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 absolutely p-summable, bounded, convergent and null sequences, respectively, where \(1\leq p<\infty\). A Banach sequence space Z is called a BK-space [1] provided each of the maps \(p_{n}:Z\rightarrow\mathbb {C}\) defined by \(p_{n}(x)=x_{n}\) is continuous for all \(n\in\mathbb {N}\), which is of great importance in the characterization of matrix transformations between sequence spaces. One can prove that the sequence spaces \(\ell_{\infty },c\) and \(c_{0}\) are BK-spaces with their usual sup-norm.

Let Z be a sequence space, then Kizmaz [2] introduced the following difference sequence spaces:

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

for \(Z\in\{\ell_{\infty},c,c_{0}\}\), where \(\Delta x_{k}=x_{k}-x_{k+1}\) for each \(k\in\mathbb{N}\). Et and Colak [3] defined the generalization of the difference sequence spaces

$$\begin{aligned}& Z\bigl(\Delta^{m}\bigr)=\bigl\{ (x_{k})\in w:\bigl( \Delta^{m} x_{k}\bigr)\in Z\bigr\} \end{aligned}$$

for \(Z\in\{\ell_{\infty},c,c_{0}\}\), where \(m\in\mathbb{N}\), \(\Delta ^{0}x_{k}=x_{k}\) and \(\Delta^{m} x_{k}=\Delta^{m-1}x_{k}-\Delta ^{m-1}x_{k+1}\) for each \(k\in\mathbb{N}\), which is equivalent to the binomial representation \(\Delta^{m} x_{k}=\sum_{i=0}^{m}(-1)^{i}\bigl( {\scriptsize\begin{matrix}{}m \cr i\end{matrix}} \bigr) x_{k+i}\). Since then, many authors have studied further generalization of the difference sequence spaces [4–7]. Moreover, Altay and Polat [8], Başarir and Kara [9–13], Başarir, Kara and Konca [14], Kara [15], Kara and İlkhan [16, 17], Polat and Başar [18], Song and Meng [19] 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 \(x=(x_{k})\in w\), 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 and Y and an infinite matrix \(A=(a_{n,k})\), the sequence space \(X_{A}\) is defined by

$$ X_{A}=\bigl\{ x=(x_{k})\in w:Ax \in X\bigr\} , $$
(1.1)

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

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

The Euler sequence spaces \(e^{r}_{0}\), \(e^{r}_{c}\) and \(e^{r}_{\infty}\) were defined by Altay and Başar [20] and Altay, Başar and Mursaleen [21] as follows:

$$\begin{aligned}& e^{r}_{0}=\left\{x=(x_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k=0}^{n} \begin{pmatrix} n\\ k \end{pmatrix} (1-r)^{n-k}r^{k}x_{k}=0\right\}, \\& e^{r}_{c}=\left\{x=(x_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k=0}^{n} \begin{pmatrix} n\\ k \end{pmatrix} (1-r)^{n-k}r^{k}x_{k} \text{ exists}\right\}, \end{aligned}$$

and

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

Altay and Polat [8] defined further generalization of the Euler sequence spaces \(e^{r}_{0}(\nabla)\), \(e^{r}_{c}(\nabla)\) and \(e^{r}_{\infty}(\nabla)\) by

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

for \(Z\in\{e_{0}^{r}, e_{c}^{r}, e_{\infty}^{r}\}\), 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.

Polat and Başar [18] employed the matrix domain technique of the triangle limitation method for obtaining the following sequence spaces:

$$Z\bigl(\nabla^{(m)}\bigr)=\bigl\{ x=(x_{k})\in w: \bigl( \nabla^{(m)} x_{k}\bigr)\in Z\bigr\} $$

for \(Z\in\{e_{0}^{r}, e_{c}^{r}, e_{\infty}^{r}\}\), where \(\nabla ^{(m)}=(\delta_{n,k}^{(m)})\) is a triangle matrix defined by

$$\delta_{n,k}^{(m)}= \textstyle\begin{cases} (-1)^{n-k} \bigl( {\scriptsize\begin{matrix}{} m\cr n-k \end{matrix}} \bigr)& \text{if $\max\{0,n-m\}\leq k\leq n$},\\ 0& \text{if $0\leq k< \max\{0,n-m\}$ or $k>n$}, \end{cases} $$

for all \(k,n,m\in\mathbb{N}\). Also, Başarir and Kayikçi [22] defined the matrix \(B^{(m)}=(b_{n,k}^{(m)})\) by

$$b_{n,k}^{(m)}= \textstyle\begin{cases} \bigl( {\scriptsize\begin{matrix}{} m\cr n-k \end{matrix}} \bigr)r^{m-n+k}s^{n-k}& \text{if $\max\{0,n-m\}\leq k\leq n$},\\ 0& \text{if $0\leq k< \max\{0,n-m\}$ or $k>n$}, \end{cases} $$

which is reduced to the matrix \(\nabla^{(m)}\) in the case \(r=1\), \(s=-1\). Kara and Başarir [23] introduced the spaces \(e^{r}_{0}(B^{(m)})\), \(e^{r}_{c}(B^{(m)})\) and \(e^{r}_{\infty }(B^{(m)})\) of \(B^{(m)}\)-difference sequences.

