Theory and Modern Applications

# Binomial difference sequence spaces of order m

## Abstract

In this paper, we introduce the binomial sequence spaces $$b^{r,s}_{0}(\nabla^{(m)})$$, $$b^{r,s}_{c}(\nabla^{(m)})$$ and $$b^{r,s}_{\infty}(\nabla^{(m)})$$ by combining the binomial transformation and mth order difference operator. We prove the BK-property and some inclusion relations. Also, we obtain the Schauder bases and compute the α-, β- and γ-duals of these sequence spaces.

## Introduction and preliminaries

Let w denote the space of all sequences. By $$\ell_{\infty}$$, c and $$c_{0}$$, we denote the spaces of bounded, convergent and null sequences, respectively. We write bs, cs and $$\ell_{p}$$ for the spaces of all bounded, convergent and p-absolutely summable series, respectively; $$1\leq p<\infty$$. 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. It is well known 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:

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

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

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

for $$Z\in\{\ell_{\infty},c,c_{0}\}$$, where $$m\in\mathbb{N}$$, $$\Delta ^{0}x_{k}=x_{k}$$, $$\Delta^{m} x_{k}=\Delta^{m-1}x_{k}-\Delta ^{m-1}x_{k+1}$$ for each $$k\in\mathbb{N}$$, which is equivalent to the binomial representation $$\Delta^{m} x_{k}=\sum_{i=0}^{m}(-1)^{i}\binom {m}{i}x_{k+i}$$. Since then, many authors have studied further generalization of the difference sequence spaces [48]. Moreover, Altay and Polat [9], Başarir [10], Başarir, Kara and Konca [11], Başarir and Kara [1217], Başarir, Öztürk and Kara [18], Polat and Başarir [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 $$(Ax)_{n}=\sum_{k=0}^{\infty }a_{n,k}x_{k}$$ and is supposed to be convergent for all $$n\in\mathbb {N}$$. For two sequence spaces X, 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. 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

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

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}=\Biggl\{ x=(x_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k=0}^{n} \binom{n}{k}(1-r)^{n-k}r^{k}x_{k}=0\Biggr\} , \\& e^{r}_{c}=\Biggl\{ x=(x_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k=0}^{n} \binom{n}{k}(1-r)^{n-k}r^{k}x_{k} \text{ exists}\Biggr\} , \end{aligned}

and

$$e^{r}_{\infty}=\Biggl\{ x=(x_{k})\in w: \sup _{n\in\mathbb{N}} \Biggl\vert \sum_{k=0}^{n} \binom{n}{k}(1-r)^{n-k}r^{k}x_{k} \Biggr\vert < \infty\Biggr\} .$$

Altay and Polat [9] 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 [19] employed the technique matrix domain of 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

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

for all $$k,n,m\in\mathbb{N}$$.

Recently Bişgin [22, 23] defined another 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

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

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

1. (i)

$$\| B^{r,s}\|<\infty$$,

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 defined the following spaces of binomial sequences:

\begin{aligned}& b^{r,s}_{0}=\Biggl\{ x=(x_{k})\in w: \lim _{n\rightarrow\infty}\frac {1}{(s+r)^{n}}\sum_{k=0}^{n} \binom{n}{k}s^{n-k}r^{k}x_{k}=0\Biggr\} , \\& b^{r,s}_{c}=\Biggl\{ x=(x_{k})\in w: \lim _{n\rightarrow\infty}\frac {1}{(s+r)^{n}}\sum_{k=0}^{n} \binom{n}{k}s^{n-k}r^{k}x_{k} \mbox{ exists} \Biggr\} , \end{aligned}

and

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

The purpose of the present paper is to study the difference spaces $$b^{r,s}_{0}(\nabla^{(m)})$$, $$b^{r,s}_{c}(\nabla^{(m)})$$ and $$b^{r,s}_{\infty}(\nabla^{(m)})$$ of the binomial sequence whose $$B^{r,s}(\nabla^{(m)})$$-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 [22, 23] and [19]. Also, we give some inclusion relations and compute the bases and α-, β- and γ-duals of these sequence spaces.

