r,s}_{p}(\nabla)\cong\ell_{p}\). □

3 The Schauder basis and α-, β- and γ-duals

For a normed space \((X, \Vert \cdot \Vert )\), a sequence \(\{ x_{k}:x_{k}\in X\}_{k\in\mathbb{N}}\) is called a Schauder basis [21] if for every \(x\in X\), there is a unique scalar sequence \((\lambda_{k})\) such that \(\Vert x-\sum_{k=0}^{n}\lambda _{k}x_{k} \Vert \rightarrow0 \text{ as } n\rightarrow\infty\). Next, we shall give a Schauder basis for the sequence space \(b_{p}^{r,s}(\nabla)\).

We define the sequence \(g^{(k)}(r,s)=\{g^{(k)}_{i}(r,s)\}_{i \in\mathbb {N}}\) by

$$g^{(k)}_{i}(r,s)= \textstyle\begin{cases} 0& \text{if $0\leq i < k$},\\ (s+r)^{k}\sum_{j=k}^{i}\bigl({\scriptsize\begin{matrix}{} j\cr k\end{matrix}} \bigr)r^{-j}(-s)^{j-k}& \text{if $i\geq k$}, \end{cases} $$

for each \(k\in\mathbb{N}\).

Theorem 3.1

The sequence \((g^{(k)}(r,s))_{k\in\mathbb{N}}\) is a Schauder basis for the binomial sequence spaces \(b_{p}^{r,s}(\nabla)\) and every \(x=(x_{i})\in b_{p}^{r,s}(\nabla)\) has a unique representation by

$$ x=\sum_{k} \lambda_{k}(r,s) g^{(k)}(r,s), $$
(3.1)

where \(1\leq p<\infty\) and \(\lambda_{k}(r,s)= [B^{r,s}(\nabla x_{i})]_{k}\) for each \(k\in\mathbb{N}\).

Proof

Obviously, \(B^{r,s}(\nabla g^{(k)}_{i}(r,s))=e_{k}\in\ell_{p}\), where \(e_{k}\) is the sequence with 1 in the kth place and zeros elsewhere for each \(k\in\mathbb{N}\). This implies that \(g^{(k)}(r,s)\in b_{p}^{r,s}(\nabla)\) for each \(k\in\mathbb{N}\).

For \(x \in b_{p}^{r,s}(\nabla)\) and \(m\in\mathbb{N}\), we put

$$x^{(m)}=\sum_{k=0}^{m} \lambda_{k}(r,s) g^{(k)}(r,s). $$

By the linearity of \(B^{r,s}(\nabla)\), we have

$$B^{r,s} \bigl(\nabla x^{(m)}_{i} \bigr)=\sum _{k=0}^{m}\lambda _{k}(r,s)B^{r,s} \bigl(\nabla g^{(k)}_{i}(r,s) \bigr)=\sum _{k=0}^{m}\lambda_{k}(r,s)e_{k} $$

and

$$\bigl[B^{r,s} \bigl(\nabla \bigl(x_{i}-x_{i}^{(m)} \bigr) \bigr) \bigr]_{k}= \textstyle\begin{cases} 0& \text{if $0\leq k \leq m$},\\ [B^{r,s}(\nabla x_{i})]_{k}& \text{if $k> m$}, \end{cases} $$

for each \(k\in\mathbb{N}\).

For any given \(\varepsilon>0\), there is a positive integer \(m_{0}\) such that

$$\sum_{k=m_{0}+1}^{\infty} \bigl\vert \bigl[B^{r,s}(\nabla x_{i}) \bigr]_{k} \bigr\vert ^{p}< \biggl(\frac {\varepsilon}{2} \biggr)^{p} $$

for all \(k\geq m_{0}\). Then we have

$$\begin{aligned} f_{b^{r,s}_{p}(\nabla)} \bigl( x-x^{(m)} \bigr) =& \Biggl(\sum _{k=m+1 }^{\infty} \bigl\vert \bigl[B^{r,s}(\nabla x_{i}) \bigr]_{k} \bigr\vert ^{p} \Biggr)^{\frac{1}{p}} \\ \leq& \Biggl(\sum_{k=m_{0}+1 }^{\infty} \bigl\vert \bigl[B^{r,s}(\nabla x_{i}) \bigr]_{k} \bigr\vert ^{p} \Biggr)^{\frac {1}{p}} \\ < & \frac{\varepsilon}{2}< \varepsilon, \end{aligned}$$

which implies that \(x \in b_{p}^{r,s}(\nabla)\) is represented as (3.1).