Recently Bişgin [24, 25] defined another generalization of the Euler sequence spaces and introduced the binomial sequence spaces \(b^{a,b}_{0}\), \(b^{a,b}_{c}\), \(b^{a,b}_{\infty}\) and \(b^{a,b}_{p}\). Let \(a,b\in\mathbb{R}\) and \(a,b\neq0\). Then the binomial matrix \(B^{a,b}=(b_{n,k}^{a,b})\) is defined by

$$b_{n,k}^{a,b}= \textstyle\begin{cases} \frac{1}{(a+b)^{n}}\bigl( {\scriptsize\begin{matrix}{} n\cr k \end{matrix}} \bigr)a^{n-k}b^{k}& \text{if $0\leq k\leq n$},\\ 0& \text{if $k>n$}, \end{cases} $$

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

  1. (i)

    \(\Vert B^{a,b}\Vert <\infty\),

  2. (ii)

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

  3. (iii)

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

Thus, the binomial matrix \(B^{a,b}\) is regular for \(ab>0\). Unless stated otherwise, we assume that \(ab >0\). If we take \(a+b =1\), we obtain the Euler matrix \(E^{r}\), so the binomial matrix generalizes the Euler matrix. Bişgin defined the following binomial sequence spaces:

$$\begin{aligned}& b^{a,b}_{0}=\left\{x=(x_{k})\in w: \lim _{n\rightarrow\infty}\frac {1}{(a+b)^{n}}\sum_{k=0}^{n} \left ( \begin{matrix} n\\ k \end{matrix} \right )a^{n-k}b^{k}x_{k}=0\right\}, \\& b^{a,b}_{c}=\left\{x=(x_{k})\in w: \lim _{n\rightarrow\infty}\frac {1}{(a+b)^{n}}\sum_{k=0}^{n} \left ( \begin{matrix} n\\ k \end{matrix} \right )a^{n-k}b^{k}x_{k} \text{ exists}\right\}, \end{aligned}$$

and

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

The purpose of the present paper is to study the binomial difference spaces \(b^{a,b}_{0}(B^{(m)})\), \(b^{a,b}_{c}(B^{(m)})\) and \(b^{a,b}_{\infty}(B^{(m)})\) whose \(B^{a,b}(B^{(m)})\)-transforms are in the spaces \(c_{0}\), c and \(\ell_{\infty}\), respectively. These new sequence spaces are the generalization of the sequence spaces defined in [24, 25] and [23]. Also, we give some inclusion relations and 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 \(b^{a,b}_{0}(B^{(m)})\), \(b^{a,b}_{c}(B^{(m)})\) and \(b^{a,b}_{\infty}(B^{(m)})\) and prove the BK-property and inclusion relations.

We first define the binomial difference sequence spaces \(b^{a,b}_{0}(B^{(m)})\), \(b^{a,b}_{c}(B^{(m)})\) and \(b^{a,b}_{\infty }(B^{(m)})\) by

$$\begin{aligned}& Z\bigl(B^{(m)}\bigr)=\bigl\{ x=(x_{k})\in w: \bigl(B^{(m)} x_{k}\bigr)\in Z\bigr\} \end{aligned}$$

for \(Z\in\{b^{a,b}_{0}, b^{a,b}_{c}, b^{a,b}_{\infty}\}\). By using the notion of (1.1), the sequence spaces \(b^{a,b}_{0}(B^{(m)})\), \(b^{a,b}_{c}(B^{(m)})\) and \(b^{a,b}_{\infty }(B^{(m)})\) can be redefined by

$$ b^{a,b}_{0}\bigl(B^{(m)}\bigr)= \bigl(b^{a,b}_{0}\bigr)_{B^{(m)}}, \qquad b^{a,b}_{c} \bigl(B^{(m)}\bigr)=\bigl(b^{a,b}_{c} \bigr)_{B^{(m)}},\qquad b^{a,b}_{\infty }\bigl(B^{(m)}\bigr)= \bigl(b^{a,b}_{\infty}\bigr)_{B^{(m)}}. $$
(2.1)

It is obvious that the sequence spaces \(b^{a,b}_{0}(B^{(m)})\), \(b^{a,b}_{c}(B^{(m)})\) and \(b^{a,b}_{\infty}(B^{(m)})\) may be reduced to some sequence spaces in the special cases of \(a, b, s, r\) and \(m\in\mathbb{N}\). For instance, if we take \(a+b=1\), then we obtain the spaces \(e^{r}_{0}(B^{(m)}) \), \(e^{r}_{c}(B^{(m)})\) and \(e^{r}_{\infty}(B^{(m)}) \), defined by Kara and Başarir [23]. If we take \(a+b=1\), \(r=1\) and \(s=-1\), then we obtain the spaces \(e^{r}_{0}(\nabla^{(m)}), e^{r}_{c}(\nabla^{(m)})\) and \(e^{r}_{\infty}(\nabla^{(m)})\), defined by Polat and Başar [18]. Especially, taking \(r=1\) and \(s=-1\), we obtain the new binomial difference sequence spaces \(b^{a,b}_{0}(\nabla^{(m)}), b^{a,b}_{c}(\nabla^{(m)})\) and \(b^{a,b}_{\infty}(\nabla^{(m)})\).