## The binomial difference sequence spaces

In this section, we introduce the spaces $$b^{r,s}_{0}(\nabla^{(m)})$$, $$b^{r,s}_{c}(\nabla^{(m)})$$, $$b^{r,s}_{\infty}(\nabla^{(m)})$$ and prove the BK-property and inclusion relations.

We first define the binomial difference sequence spaces $$b^{r,s}_{0}(\nabla^{(m)})$$, $$b^{r,s}_{c}(\nabla^{(m)})$$ and $$b^{r,s}_{\infty}(\nabla^{(m)})$$ by

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

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

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

It is obvious that the sequence spaces $$b^{r,s}_{0}(\nabla^{(m)})$$, $$b^{r,s}_{c}(\nabla^{(m)})$$ and $$b^{r,s}_{\infty}(\nabla^{(m)})$$ may be reduced to some sequence spaces in the special cases of s, r and $$m\in\mathbb{N}$$. For instance, we take $$m=0$$, then obtain the spaces $$b^{r,s}_{0}$$, $$b^{r,s}_{c}$$ and $$b^{r,s}_{\infty}$$ defined by Bişgin [22, 23]. On taking $$s+r=1$$, 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 [19].

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

$$y_{n}=\bigl[B^{r,s}\bigl(\nabla^{(m)} x_{k}\bigr)\bigr]_{n}=\frac{1}{(s+r)^{n}}\sum _{k=0}^{n}\binom{n}{k}s^{n-k}r^{k} \bigl(\nabla^{(m)} x_{k}\bigr)$$
(2.2)

for each $$n\in\mathbb{N}$$, where

$$\nabla^{(m)} x_{k}=\sum_{i=0}^{m}(-1)^{i} \binom{m}{i}x_{k-i}=\sum_{i=\max\{0,k-m\}}^{m}(-1)^{k-i} \binom{m}{k-i}x_{i}.$$

Then the binomial difference sequence spaces $$b^{r,s}_{0}(\nabla ^{(m)})$$, $$b^{r,s}_{c}(\nabla^{(m)})$$ and $$b^{r,s}_{\infty}(\nabla ^{(m)})$$ can be redefined by all sequences whose $$B^{r,s}(\nabla ^{(m)})$$-transforms are in the spaces $$c_{0}$$, c and $$\ell_{\infty}$$.

### Theorem 2.1

Let $$Z\in\{b^{r,s}_{0}, b^{r,s}_{c}, b^{r,s}_{\infty}\}$$. Then $$Z(\nabla^{(m)})$$ is a BK-space with the norm $$\| x\|_{Z(\nabla^{(m)})}=\|(\nabla^{(m)} x_{k})\|_{Z}$$.

### Proof

The sequence spaces $$b^{r,s}_{0}$$, $$b^{r,s}_{c}$$ and $$b^{r,s}_{\infty}$$ are BK-spaces (see [22], Theorem 2.1 and [23], Theorem 2.1). Moreover, $$\nabla^{(m)}$$ is a triangle matrix and (2.1) holds. By using Theorem 4.3.12 of Wilansky [24], we deduce that the binomial sequence spaces $$b^{r,s}_{0}(\nabla^{(m)})$$, $$b^{r,s}_{c}(\nabla^{(m)})$$ and $$b^{r,s}_{\infty}(\nabla^{(m)})$$ are BK-spaces. □

### Theorem 2.2

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

### Proof

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

Consider $$T:b^{r,s}_{0}(\nabla^{(m)})\rightarrow c_{0}$$ by $$T(x)=B^{r,s}(\nabla^{(m)} x_{k})$$. 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

$$x_{k}=\sum_{i=0}^{k}(s+r)^{i} \sum_{j=i}^{k}\binom{m+k-j-1}{ k-j} \binom{j}{i}r^{-j}(-s)^{j-i}y_{i}$$
(2.3)