To prove the uniqueness of this representation, we assume that

$$x=\sum_{k} \mu_{k}(r,s) g^{(k)}(r,s). $$

Then we have

$$\bigl[B^{r,s}(\nabla x_{i}) \bigr]_{k}=\sum _{k}\mu_{k}(r,s) \bigl[B^{r,s} \bigl(\nabla g^{(k)}_{i}(r,s) \bigr) \bigr]_{k}=\sum _{k}\mu_{k}(r,s) (e_{k})_{k}= \mu_{k}(r,s), $$

which is a contradiction with the assumption that \(\lambda _{k}(r,s)=[B^{r,s}(\nabla x_{i})]_{k}\) for each \(k \in\mathbb{N}\). This shows the uniqueness of this representation. □

Corollary 3.2

The sequence space \(b_{p}^{r,s}(\nabla)\) is separable, where \(1\leq p<\infty\).

For the duality theory, the study of sequence spaces is more useful when we investigate them equipped with linear topologies. Köthe and Toeplitz [22] first computed duals whose elements can be represented as sequences and defined the α-dual (or Köthe-Toeplitz dual).

For the sequence spaces X and Y, define the multiplier space \(M(X,Y)\) by

$$M(X,Y)= \bigl\{ u=(u_{k})\in w:ux=(u_{k}x_{k})\in Y \text{ for all } x=(x_{k})\in X \bigr\} . $$

Then the α-, β- and γ-duals of a sequence space X are defined by

$$X^{\alpha}=M(X,\ell_{1}),\qquad X^{\beta}=M(X,c)\quad \text{and}\quad X^{\gamma }=M(X,\ell_{\infty}), $$

respectively.

We give the following properties:

$$\begin{aligned} &\sup_{n\in\mathbb{N}} \sum_{k} \vert a_{n,k} \vert ^{q}< \infty, \end{aligned}$$
(3.2)
$$\begin{aligned} &\sup_{k\in\mathbb{N}} \sum_{n} \vert a_{n,k} \vert < \infty, \end{aligned}$$
(3.3)
$$\begin{aligned} &\sup_{n,k\in\mathbb{N}} \vert a_{n,k} \vert < \infty, \end{aligned}$$
(3.4)
$$\begin{aligned} &\lim_{n\rightarrow\infty}a_{n,k}=a_{k} \quad\text{for each } k \in \mathbb{N}, \end{aligned}$$
(3.5)
$$\begin{aligned} &\sup_{K\in\Gamma} \sum_{k} \biggl\vert \sum_{n\in K} a_{n,k} \biggr\vert ^{q}< \infty , \end{aligned}$$
(3.6)
$$\begin{aligned} &\lim_{n\rightarrow\infty}\sum_{k} \vert a_{n,k} \vert =\sum_{k} \Bigl\vert \lim _{n\rightarrow\infty}a_{n,k} \Bigr\vert , \end{aligned}$$
(3.7)

where Γ is the collection of all finite subsets of \(\mathbb{N}\), \(\frac{1}{p}+\frac{1}{q}=1\) and \(1< p\leq\infty\).

Lemma 3.3

[23]

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

  1. (i)

    \(A\in(\ell_{1}:\ell_{1})\) if and only if (3.3) holds.

  2. (ii)

    \(A\in(\ell_{1}:c)\) if and only if (3.4) and (3.5) hold.

  3. (iii)

    \(A\in(\ell_{1}:\ell_{\infty})\) if and only if (3.4) holds.

  4. (iv)

    \(A\in(\ell_{p}:\ell_{1})\) if and only if (3.6) holds with \(\frac {1}{p}+\frac{1}{q}=1\) and \(1< p\leq\infty\).

  5. (v)

    \(A\in( \ell_{p}:c)\) if and only if (3.2) and (3.5) hold with \(\frac{1}{p}+\frac{1}{q}=1\) and \(1< p<\infty\).

  6. (vi)

    \(A\in( \ell_{p}:\ell_{\infty} )\) if and only if (3.2) holds with \(\frac{1}{p}+\frac{1}{q}=1\) and \(1< p<\infty\).

  7. (vii)

    \(A\in( \ell_{\infty}:c )\) if and only if (3.5) and (3.7) hold with \(\frac{1}{p}+\frac{1}{q}=1\) and \(1< p<\infty\).

  8. (viii)

    \(A\in( \ell_{\infty}:\ell_{\infty} )\) if and only if (3.2) holds with \(q=1\).