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

$$ y_{n}=\bigl[B^{a,b}\bigl(B^{(m)} x_{k}\bigr)\bigr]_{n} =\frac{1}{(a+b)^{n}}\sum _{k=0}^{n}\sum_{i=k}^{n} \left ( \begin{matrix} m\\ i-k \end{matrix} \right )\left ( \begin{matrix} n\\ i \end{matrix} \right )a^{n-i}b^{i}r^{m+k-i}s^{i-k}x_{k} $$
(2.2)

for each \(n\in\mathbb{N}\). Then the sequence spaces \(b^{a,b}_{0}(B^{(m)})\), \(b^{a,b}_{c}(B^{(m)})\) and \(b^{a,b}_{\infty }(B^{(m)})\) can be redefined by all sequences whose \(B^{a,b}(B^{(m)})\)-transforms are in the spaces \(c_{0}\), c and \(\ell _{\infty}\).

Theorem 2.1

The sequence spaces \(b^{a,b}_{0}(B^{(m)})\), \(b^{a,b}_{c}(B^{(m)})\) and \(b^{a,b}_{\infty}(B^{(m)})\) are BK-spaces with their sup-norm defined by

$$\begin{aligned}& \Vert x\Vert _{b^{a,b}_{0}(B^{(m)})}=\Vert x\Vert _{b^{a,b}_{c}(B^{(m)})}=\Vert x \Vert _{b^{a,b}_{\infty }(B^{(m)})}=\sup_{n\in\mathbb{N}}\bigl\vert \bigl[B^{a,b}\bigl(B^{(m)} x_{k}\bigr) \bigr]_{n}\bigr\vert . \end{aligned}$$

Proof

The sequence spaces \(b^{a,b}_{0}\), \(b^{a,b}_{c}\) and \(b^{a,b}_{\infty}\) are BK-spaces with their sup-norm (see [24], Theorem 2.1 and [25], Theorem 2.1). Moreover, \(B^{(m)}\) is a triangle matrix and (2.1) holds. By using Theorem 4.3.12 of Wilansky [26], we deduce that the binomial sequence spaces \(b^{a,b}_{0}(B^{(m)})\), \(b^{a,b}_{c}(B^{(m)})\) and \(b^{a,b}_{\infty}(B^{(m)})\) are BK-spaces. □

Theorem 2.2

The sequence spaces \(b^{a,b}_{0}(B^{(m)})\), \(b^{a,b}_{c}(B^{(m)})\) and \(b^{a,b}_{\infty}(B^{(m)})\) are linearly isomorphic to the spaces \(c_{0}\), c and \(\ell_{\infty}\), respectively.

Proof

Similarly, we prove the theorem only for the space \(b^{a,b}_{0}(B^{(m)})\). To prove \(b^{a,b}_{0}(B^{(m)})\cong c_{0}\), we must show the existence of a linear bijection between the spaces \(b^{a,b}_{0}(B^{(m)})\) and \(c_{0}\).

Consider \(b^{a,b}_{0}(B^{(m)})\rightarrow c_{0}\) by \(T(x)=B^{a,b}(B^{(m)} x_{k})\). The linearity of T is obvious and \(x=0\) whenever \(T(x)=0\). Therefore, T is injective.

Let \(y=(y_{n})\in c_{0} \) and define the sequence \(x=(x_{k})\) by

$$ x_{k}=\sum_{i=0}^{k}(a+b)^{i} \sum_{j=i}^{k}\left ( \begin{matrix} m+k-j-1\\ k-j \end{matrix} \right ) \left ( \begin{matrix} j\\ i \end{matrix} \right )\frac{(-s)^{k-j}}{r^{m+k-j}}(-a)^{j-i}b^{-j}y_{i} $$
(2.3)

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

$$\lim_{n\rightarrow\infty}\bigl[B^{a,b}\bigl(B^{(m)} x_{k}\bigr)\bigr]_{n}=\lim_{n\rightarrow \infty} \frac{1}{(a+b)^{n}}\sum_{k=0}^{n}\left ( \begin{matrix} n\\ k \end{matrix} \right )a^{n-k}b^{k} \bigl(B^{(m)} x_{k}\bigr)=\lim_{n\rightarrow\infty}y_{n}=0, $$

which implies that \(x\in b^{a,b}_{0}(B^{(m)})\) and \(T(x)=y\). Consequently, T is surjective and is norm preserving. Thus, \(b^{a,b}_{0}(B^{(m)})\cong c_{0}\). □

The following theorems give some inclusion relations for this class of sequence spaces. We have the well known inclusion \(c_{0}\subseteq c\subseteq\ell_{\infty}\), then the corresponding extended versions also preserve this inclusion.