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

$$\lim_{n\rightarrow\infty}\bigl[B^{r,s}\bigl(\nabla^{(m)} x_{k}\bigr)\bigr]_{n}=\lim_{n\rightarrow\infty} \frac{1}{(s+r)^{n}}\sum_{k=0}^{n} \binom{n}{k} s^{n-k}r^{k}\bigl(\nabla^{(m)} x_{k}\bigr)=\lim_{n\rightarrow\infty}y_{n}=0,$$

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

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 $$b^{r,s}_{0}(\nabla^{(m)})\subseteq b^{r,s}_{c}(\nabla ^{(m)})\subseteq b^{r,s}_{\infty}(\nabla^{(m)})$$ holds.

### Theorem 2.4

The inclusions $$b^{r,s}_{0}(\nabla^{(m)})\subseteq b^{r,s}_{0}(\nabla ^{(m+1)})$$, $$b^{r,s}_{c}(\nabla^{(m)})\subseteq b^{r,s}_{c}(\nabla ^{(m+1)})$$ and $$b^{r,s}_{\infty}(\nabla^{(m)})\subseteq b^{r,s}_{\infty }(\nabla^{(m+1)})$$ hold.

### Proof

Let $$x=(x_{k})\in b^{r,s}_{0}(\nabla^{(m)})$$, then the inequality

\begin{aligned} \bigl\vert \bigl[B^{r,s}\bigl(\nabla^{(m+1)} x_{k} \bigr)\bigr]_{n} \bigr\vert =& \bigl\vert \bigl[B^{r,s} \bigl(\nabla ^{(m)}(\nabla x_{k})\bigr)\bigr]_{n} \bigr\vert \\ =& \bigl\vert \bigl[B^{r,s}\bigl(\nabla^{(m)} x_{k}\bigr)\bigr]_{n}-\bigl[B^{r,s}\bigl( \nabla^{(m)} x_{k}\bigr)\bigr]_{n-1} \bigr\vert \\ \leq& \bigl\vert \bigl[B^{r,s}\bigl(\nabla^{(m)} x_{k}\bigr)\bigr]_{n} \bigr\vert + \bigl\vert \bigl[B^{r,s}\bigl(\nabla ^{(m)} x_{k}\bigr) \bigr]_{n-1} \bigr\vert \end{aligned}

holds and tends to 0 as $$n\rightarrow\infty$$, which implies that $$x\in b^{r,s}_{0}(\nabla^{(m+1)})$$. □

### Theorem 2.5

The inclusions $$e_{0}^{r}(\nabla^{(m)})\subseteq b^{r,s}_{0}(\nabla ^{(m)})$$, $$e_{c}^{r}(\nabla^{(m)})\subseteq b^{r,s}_{c}(\nabla^{(m)})$$ and $$e_{\infty}^{r}(\nabla^{(m)})\subseteq b^{r,s}_{\infty}(\nabla ^{(m)})$$ strictly hold.

### Proof

Similarly, we only prove the inclusion $$e_{0}^{r}(\nabla ^{(m)})\subseteq b^{r,s}_{0}(\nabla^{(m)})$$. If $$r+s=1$$, we have $$E^{r}=B^{r,s}$$. So $$e_{0}^{r}(\nabla^{(m)})\subseteq b^{r,s}_{0}(\nabla ^{(m)})$$ holds. Take $$0< r<1$$ and $$s=4$$. We define a sequence $$x=(x_{k})$$ by

$$x_{k}=\sum_{j=0}^{k} \binom{m+k-j-1}{ k-j} \biggl(-\frac{3}{r}\biggr)^{j}$$

for all $$m, k\in\mathbb{N}$$. It is clear that $$[E^{r}(\nabla^{(m)} x_{k})]_{n}=((-2-r)^{n})\notin c_{0}$$ and $$[B^{r,s}(\nabla^{(m)} x_{k})]_{n}=((\frac{1}{4+r})^{n})\in c_{0}$$. So, we have $$x\in b^{r,s}_{0}(\nabla^{(m)})\setminus e_{0}^{r}(\nabla^{(m)})$$. This shows that the inclusion $$e_{0}^{r}(\nabla^{(m)})\subseteq b^{r,s}_{0}(\nabla ^{(m)})$$ strictly holds. □

