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# On some binomial $$B^{(m)}$$-difference sequence spaces

Journal of Inequalities and Applications20172017:194

https://doi.org/10.1186/s13660-017-1470-4

• Received: 15 May 2017
• Accepted: 8 August 2017
• Published:

## Abstract

In this paper, we introduce the binomial sequence spaces $$b^{a,b}_{0}(B^{(m)})$$, $$b^{a,b}_{c}(B^{(m)})$$ and $$b^{a,b}_{\infty}(B^{(m)})$$ 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 $$b_{c}^{a,b}(B^{(m)})$$.

## Keywords

• sequence space
• matrix domain
• Schauder basis
• α-, β- and γ-duals

## 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  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  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  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 . Moreover, Altay and Polat , Başarir and Kara , Başarir, Kara and Konca , Kara , Kara and İlkhan [16, 17], Polat and Başar , Song and Meng  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  and Altay, Başar and Mursaleen  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  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  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  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  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 . 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 . 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 . 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 , Theorem 2.1 and , Theorem 2.1). Moreover, $$B^{(m)}$$ is a triangle matrix and (2.1) holds. By using Theorem 4.3.12 of Wilansky , 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 $$Schauder$$ $$basis$$  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  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



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



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, 2325]. 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  is added to the list of references.

## Declarations

### Acknowledgements

We wish to thank the referee for his/her constructive comments and suggestions. 