Theorem 2.3

The inclusion \(b^{a,b}_{0}(B^{(m)})\subseteq b^{a,b}_{c}(B^{(m)})\subseteq b^{a,b}_{\infty}(B^{(m)})\) holds.

Theorem 2.4

The inclusions \(e_{0}^{a}(B^{(m)})\subseteq b^{a,b}_{0}(B^{(m)})\), \(e_{c}^{a}(B^{(m)})\subseteq b^{a,b}_{c}(B^{(m)})\) and \(e_{\infty }^{a}(B^{(m)})\subseteq b^{a,b}_{\infty}(B^{(m)})\) strictly hold.

Proof

Similarly, we only prove the inclusion \(e_{0}^{a}(B^{(m)})\subseteq b^{a,b}_{0}(B^{(m)})\). If \(a+b=1\), we have \(E^{a}=B^{a,b}\). So \(e_{0}^{a}(B^{(m)})\subseteq b^{a,b}_{0}(B^{(m)})\) holds. Let \(0< a<1\) and \(b=4\). We define a sequence \(x=(x_{k})\) by \(x_{k}=(-\frac {3}{a})^{k}\) for each \(k\in\mathbb{N}\). It is clear that

$$E^{a}\bigl(B^{(m)} x_{k}\bigr)=\left(\sum _{i=0}^{m}\left ( \begin{matrix} m\\ i \end{matrix} \right )s^{i}r^{m-i} \biggl(-\frac{a}{3}\biggr)^{i}(-2-a)^{n}\right)\notin c_{0} $$

and

$$B^{a,b}\bigl(B^{(m)} x_{k}\bigr) =\left(\sum _{i=0}^{m}\left ( \begin{matrix} m\\ i \end{matrix} \right )s^{i}r^{m-i} \biggl(-\frac{a}{3}\biggr)^{i}\biggl(\frac{1}{4+a} \biggr)^{n}\right)\in c_{0}. $$

So, we have \(x\in b^{a,b}_{0}(B^{(m)})\setminus e_{0}^{a}(B^{(m)})\). This shows that the inclusion \(e_{0}^{a}(B^{(m)})\subseteq b^{a,b}_{0}(B^{(m)})\) strictly holds. □

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 \(Schauder\) \(basis\) [1] if for every \(x\in X\), there is a unique scalar sequence \((\lambda_{k})\) such that \(\Vert x-\sum_{k=0}^{n}\lambda _{k}x_{k}\Vert \rightarrow0\) as \(n\rightarrow\infty\). We shall construct Schauder bases for the sequence spaces \(b_{0}^{a,b}(B^{(m)})\) and \(b_{c}^{a,b}(B^{(m)})\).

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

$$g^{(k)}_{i}(a,b)= \textstyle\begin{cases} 0& \text{if $0\leq i < k$},\\ (a+b)^{k}\sum_{j=k}^{i}\bigl( {\scriptsize\begin{matrix}{} m+i-j-1\\ i-j \end{matrix}} \bigr)\bigl( {\scriptsize\begin{matrix}{} j\cr k \end{matrix}} \bigr)\frac{(-s)^{i-j}}{r^{m+i-j}}b^{-j}(-a)^{j-k}& \text{if $i\geq k$}, \end{cases} $$

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

Theorem 3.1

The sequence \((g^{(k)}(a,b))_{k\in\mathbb{N}}\) is a Schauder basis for the binomial sequence space \(b_{0}^{a,b}(B^{(m)})\) and every \(x=(x_{i})\in b_{0}^{a,b}(B^{(m)})\) has a unique representation by

$$ x=\sum_{k} \lambda_{k}(a,b) g^{(k)}(a,b), $$
(3.1)

where \(\lambda_{k}(a,b)= [B^{a,b}(B^{(m)} x_{i})]_{k}\) for each \(k\in \mathbb{N}\).

Proof

Obviously, \(B^{a,b}(B^{(m)} g^{(k)}_{i}(a,b))=e_{k}\in c_{0}\), where \(e_{k}\) is the sequence with 1 in the kth place and zeros elsewhere for each \(k\in\mathbb{N}\). This implies that \(g^{(k)}(a,b)\in b_{0}^{a,b}(B^{(m)})\) for each \(k\in\mathbb{N}\).

For \(x \in b_{0}^{a,b}(B^{(m)})\) and \(n\in\mathbb{N}\), we put

$$x^{(n)}=\sum_{k=0}^{n} \lambda_{k}(a,b) g^{(k)}(a,b). $$

By the linearity of \(B^{a,b}(B^{(m)})\), we have

$$B^{a,b}\bigl(B^{(m)} x^{(n)}_{i}\bigr)=\sum _{k=0}^{n}\lambda _{k}(a,b)B^{a,b} \bigl(B^{(m)} g^{(k)}_{i}(a,b)\bigr)=\sum _{k=0}^{n}\lambda_{k}(a,b)e_{k} $$

and

$$\bigl[B^{a,b}\bigl(B^{(m)}\bigl(x_{i}-x_{i}^{(n)} \bigr)\bigr)\bigr]_{k}= \textstyle\begin{cases} 0& \text{if $0\leq k < n$},\\ [B^{a,b}(B^{(m)} x_{i})]_{k}& \text{if $k\geq n$}, \end{cases} $$