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

For a normed space $$(X, \|\cdot\|)$$, 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 an unique scalar sequence $$(\lambda_{k})$$ such that $$\| x-\sum_{k=0}^{n}\lambda _{k}x_{k}\|\rightarrow0$$ as $$n\rightarrow\infty$$. We shall construct the Schauder bases for the sequence spaces $$b_{0}^{r,s}(\nabla ^{(m)})$$ and $$b_{c}^{r,s}(\nabla^{(m)})$$.

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

$$g^{(k)}_{i}(r,s)= \textstyle\begin{cases} 0& \text{if }0\leq i < k, \\ (s+r)^{k}\sum_{j=k}^{i}\binom{m+i-j-1}{i-j}\binom{j}{k} r^{-j}(-s)^{j-k}& \text{if }i\geq k, \end{cases}$$

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 space $$b_{0}^{r,s}(\nabla^{(m)})$$ and every $$x=(x_{i})\in b_{0}^{r,s}(\nabla^{(m)})$$ has an unique representation by

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

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

### Proof

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

For $$x \in b_{0}^{r,s}(\nabla^{(m)})$$ and $$n\in\mathbb{N}$$, we put

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

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

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

and

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

for each $$k\in\mathbb{N}$$.

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

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

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

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

which implies $$x \in b_{0}^{r,s}(\nabla^{(m)})$$ is represented as in (3.1).

To show the uniqueness of this representation, we assume that

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

Then we have

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

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

### Theorem 3.2

We define $$g=(g_{n})$$ by

$$g_{n}=\sum_{k=0}^{n}(s+r)^{k} \sum_{j=k}^{n}\binom{m+n-j-1}{n-j} \binom{j}{k}r^{-j}(-s)^{j-k}$$

for all $$n\in\mathbb{N}$$ and $$\lim_{k\rightarrow\infty}\lambda _{k}(r,s)=l$$. The set $$\{g, g^{(0)}(r,s), g^{(1)}(r,s),\ldots ,g^{(k)}(r,s),\ldots\}$$ is a Schauder basis for the space $$b_{c}^{r,s}(\nabla^{(m)})$$ and every $$x\in b_{c}^{r,s}(\nabla^{(m)})$$ has an unique representation by

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

### Proof

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

From Theorem 2.1, we know that $$b_{0}^{r,s}(\nabla^{(m)})$$ and $$b_{c}^{r,s}(\nabla^{(m)})$$ are Banach spaces. By combining this fact with Theorem 3.1 and Theorem 3.2, we can give the following corollary.

### Corollary 3.3

The sequence spaces $$b_{0}^{r,s}(\nabla^{(m)})$$ and $$b_{c}^{r,s}(\nabla ^{(m)})$$ are separable.

Köthe and Toeplitz [25] first computed the dual whose elements can be represented as sequences and defined the α-dual (or Köthe-Toeplitz dual). Chandra and Tripathy [26] generalized the notion of Köthe-Toeplitz dual of sequence spaces. Next, we compute the α-, β- and γ-duals of the sequence spaces $$b_{0}^{r,s}(\nabla^{(m)})$$, $$b_{c}^{r,s}(\nabla^{(m)})$$ and $$b_{\infty }^{r,s}(\nabla^{(m)})$$.