Theorem 3.4

We define the set \(U_{1}^{r,s}\) and \(U_{2}^{r,s}\) by

$$U_{1}^{r,s}=\left \{u=(u_{k})\in w:\sup _{i\in\mathbb{N}}\sum_{k}\left \vert (s+r)^{i}\sum_{j=i}^{k}\left ( \begin{matrix} j\\ i \end{matrix} \right )r^{-j}(-s)^{j-i}u_{k} \right \vert < \infty \right \} $$

and

$$U_{2}^{r,s}=\left \{u=(u_{k})\in w:\sup _{K\in\Gamma}\sum_{i}\left \vert \sum_{k\in K}(s+r)^{i}\sum _{j=i}^{k}\left ( \begin{matrix} j\\ i \end{matrix} \right )r^{-j}(-s)^{j-i}u_{k} \right \vert ^{q}< \infty \right \}. $$

Then \([b^{r,s}_{1}(\nabla)]^{\alpha}=U_{1}^{r,s}\) and \([b^{r,s}_{p}(\nabla)]^{\alpha}=U_{2}^{r,s}\), where \(1< p\leq\infty\).

Proof

Let \(u=(u_{k})\in w\) and \(x=(x_{k})\) be defined by (2.3), then we have

$$u_{k}x_{k}=\sum_{i=0}^{k}(s+r)^{i} \sum_{j=i}^{k}\left ( \begin{matrix} j\\ i \end{matrix} \right )r^{-j}(-s)^{j-i}u_{k}y_{i}= \bigl(G^{r,s}y \bigr)_{k} $$

for each \(k\in\mathbb{N}\), where \(G^{r,s}=(g^{r,s}_{k,i})\) is defined by

$$g^{r,s}_{k,i}= \textstyle\begin{cases} (s+r)^{i}\sum_{j=i}^{k}\left({\scriptsize\begin{matrix}{} j\cr i \end{matrix}} \right)r^{-j}(-s)^{j-i}u_{k}& \text{if $0\leq i\leq k$},\\ 0& \text{if $i>k$}. \end{cases} $$

Therefore, we deduce that \(ux= (u_{k}x_{k})\in\ell_{1}\) whenever \(x\in b_{1}^{r,s}(\nabla)\) or \(b_{p}^{r,s}(\nabla)\) if and only if \(G^{r,s}y\in\ell_{1}\) whenever \(y\in \ell_{1}\) or \(\ell_{p}\), which implies that \(u=(u_{k})\in[b_{1}^{r,s}(\nabla )]^{\alpha} \text{ or } [b_{p}^{r,s}(\nabla)]^{\alpha}\) if and only if \(G^{r,s}\in(\ell_{1}:\ell_{1})\) and \(G^{r,s}\in(\ell_{p}:\ell_{1})\) by parts (i) and (iv) of Lemma 3.3, we obtain \(u=(u_{k})\in [b_{1}^{r,s}(\nabla)]^{\alpha}\) if and only if

$$\sup_{i\in\mathbb{N}}\sum_{k}\left \vert (s+r)^{i}\sum_{j=i}^{k} \left ( \begin{matrix} j\\ i \end{matrix} \right )r^{-j}(-s)^{j-i}u_{k} \right \vert < \infty $$

and \(u=(u_{k})\in[b_{p}^{r,s}(\nabla)]^{\alpha}\) if and only if

$$\sup_{K\in\Gamma}\sum_{i}\left \vert \sum_{k\in K}(s+r)^{i}\sum _{j=i}^{k}\left ( \begin{matrix} j\\ i \end{matrix} \right )r^{-j}(-s)^{j-i}u_{k} \right \vert ^{q}< \infty. $$

Thus, we have \([b^{r,s}_{1}(\nabla)]^{\alpha}=U_{1}^{r,s}\) and \([b^{r,s}_{p}(\nabla)]^{\alpha}=U_{2}^{r,s}\), where \(1< p\leq\infty\). □

Now, we define the sets \(U_{3}^{r,s}\), \(U_{4}^{r,s}\), \(U_{5}^{r,s}\), \(U_{6}^{r,s}\) and \(U_{7}^{r,s}\) by

$$\begin{aligned} &U_{3}^{r,s}=\left \{u=(u_{k})\in w: \lim _{n\rightarrow\infty} (s+r)^{k}\sum_{i=k}^{n} \sum_{j=k}^{i}\left ( \begin{matrix} j\\ k \end{matrix} \right )r^{-j}(-s)^{j-k}u_{i} \text{ exists for each } k \in\mathbb {N} \right \}, \\ &U_{4}^{r,s}=\left \{u=(u_{k})\in w: \sup _{n,k\in\mathbb{N}}\left \vert (s+r)^{k}\sum _{i=k}^{n}\sum_{j=k}^{i} \left ( \begin{matrix} j\\ k \end{matrix} \right )r^{-j}(-s)^{j-k}u_{i} \right \vert < \infty \right \}, \\ &U_{5}^{r,s}= \left\{u=(u_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k}\left \vert (s+r)^{k}\sum_{i=k}^{n}\sum _{j=k}^{i}\left ( \begin{matrix} j\\ k \end{matrix} \right )r^{-j}(-s)^{j-k}u_{i} \right \vert \right. \\ &\left.\phantom{U_{5}^{r,s}=}=\sum_{k}\left \vert \lim _{n\rightarrow\infty}(s+r)^{k}\sum_{i=k}^{n} \sum_{j=k}^{i}\left ( \begin{matrix} j\\ k \end{matrix} \right )r^{-j}(-s)^{j-k}u_{i} \right \vert \right\}, \\ &U_{6}^{r,s}=\left \{u=(u_{k})\in w: \sup _{n\in\mathbb{N}}\sum_{k=0}^{n}\left \vert (s+r)^{k}\sum_{i=k}^{n} \sum_{j=k}^{i}\left ( \begin{matrix} j\\ k \end{matrix} \right )r^{-j}(-s)^{j-k}u_{i} \right \vert ^{q}< \infty \right \},\quad 1< q< \infty, \end{aligned}$$