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

For every \(\varepsilon>0\), there is a positive integer \(n_{0}\) such that

$$\bigl\vert \bigl[B^{a,b}\bigl(B^{(m)} x_{i}\bigr) \bigr]_{k}\bigr\vert < \frac{\varepsilon}{2} $$

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

$$\bigl\Vert x-x^{(n)}\bigr\Vert _{b_{0}^{a,b}(B^{(m)})}=\sup _{k\geq n}\bigl\vert \bigl[B^{a,b}\bigl(B^{(m)} x_{i}\bigr)\bigr]_{k}\bigr\vert \leq\sup _{k\geq n_{0}}\bigl\vert \bigl[B^{a,b}\bigl(B^{(m)} x_{i}\bigr)\bigr]_{k}\bigr\vert < \frac{\varepsilon}{2}< \varepsilon, $$

which implies that \(x \in b_{0}^{a,b}(B^{(m)})\) is represented as in (3.1).

To show the uniqueness of this representation, we assume that

$$x=\sum_{k} \mu_{k}(a,b) g^{(k)}(a,b). $$

Then we have

$$\bigl[B^{a,b}\bigl(B^{(m)} x_{i}\bigr) \bigr]_{k}=\sum_{k}\mu_{k}(a,b) \bigl[B^{a,b}\bigl(B^{(m)} g^{(k)}_{i}(a,b) \bigr)\bigr]_{k}=\sum_{k} \mu_{k}(a,b) (e_{k})_{k}=\mu_{k}(a,b), $$

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

Theorem 3.2

Let \(g=(1,1,1,1,\ldots)\) and \(lim_{k\rightarrow\infty}\lambda_{k}(a,b)=l\). The set \(\{g, g^{(0)}(a,b), g^{(1)}(a,b),\ldots, g^{(k)}(a,b),\ldots\}\) is a Schauder basis for the space \(b_{c}^{a,b}(B^{(m)})\) and every \(x\in b_{c}^{a,b}(B^{(m)})\) has a unique representation by

$$ x=lg+\sum_{k} \bigl[ \lambda_{k}(a,b)-l\bigr] g^{(k)}(a,b). $$
(3.2)

Proof

Obviously, \(B^{a,b}(B^{(m)} g^{k}_{i}(a,b))=e_{k}\in c\) and \(g\in b_{c}^{a,b}(B^{(m)})\). For \(x \in b_{c}^{a,b}(B^{(m)})\), we put \(y=x-lg\) and we have \(y\in b_{0}^{a,b}(B^{(m)})\). Hence, we deduce that y has a unique representation by (3.1), which implies that x has a unique representation by (3.2). Thus, we complete the proof. □

Corollary 3.3

The sequence spaces \(b_{0}^{a,b}(B^{(m)})\) and \(b_{c}^{a,b}(B^{(m)})\) are separable.

Köthe and Toeplitz [27] 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-,β- and γ-duals of the sequence spaces \(b_{0}^{a,b}(B^{(m)})\), \(b_{c}^{a,b}(B^{(m)})\) and \(b_{\infty }^{a,b}(B^{(m)})\).

For the sequence spaces X and Y, define 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

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

respectively.

Let us give the following properties:

$$\begin{aligned}& \sup_{K\in\Gamma} \sum_{n} \biggl\vert \sum_{k\in K} a_{n,k}\biggr\vert < \infty, \end{aligned}$$
(3.3)
$$\begin{aligned}& \sup_{n\in\mathbb{N}} \sum_{k} \vert a_{n,k}\vert < \infty, \end{aligned}$$
(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)
$$\begin{aligned}& \lim_{n\rightarrow\infty}\sum_{k}a_{n,k}=a, \end{aligned}$$
(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 \(\mathbb{N}\).

Lemma 3.4

[28]

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

  1. (i)

    \(A\in(c_{0}:\ell_{1})=(c:\ell_{1})=(\ell_{\infty}:\ell_{1})\) if and only if (3.3) holds.

  2. (ii)

    \(A\in(c_{0}:c)\) if and only if (3.4) and (3.5) hold.

  3. (iii)

    \(A\in(c:c)\) if and only if (3.4), (3.5) and (3.6) hold.

  4. (iv)

    \(A\in(\ell_{\infty}:c)\) if and only if (3.5) and (3.7) hold.

  5. (v)

    \(A\in(c_{0}:\ell_{\infty})=(c:\ell_{\infty})=(\ell_{\infty}:\ell _{\infty})\) if and only if (3.4) holds.