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

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

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

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

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 \mbox{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

[27]

Let $$A=(a_{n,k})$$ be an infinite matrix, then:

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 $$b_{0}^{r,s}(\nabla^{(m)})$$, $$b_{c}^{r,s}(\nabla^{(m)})$$ and $$b_{\infty}^{r,s}(\nabla^{(m)})$$ is the set

$$U^{r,s}_{1}=\Biggl\{ u=(u_{k})\in w:\sup _{K\in\Gamma}\sum_{k} \Biggl\vert \sum _{i\in K} (s+r)^{i}\sum _{j=i}^{k}\binom{m+k-j-1}{ k-j}\binom{j}{i}r^{-j}(-s)^{j-i}u_{k} \Biggr\vert < \infty\Biggr\} .$$

### Proof

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

$$u_{k}x_{k}=\sum_{i=0}^{k}(s+r)^{i} \sum_{j=i}^{k}\binom{m+k-j-1}{ k-j} \binom{j}{i}r^{-j}(-s)^{j-i}u_{k}y_{i}= \bigl(G^{r,s}y\bigr)_{k}$$

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

$$g^{r,s}_{k,i}= \textstyle\begin{cases} (s+r)^{i}\sum_{j=i}^{k}\binom{m+k-j-1}{k-j} \binom{j}{i} r^{-j}(-s)^{j-i}u_{k}& \text{if }0\leq i\leq k, \\ 0& \text{if }i>k. \end{cases}$$

Therefore, we deduce that $$ux= (u_{k}x_{k})\in\ell_{1}$$ whenever $$x\in b_{0}^{r,s}(\nabla^{(m)})$$, $$b_{c}^{r,s}(\nabla^{(m)})$$ or $$b_{\infty }^{r,s}(\nabla^{(m)})$$ if and only if $$G^{r,s}y\in\ell_{1}$$ whenever $$y\in c_{0}, c$$ or $$\ell _{\infty}$$, which implies that $$u=(u_{k})\in[b_{0}^{r,s}(\nabla ^{(m)})]^{\alpha}, [b_{c}^{r,s}(\nabla^{(m)})]^{\alpha} \mbox{ or } [b_{\infty}^{r,s}(\nabla^{(m)})]^{\alpha}$$ if and only if $$G^{r,s}\in (c_{0}:\ell_{1})=(c:\ell_{1})=(\ell_{\infty}:\ell_{1})$$. By Lemma 3.4(i), we obtain

$$u=(u_{k})\in\bigl[b_{0}^{r,s}\bigl( \nabla^{(m)}\bigr)\bigr]^{\alpha}=\bigl[b_{c}^{r,s} \bigl(\nabla ^{(m)}\bigr)\bigr]^{\alpha} =\bigl[b_{\infty}^{r,s} \bigl(\nabla^{(m)}\bigr)\bigr]^{\alpha}$$

if and only if

$$\sup_{K\in\Gamma}\sum_{k} \Biggl\vert \sum_{i\in K}(s+r)^{i}\sum _{j=i}^{k}\binom{m+k-j-1}{k-j} \binom{j}{i}r^{-j}(-s)^{j-i}u_{k} \Biggr\vert < \infty.$$

Thus, we have $$[b_{0}^{r,s}(\nabla^{(m)})]^{\alpha}=[b_{c}^{r,s}(\nabla ^{(m)})]^{\alpha} =[b_{\infty}^{r,s}(\nabla^{(m)})]^{\alpha}=U^{r,s}_{1}$$. □

Now, we define the sets $$U_{2}^{r,s}$$, $$U_{3}^{r,s}$$, $$U_{4}^{r,s}$$ and $$U_{5}^{r,s}$$ by

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

$$U_{5}^{r,s}=\biggl\{ u=(u_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k}u_{n,k} \mbox{ exists}\biggr\} ,$$

where

$$u_{n,k}=(s+r)^{k}\sum_{i=k}^{n} \sum_{j=k}^{i}\binom{m+i-j-1}{ i-j} \binom{j}{k} r^{-j}(-s)^{j-k}u_{i}.$$

### Theorem 3.6

The following equations hold:

1. (i)

$$[b_{0}^{r,s}(\nabla^{(m)})]^{ \beta}=U_{2}^{r,s}\cap U_{3}^{r,s}$$,

2. (ii)

$$[b_{c}^{r,s}(\nabla^{(m)})]^{ \beta}=U_{2}^{r,s}\cap U_{3}^{r,s}\cap U_{5}^{r,s}$$,

3. (iii)

$$[b_{\infty}^{r,s}(\nabla^{(m)})]^{ \beta}=U_{3}^{r,s}\cap U_{4}^{r,s}$$.

