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Binomial difference sequence spaces of order m

Advances in Difference Equations20172017:241

  • Received: 4 May 2017
  • Accepted: 20 July 2017
  • Published:


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.


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

1 Introduction and preliminaries

Let w denote the space of all sequences. By \(\ell_{\infty}\), 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
$$ X_{A}=\bigl\{ x=(x_{k})\in w:Ax \in X\bigr\} , $$
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:
$$\begin{aligned}& e^{r}_{0}=\Biggl\{ x=(x_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k=0}^{n} \binom{n}{k}(1-r)^{n-k}r^{k}x_{k}=0\Biggr\} , \\& e^{r}_{c}=\Biggl\{ x=(x_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k=0}^{n} \binom{n}{k}(1-r)^{n-k}r^{k}x_{k} \text{ exists}\Biggr\} , \end{aligned}$$
$$ e^{r}_{\infty}=\Biggl\{ x=(x_{k})\in w: \sup _{n\in\mathbb{N}} \Biggl\vert \sum_{k=0}^{n} \binom{n}{k}(1-r)^{n-k}r^{k}x_{k} \Biggr\vert < \infty\Biggr\} . $$
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)


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

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.

2 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
$$ b^{r,s}_{0}\bigl(\nabla^{(m)}\bigr)= \bigl(b^{r,s}_{0}\bigr)_{\nabla^{(m)}},\qquad b^{r,s}_{c}\bigl(\nabla^{(m)}\bigr)= \bigl(b^{r,s}_{c}\bigr)_{\nabla^{(m)}},\qquad b^{r,s}_{\infty }\bigl(\nabla^{(m)}\bigr)= \bigl(b^{r,s}_{\infty}\bigr)_{\nabla^{(m)}}. $$

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


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.


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


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.


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. □

3 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
$$ x=\sum_{k} \lambda_{k}(r,s) g^{(k)}(r,s), $$
where \(\lambda_{k}(r,s)= [B^{r,s}(\nabla^{(m)} x_{i})]_{k}\) for each \(k\in\mathbb{N}\).


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
$$ B^{r,s}\bigl(\nabla^{(m)} x^{(n)}_{i}\bigr)= \sum_{k=0}^{n}\lambda _{k}(r,s)B^{r,s} \bigl(\nabla^{(m)} g^{(k)}_{i}(r,s)\bigr)=\sum _{k=0}^{n}\lambda _{k}(r,s)e_{k} $$
$$\bigl[B^{r,s}\bigl(\nabla^{(m)}\bigl(x_{i}-x_{i}^{(n)} \bigr)\bigr)\bigr]_{k}= \textstyle\begin{cases} 0& \text{if }0\leq k < n, \\ [B^{r,s}(\nabla^{(m)} x_{i})]_{k}& \text{if }k\geq n, \end{cases} $$
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
$$ x=lg+\sum_{k} \bigl[ \lambda_{k}(r,s)-l\bigr] g^{(k)}(r,s). $$


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
$$ X^{\alpha}=M(X,\ell_{1}),\qquad X^{\beta}=M(X,cs)\quad \mbox{and}\quad X^{\gamma}=M(X, bs), $$
Let us give the following properties:
$$\begin{aligned}& \sup_{K\in\Gamma} \sum_{n} \biggl\vert \sum_{k\in K} a_{n,k} \biggr\vert < \infty, \end{aligned}$$
$$\begin{aligned}& \sup_{n\in\mathbb{N}} \sum_{k} \vert a_{n,k} \vert < \infty, \end{aligned}$$
$$\begin{aligned}& \lim_{n\rightarrow\infty}a_{n,k}=a_{k}\quad \mbox{for each } k\in\mathbb {N}, \end{aligned}$$
$$\begin{aligned}& \lim_{n\rightarrow\infty}\sum_{k}a_{n,k}=a, \end{aligned}$$
$$\begin{aligned}& \lim_{n\rightarrow\infty}\sum_{k} \vert a_{n,k} \vert =\sum_{k} \Bigl\vert \lim_{n\rightarrow\infty}a_{n,k} \Bigr\vert , \end{aligned}$$
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:
  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
$$ U^{r,s}_{1}=\Biggl\{ u=(u_{k})\in w:\sup _{K\in\Gamma}\sum_{k} \Biggl\vert \sum _{i\in K} (s+r)^{i}\sum _{j=i}^{k}\binom{m+k-j-1}{ k-j}\binom{j}{i}r^{-j}(-s)^{j-i}u_{k} \Biggr\vert < \infty\Biggr\} . $$


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
$$\begin{aligned}& U_{2}^{r,s}=\biggl\{ u=(u_{k})\in w: \sup _{n\in\mathbb{N}}\sum_{k} \vert u_{n,k} \vert < \infty\biggr\} , \\& U_{3}^{r,s}=\Bigl\{ u=(u_{k})\in w: \lim _{n\rightarrow\infty} u_{n,k} \mbox{ exists for each } k \in\mathbb{N} \Bigr\} , \\& U_{4}^{r,s}=\biggl\{ u=(u_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k} \vert u_{n,k} \vert =\sum_{k} \Bigl\vert \lim_{n\rightarrow\infty}u_{n,k} \Bigr\vert \biggr\} , \end{aligned}$$
$$ U_{5}^{r,s}=\biggl\{ u=(u_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k}u_{n,k} \mbox{ exists}\biggr\} , $$
$$ u_{n,k}=(s+r)^{k}\sum_{i=k}^{n} \sum_{j=k}^{i}\binom{m+i-j-1}{ i-j} \binom{j}{k} r^{-j}(-s)^{j-k}u_{i}. $$

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)

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



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 \(U_{2}^{r,s}\).


Using Lemma 3.4(v) instead of (ii), the proof can be given 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 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.



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

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

Department of Mathematics, Tianjin University of Technology, Tianjin, 300000, P.R. China


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