Theorem 3.5

The α-dual of the spaces \(b_{0}^{a,b}(B^{(m)})\), \(b_{c}^{a,b}(B^{(m)})\) and \(b_{\infty}^{a,b}(B^{(m)})\) is the set

$$\begin{aligned} U^{a,b}_{1} =& \Biggl\{ u=(u_{k})\in w: \sup _{K\in\Gamma}\sum_{k}\Biggl\vert \sum _{i\in K} (a+b)^{i}\sum _{j=i}^{k}\left ( \begin{matrix} m+k-j-1\\ k-j \end{matrix} \right )\left ( \begin{matrix} j\\ i \end{matrix} \right ) \\ \phantom{\sum _{j=i}^{k}\begin{matrix} j\\ i \end{matrix}} & {}\times \frac{(-s)^{k-j}}{r^{m+k-j}}b^{-j}(-a)^{j-i}u_{k} \Biggr\vert < \infty\Biggr\} . \end{aligned}$$

Proof

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

$$\begin{aligned}& u_{k}x_{k}=\sum_{i=0}^{k}(a+b)^{i} \sum_{j=i}^{k}\left ( \begin{matrix} m+k-j-1\\ k-j \end{matrix} \right )\left ( \begin{matrix} j\\ i \end{matrix} \right )\frac{(-s)^{k-j}}{r^{m+k-j}}b^{-j}(-a)^{j-i}u_{k}y_{i}= \bigl(G^{a,b}y\bigr)_{k} \end{aligned}$$

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

$$g^{a,b}_{k,i}= \textstyle\begin{cases} (a+b)^{i}\sum_{j=i}^{k}\bigl( {\scriptsize\begin{matrix}{} m+k-j-1\cr k-j \end{matrix}} \bigr)\bigl( {\scriptsize\begin{matrix}{} j\cr i \end{matrix}} \bigr)\frac{(-s)^{k-j}}{r^{m+k-j}}b^{-j}(-a)^{j-i}u_{k}& \text{if $0\leq i\leq k$},\\ 0& \text{if $i>k$}. \end{cases} $$

Therefore, we deduce that \(ux= (u_{k}x_{k})\in\ell_{1}\) whenever \(x\in b_{0}^{a,b}(B^{(m)})\), \(b_{c}^{a,b}(B^{(m)})\) or \(b_{\infty}^{a,b}(B^{(m)})\), if and only if \(G^{a,b}y\in\ell_{1}\), whenever \(y\in c_{0}, c\) or \(\ell_{\infty}\). This implies that \(u=(u_{k})\in [b_{0}^{a,b}(B^{(m)})]^{\alpha}, [b_{c}^{a,b}(B^{(m)})]^{\alpha}\) or \([b_{\infty}^{a,b}(B^{(m)})]^{\alpha}\) if and only if \(G^{a,b}\in (c_{0}:\ell_{1})\), \(G^{a,b}\in(c:\ell_{1})\) or \(G^{a,b}\in(\ell_{\infty }:\ell_{1})\) by Parts (i) of Lemma 3.4. So we obtain

$$u=(u_{k})\in\bigl[b_{0}^{a,b}\bigl(B^{(m)} \bigr)\bigr]^{\alpha }=\bigl[b_{c}^{a,b} \bigl(B^{(m)}\bigr)\bigr]^{\alpha} =\bigl[b_{\infty}^{a,b} \bigl(B^{(m)}\bigr)\bigr]^{\alpha} $$

if and only if

$$\sup_{K\in\Gamma}\sum_{k}\left \vert \sum_{i\in K}(a+b)^{i}\sum _{j=i}^{k}\left ( \begin{matrix} m+k-j-1\\ k-j \end{matrix} \right )\left ( \begin{matrix} j\\ i \end{matrix} \right )\frac{(-s)^{k-j}}{r^{m+k-j}}b^{-j}(-a)^{j-i}u_{k} \right \vert < \infty. $$

Thus, we have \([b_{0}^{a,b}(B^{(m)})]^{\alpha }=[b_{c}^{a,b}(B^{(m)})]^{\alpha} =[b_{\infty}^{a,b}(B^{(m)})]^{\alpha }=U^{a,b}_{1}\). □

Now, we define the sets \(U_{2}^{a,b}\), \(U_{3}^{a,b}\), \(U_{4}^{a,b}\) and \(U_{5}^{a,b}\) by

$$\begin{aligned}& U_{2}^{a,b}=\biggl\{ u=(u_{k})\in w: \sup _{n\in\mathbb{N}}\sum_{k}\vert u_{n,k}\vert < \infty\biggr\} , \\& U_{3}^{a,b}=\Bigl\{ u=(u_{k})\in w: \lim _{n\rightarrow\infty} u_{n,k} \text{ exists for each } k \in\mathbb{N} \Bigr\} , \\& U_{4}^{a,b}=\biggl\{ u=(u_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k}\vert u_{n,k}\vert =\sum_{k}\Bigl\vert \lim _{n\rightarrow\infty}u_{n,k}\Bigr\vert \biggr\} , \end{aligned}$$

and

$$\begin{aligned}& U_{5}^{a,b}=\biggl\{ u=(u_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k}u_{n,k} \text{ exists}\biggr\} , \end{aligned}$$

where

$$\begin{aligned} u_{n,k}=(a+b)^{k}\sum_{i=k}^{n} \sum_{j=k}^{i}\left ( \begin{matrix} m+i-j-1\\ i-j \end{matrix} \right )\left ( \begin{matrix} j\\ k \end{matrix} \right )\frac{(-s)^{i-j}}{r^{m+i-j}}(-a)^{j-k}(b)^{-j}u_{i}. \end{aligned}$$

Theorem 3.6

We have the following relations:

  1. (i)

    \([b_{0}^{a,b}(B^{(m)})]^{ \beta}=U_{2}^{a,b}\cap U_{3}^{a,b}\),

  2. (ii)

    \([b_{c}^{a,b}(B^{(m)})]^{ \beta}=U_{2}^{a,b}\cap U_{3}^{a,b}\cap U_{5}^{a,b}\),

  3. (iii)

    \([b_{\infty}^{a,b}(B^{(m)})]^{ \beta}=U_{3}^{a,b}\cap U_{4}^{a,b}\).