### Proof

Since the proof may be obtained in the same way for (ii) and (iii), we only prove (i). 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}\Biggl[\sum _{i=0}^{k}(s+r)^{i}\sum _{j=i}^{k}\binom{m+k-j-1}{ k-j}\binom{j}{i} r^{-j}(-s)^{j-i}y_{i}\Biggr] \\ =&\sum_{k=0}^{n}\Biggl[(s+r)^{k} \sum_{i=k}^{n}\sum _{j=k}^{i}\binom {m+i-j-1}{ i-j} \binom{j}{k} r^{-j}(-s)^{j-k}u_{i}\Biggr]y_{k} \\ =&\bigl(U^{r,s}y\bigr)_{n}, \end{aligned}

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

$$u_{n,k}= \textstyle\begin{cases} (s+r)^{k}\sum_{i=k}^{n}\sum_{j=k}^{i}\binom{m+i-j-1}{i-j} \binom{j}{k} r^{-j}(-s)^{j-k}u_{i}& \text{if }0\leq k \leq n, \\ 0& \text{if }k> n. \end{cases}$$

Therefore, we deduce that $$ux= (u_{k}x_{k})\in cs$$ whenever $$x\in b_{0}^{r,s}(\nabla^{(m)})$$ if and only if $$U^{r,s}y\in c$$ whenever $$y\in c_{0}$$, which implies that $$u=(u_{k})\in[b_{0}^{r,s}(\nabla^{(m)})]^{ \beta}$$ if and only if $$U^{r,s}\in(c_{0}:c)$$. By Lemma 3.4(ii), we obtain $$[b_{0}^{r,s}(\nabla^{(m)})]^{ \beta}=U_{2}^{r,s}\cap U_{3}^{r,s}$$. □

### Theorem 3.7

The γ-dual of the spaces $$b_{0}^{r,s}(\nabla^{(m)})$$, $$b_{c}^{r,s}(\nabla^{(m)})$$ and $$b_{\infty}^{r,s}(\nabla^{(m)})$$ is the set $$U_{2}^{r,s}$$.

### Proof

Using Lemma 3.4(v) instead of (ii), the proof can be given in a similar way. So, we omit the details. □

## Conclusion

By considering the definitions of the binomial matrix $$B^{r,s}=(b^{r,s}_{n,k})$$ and mth order difference operator, we introduce the sequence spaces $$b_{0}^{r,s}(\nabla^{(m)})$$, $$b_{c}^{r,s}(\nabla^{(m)})$$ and $$b_{\infty}^{r,s}(\nabla^{(m)})$$. These spaces are the natural continuation of [3, 19, 22, 23]. Our results are the generalization of the matrix domain of the Euler matrix of order r.

## References

1. Choudhary, B, Nanda, S: Functional Analysis with Applications. Wiley, New Delhi (1989)

2. Kizmaz, H: On certain sequence spaces. Can. Math. Bull. 24, 169-176 (1981)

3. Et, M, Colak, R: On generalized difference sequence spaces. Soochow J. Math. 21, 377-386 (1995)

4. Bektas, C, Et, M, Colak, R: Generalized difference sequence spaces and their dual spaces. J. Math. Anal. Appl. 292, 423-432 (2004)

5. Dutta, H: Characterization of certain matrix classes involving generalized difference summability spaces. Appl. Sci. 11, 60-67 (2009)