and

$$U_{7}^{r,s}=\left \{u=(u_{k})\in w: \sup _{n\in\mathbb{N}}\sum_{k=0}^{n}\left \vert (s+r)^{k}\sum_{i=k}^{n} \sum_{j=k}^{i}\left ( \begin{matrix} j\\ k \end{matrix} \right )r^{-j}(-s)^{j-k}u_{i} \right \vert < \infty \right \}. $$

Theorem 3.5

We have the following relations:

  1. (i)

    \([b_{1}^{r,s}(\nabla)]^{ \beta}=U_{3}^{r,s}\cap U_{4}^{r,s}\),

  2. (ii)

    \([b_{p}^{r,s}(\nabla)]^{ \beta}=U_{3}^{r,s}\cap U_{6}^{r,s}\), where \(1< p<\infty\),

  3. (iii)

    \([b_{\infty}^{r,s}(\nabla)]^{ \beta}=U_{3}^{r,s}\cap U_{5}^{r,s}\),

  4. (iv)

    \([b_{1}^{r,s}(\nabla)]^{ \gamma}=U_{4}^{r,s}\),

  5. (v)

    \([b_{p}^{r,s}(\nabla)]^{ \gamma}=U_{6}^{r,s}\), where \(1< p<\infty \),

  6. (vi)

    \([b_{\infty}^{r,s}(\nabla)]^{ \gamma}=U_{7}^{r,s}\).

Proof

Let \(u=(u_{k})\in w\) and \(x=(x_{k})\) be defined by (2.3), then we consider the following equation:

$$\begin{aligned} \sum_{k=0}^{n}u_{k}x_{k}&= \sum_{k=0}^{n}u_{k}\left [ \sum_{i=0}^{k}(s+r)^{i}\sum _{j=i}^{k}\left ( \begin{matrix} j\\ i \end{matrix} \right )r^{-j}(-s)^{j-i}y_{i} \right ] \\ &=\sum_{k=0}^{n}\left [(s+r)^{k} \sum_{i=k}^{n}\sum _{j=k}^{i}\left ( \begin{matrix} j\\ k \end{matrix} \right )r^{-j}(-s)^{j-k}u_{i} \right ]y_{k} = \bigl(U^{r,s}y \bigr)_{n}, \end{aligned}$$

where \(U^{r,s}=(u^{r,s}_{n,k})\) is defined by

$$u_{n,k}= \textstyle\begin{cases} (s+r)^{k}\sum_{i=k}^{n}\sum_{j=k}^{i}\bigl({\scriptsize\begin{matrix}{} j\cr k\end{matrix}} \bigr)r^{-j}(-s)^{j-k}u_{i}&\text{if $0\leq k \leq n$},\\ 0&\text{if $k> n$}. \end{cases} $$

Therefore, we deduce that \(ux= (u_{k}x_{k})\in c\) whenever \(x\in b_{1}^{r,s}(\nabla)\) if and only if \(U^{r,s}y\in c\) whenever \(y\in\ell_{1}\), which implies that \(u=(u_{k})\in[b_{1}^{r,s}(\nabla)]^{ \beta}\) if and only if \(U^{r,s}\in(\ell_{1}:c)\). By Lemma 3.3(ii), we obtain \([b_{1}^{r,s}(\nabla)]^{ \beta}=U_{3}^{r,s}\cap U_{4}^{r,s}\). Using Lemma 3.3(i) and (iii)-(viii) instead of (ii), the proof can be completed in a similar way. So, we omit the details. □

4 Conclusion

By considering the definitions of the binomial matrix \(B^{r,s}=(b^{r,s}_{n,k})\) and the difference operator, we introduce the sequence spaces \(b^{r,s}_{p}(\nabla)\) and \(b^{r,s}_{\infty}(\nabla)\). These spaces are the natural continuations of [1, 7, 20]. Our results are the generalizations of the matrix domain of the Euler matrix of order r. In order to give fully inform the reader on related topics with applications and a possible line of further investigation, the e-book [24] is added to the list of references.