Proof

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

$$\begin{aligned} \sum_{k=0}^{n}u_{k}x_{k} =& \sum_{k=0}^{n}u_{k}\left[\sum _{i=0}^{k}(a+b)^{i}\sum _{j=i}^{k}\left ( \begin{matrix} m+k-j-1\\ k-j \end{matrix} \right ) \left ( \begin{matrix} j\\ i \end{matrix} \right )\frac{(-s)^{k-j}}{r^{m+k-j}}(-a)^{j-i}b^{-j}y_{i} \right] \\ =&\sum_{k=0}^{n}\left[(a+b)^{k} \sum_{i=k}^{n}\sum _{j=k}^{i}\left ( \begin{matrix} m+i-j-1\\ i-j \end{matrix} \right )\left ( \begin{matrix} j\\ k \end{matrix} \right )\frac{(-s)^{i-j}}{r^{m+i-j}}(-a)^{j-k}b^{-j}u_{i} \right]y_{k} \\ =&\bigl(U^{a,b}y\bigr)_{n}, \end{aligned}$$

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

$$u_{n,k}^{a,b}= \textstyle\begin{cases} (a+b)^{k}\sum_{i=k}^{n}\sum_{j=k}^{i}\bigl( {\scriptsize\begin{matrix}{} m+i-j-1\cr i-j \end{matrix}} \bigr)\bigl( {\scriptsize\begin{matrix}{} j\cr k \end{matrix}} \bigr)\frac{(-s)^{i-j}}{r^{m+i-j}}(-a)^{j-k}(b)^{-j}u_{i}& \text{if $0\leq k \leq n$},\\ 0& \text{if $k> n$}. \end{cases} $$

Therefore, we deduce that \(ux= (u_{k}x_{k})\in c\) whenever \(x\in b_{0}^{a,b}(B^{(m)})\) if and only if \(U^{a,b}y\in c\) whenever \(y\in c_{0}\), which implies that \(u=(u_{k})\in[b_{0}^{a,b}(B^{(m)})]^{ \beta}\) if and only if \(U^{a,b}\in(c_{0}:c)\) by Part (ii) of Lemma 3.4. So we obtain \([b_{0}^{a,b}(B^{(m)})]^{ \beta}=U_{2}^{a,b}\cap U_{3}^{a,b}\). Using Parts (iii), (iv) instead of Part (ii) of Lemma 3.4, the proof can be completed in a similar way. □

Similarly, we give the following theorem without proof.

Theorem 3.7

The γ-dual of the spaces \(b_{0}^{a,b}(B^{(m)})\), \(b_{c}^{a,b}(B^{(m)})\) and \(b_{\infty}^{a,b}(B^{(m)})\) is the set \(U_{2}^{a,b}\).

4 Certain matrix mappings on the space \(b_{c}^{a,b}(B^{(m)})\)

In this section, we characterize matrix transformations from \(b_{c}^{a,b}(B^{(m)})\) into \(\ell_{p}\), \(\ell_{\infty}\) and c. Let us define the matrix \(\Theta=(\theta_{n,k})\) via an infinite matrix \(\Lambda=(\lambda_{n,k})\) by \(\Theta=\Lambda(B^{a,b}(B^{(m)}))^{-1}\), that is,

$$ \theta_{n,k}=(a+b)^{k}\sum _{j=k}^{\infty} \left ( \begin{matrix} m+n-j-1\\ n-j \end{matrix} \right ) \left ( \begin{matrix} j\\ k \end{matrix} \right )\frac{(-s)^{n-j}}{r^{m+n-j}}(-a)^{j-k}b^{-j} \lambda_{n,j}, $$
(4.1)

where \((B^{a,b}(B^{(m)}))^{-1}\) is the inverse of the \(B^{a,b}(B^{(m)})\)-transform. We now give the following lemmas.

Lemma 4.1

Let Z be any given sequence space and the entries of the matrices \(\Lambda=(\lambda_{n,k})\) and \(\Theta=(\theta_{n,k})\) are connected with equation (4.1). If \((\lambda_{n,k})_{k}\in [b_{c}^{a,b}(B^{(m)})]^{ \beta}\) for all \(n\in\mathbb{N}\), then \(\Lambda\in(b_{c}^{a,b}(B^{(m)}): Z)\) if and only if \(\Theta\in(c:Z)\).