6. Reddy, BS: On some generalized difference sequence spaces. Soochow J. Math. 26, 377-386 (2010)

7. Tripathy, BC, Esi, A: A new type of difference sequence spaces. Int. J. Sci. Technol. 1, 147-155 (2006)

8. Tripathy, BC, Sen, M: Characterization of some matrix classes involving paranormed sequence spaces. Tamkang J. Math. 37, 155-162 (2006)

9. Altay, B, Polat, H: On some new Euler difference sequence spaces. Southeast Asian Bull. Math. 30, 209-220 (2006)

10. Başarir, M: On the generalized Riesz B-difference sequence spaces. Filomat 24, 35-52 (2010)

11. Başarir, M, Kara, EE, Konca, S: On some new weighted Euler sequence spaces and compact operators. Math. Inequal. Appl. 17, 649-664 (2014)

12. Başarir, M, Kara, EE: On compact operators on the Riesz $${B}^{m}$$-difference sequence spaces. Iran. J. Sci. Technol. 35, 279-285 (2011)

13. Başarir, M, Kara, EE: On some difference sequence spaces of weighted means and compact operators. Ann. Funct. Anal. 2, 114-129 (2011)

14. Başarir, M, Kara, EE: On compact operators on the Riesz $${B}^{m}$$-difference sequence spaces II. Iran. J. Sci. Technol. 33, 371-376 (2012)

15. Başarir, M, Kara, EE: On the B-difference sequence space derived by generalized weighted mean and compact operators. J. Math. Anal. Appl. 391, 67-81 (2012)

16. Başarir, M, Kara, EE: On the mth order difference sequence space of generalized weighted mean and compact operators. Acta Math. Sci. 33, 797-813 (2013)

17. Kara, EE, Başarir, M: On compact operators and some Euler $${B}^{(m)}$$-difference sequence spaces. J. Math. Anal. Appl. 379, 499-511 (2011)

18. Kara, EE, Öztürk, M, Başarir, M: Some topological and geometric properties of generalized Euler sequence space. Math. Slovaca 60, 385-398 (2010)

19. Polat, H, Başar, F: Some Euler spaces of difference sequences of order m. Acta Math. Sci. 27, 254-266 (2007)

20. Altay, B, Başar, F: On some Euler sequence spaces of nonabsolute type. Ukr. Math. J. 57, 1-17 (2005)

21. Altay, B, Başar, F, Mursaleen, M: On the Euler sequence spaces which include the spaces $$\ell _{p}$$ and $$\ell_{\infty}$$ I. Inf. Sci. 176, 1450-1462 (2006)

22. Bişgin, MC: The binomial sequence spaces of nonabsolute type. J. Inequal. Appl. 2016, 309 (2016)

23. Bişgin, MC: The binomial sequence spaces which include the spaces $$\ell _{p}$$ and $$\ell_{\infty}$$ and geometric properties. J. Inequal. Appl. 2016, 304 (2016)

24. Wilansky, A: Summability through Functional Analysis. North-Holland Mathematics Studies, vol. 85. Elsevier, Amsterdam (1984)

25. Köthe, G, Toeplitz, O: Linear Raume mit unendlich vielen koordinaten and Ringe unenlicher Matrizen. J. Reine Angew. Math. 171, 193-226 (1934)

26. Chandra, P, Tripathy, BC: On generalised Köthe-Toeplitz duals of some sequence spaces. Indian J. Pure Appl. Math. 33, 1301-1306 (2002)

27. Stieglitz, M, Tietz, H: Matrixtransformationen von Folgenräumen eine Ergebnisübersicht. Math. Z. 154, 1-16 (1977)

## Acknowledgements

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

## Author information

Authors

### Corresponding author

Correspondence to Meimei Song.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

JM came up with the main ideas and drafted the manuscript. MS revised the paper. All authors read and approved the final manuscript.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Meng, J., Song, M. Binomial difference sequence spaces of order m . Adv Differ Equ 2017, 241 (2017). https://doi.org/10.1186/s13662-017-1291-2