Proof

Let \(\Lambda\in(b_{c}^{a,b}(B^{(m)}): Z)\) and \(y=(y_{n})\in c\). For every \(x=(x_{k})\in b_{c}^{a,b}(B^{(m)})\), we have \(x_{k}=[(B^{a,b}(B^{(m)}))^{-1}y_{n}]_{k}\). Since \((\lambda _{n,k})_{k}\in[b_{c}^{a,b}(B^{(m)})]^{ \beta}\) for all \(n\in\mathbb {N}\), this implies the existence of the Λ-transform of x, i.e. Λx exists. So we obtain \(\Lambda x=\Lambda (B^{a,b}(B^{(m)}))^{-1}y=\Theta y\), which implies that \(\Theta\in(c:Z)\).

Conversely, let \(\Theta\in(c:Z)\) and \(x\in b_{c}^{a,b}(B^{(m)})\). For every \(y\in c\), we have \(y_{n}=[B^{a,b}(B^{(m)}x_{k})]_{n}\). Since \((\lambda_{n,k})_{k}\in[b_{c}^{a,b}(B^{(m)})]^{ \beta}\) for all \(n\in \mathbb{N}\), this implies that Θy exists, which can be proved in a similar way to the proof of Theorem 3.6. So we have \(\Theta y=\Theta B^{a,b}(B^{(m)})x=\Lambda x\), which shows that \(\Lambda\in (b_{c}^{a,b}(B^{(m)}):Z)\). □

Lemma 4.2

[28]

Let \(A=(a_{n,k})\) be an infinite matrix. Then the following statement holds: \(A\in(c:\ell_{p})\) if and only if

$$ \sup_{K\in\Gamma}\sum_{n} \biggl\vert \sum_{k\in K}a_{n,k}\biggr\vert ^{p}< \infty,\quad 1\leq p< \infty. $$
(4.2)

For brevity of notation, we write

$$\begin{aligned}& t_{n,k}=(a+b)^{k}\sum_{i=k}^{n} \sum_{j=k}^{i}\left ( \begin{matrix} m+i-j-1\\ i-j \end{matrix} \right )\left ( \begin{matrix} j\\ k \end{matrix} \right )\frac{(-s)^{i-j}}{r^{m+i-j}}(-a)^{j-k}(b)^{-j}a_{n,j}, \\& t_{n,k}^{l}=(a+b)^{k}\sum _{i=k}^{l}\sum_{j=k}^{i} \left ( \begin{matrix} m+i-j-1\\ i-j \end{matrix} \right )\left ( \begin{matrix} j\\ k \end{matrix} \right ) \frac{(-s)^{i-j}}{r^{m+i-j}}(-a)^{j-k}(b)^{-j}a_{n,j} \end{aligned}$$

for all \(n,k\in\mathbb{N}\).

By using Lemma 4.1, there are some immediate consequences with \(t_{n,k}\) or \(t_{n,k}^{l}\) in place of \(a_{n,k}\) in Lemma 3.4 and Lemma 4.2.

Theorem 4.3

\(A\in(b_{c}^{a,b}(B^{(m)}):\ell_{p})\) if and only if

$$\begin{aligned}& \sup_{K\in\Gamma}\sum_{n} \biggl\vert \sum_{k\in K}t_{n,k}\biggr\vert ^{p}< \infty, \end{aligned}$$
(4.3)
$$\begin{aligned}& t_{n,k}\quad \textit{exists for each } k,n\in\mathbb{N}, \end{aligned}$$
(4.4)
$$\begin{aligned}& \sum_{k}t_{n,k}\quad \textit{converges for each } n\in\mathbb{N}, \end{aligned}$$
(4.5)
$$\begin{aligned}& \sup_{l\in\mathbb{N}}\sum_{k=0}^{l} \bigl\vert t_{n,k}^{l}\bigr\vert < \infty,\quad n\in \mathbb{N}. \end{aligned}$$
(4.6)

Theorem 4.4

\(A\in(b_{c}^{a,b}(B^{(m)}):\ell_{\infty})\) if and only if (4.4) and (4.6) hold, and

$$ \sup_{n\in\mathbb{N}}\sum_{k} \vert t_{n,k}\vert < \infty. $$
(4.7)

Theorem 4.5

\(A\in(b_{c}^{a,b}(B^{(m)}):c)\) if and only if (4.4), (4.6) and (4.7) hold, and

$$\begin{aligned}& \lim_{n\rightarrow\infty} t_{n,k} \quad\textit{exists for each } k\in\mathbb {N}, \end{aligned}$$
(4.8)
$$\begin{aligned}& \lim_{n\rightarrow\infty} \sum_{k}t_{n,k} \quad\textit{exists}. \end{aligned}$$
(4.9)

5 Conclusion

By considering the definitions of the binomial matrix \(B^{a,b}=(b^{a,b}_{n,k})\) and the difference operator, we introduce the sequence spaces \(b_{0}^{a,b}(B^{(m)})\), \(b_{c}^{a,b}(B^{(m)})\) and \(b_{\infty}^{a,b}(B^{(m)})\). These spaces are the natural continuation of [18, 23–25]. Our results are the generalization of the matrix 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 [29] is added to the list of references.