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

Journal of Inequalities and Applications20172017:194

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


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 \(b_{c}^{a,b}(B^{(m)})\).


  • 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 [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 [47]. Moreover, Altay and Polat [8], Başarir and Kara [913], 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
$$ X_{A}=\bigl\{ x=(x_{k})\in w:Ax \in X\bigr\} , $$
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:
$$\begin{aligned}& e^{r}_{0}=\left\{x=(x_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k=0}^{n} \begin{pmatrix} n\\ k \end{pmatrix} (1-r)^{n-k}r^{k}x_{k}=0\right\}, \\& e^{r}_{c}=\left\{x=(x_{k})\in w: \lim _{n\rightarrow\infty}\sum_{k=0}^{n} \begin{pmatrix} n\\ k \end{pmatrix} (1-r)^{n-k}r^{k}x_{k} \text{ exists}\right\}, \end{aligned}$$
$$\begin{aligned}& e^{r}_{\infty}=\left\{x=(x_{k})\in w: \sup _{n\in\mathbb{N}}\left \vert \sum_{k=0}^{n} \left ( \begin{matrix} n\\ k \end{matrix} \right ) (1-r)^{n-k}r^{k}x_{k}\right \vert < \infty\right\}. \end{aligned}$$
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)


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

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

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,
$$ y_{n}=\bigl[B^{a,b}\bigl(B^{(m)} x_{k}\bigr)\bigr]_{n} =\frac{1}{(a+b)^{n}}\sum _{k=0}^{n}\sum_{i=k}^{n} \left ( \begin{matrix} m\\ i-k \end{matrix} \right )\left ( \begin{matrix} n\\ i \end{matrix} \right )a^{n-i}b^{i}r^{m+k-i}s^{i-k}x_{k} $$
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
$$\begin{aligned}& \Vert x\Vert _{b^{a,b}_{0}(B^{(m)})}=\Vert x\Vert _{b^{a,b}_{c}(B^{(m)})}=\Vert x \Vert _{b^{a,b}_{\infty }(B^{(m)})}=\sup_{n\in\mathbb{N}}\bigl\vert \bigl[B^{a,b}\bigl(B^{(m)} x_{k}\bigr) \bigr]_{n}\bigr\vert . \end{aligned}$$


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.


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


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
$$E^{a}\bigl(B^{(m)} x_{k}\bigr)=\left(\sum _{i=0}^{m}\left ( \begin{matrix} m\\ i \end{matrix} \right )s^{i}r^{m-i} \biggl(-\frac{a}{3}\biggr)^{i}(-2-a)^{n}\right)\notin c_{0} $$
$$B^{a,b}\bigl(B^{(m)} x_{k}\bigr) =\left(\sum _{i=0}^{m}\left ( \begin{matrix} m\\ i \end{matrix} \right )s^{i}r^{m-i} \biggl(-\frac{a}{3}\biggr)^{i}\biggl(\frac{1}{4+a} \biggr)^{n}\right)\in c_{0}. $$
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
$$ x=\sum_{k} \lambda_{k}(a,b) g^{(k)}(a,b), $$
where \(\lambda_{k}(a,b)= [B^{a,b}(B^{(m)} x_{i})]_{k}\) for each \(k\in \mathbb{N}\).


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


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


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

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)

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



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


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
$$ \sup_{K\in\Gamma}\sum_{n} \biggl\vert \sum_{k\in K}a_{n,k}\biggr\vert ^{p}< \infty,\quad 1\leq p< \infty. $$
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
$$\begin{aligned}& \sup_{K\in\Gamma}\sum_{n} \biggl\vert \sum_{k\in K}t_{n,k}\biggr\vert ^{p}< \infty, \end{aligned}$$
$$\begin{aligned}& t_{n,k}\quad \textit{exists for each } k,n\in\mathbb{N}, \end{aligned}$$
$$\begin{aligned}& \sum_{k}t_{n,k}\quad \textit{converges for each } n\in\mathbb{N}, \end{aligned}$$
$$\begin{aligned}& \sup_{l\in\mathbb{N}}\sum_{k=0}^{l} \bigl\vert t_{n,k}^{l}\bigr\vert < \infty,\quad n\in \mathbb{N}. \end{aligned}$$

Theorem 4.4

\(A\in(b_{c}^{a,b}(B^{(m)}):\ell_{\infty})\) if and only if (4.4) and (4.6) hold, and
$$ \sup_{n\in\mathbb{N}}\sum_{k} \vert t_{n,k}\vert < \infty. $$

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

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 [29] is added to the list of references.



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


  1. Choudhary, B, Nanda, S: Functional Analysis with Application. Wiley, New Delhi (1989) MATHGoogle Scholar
  2. Kizmaz, H: On certain sequence spaces. Can. Math. Bull. 24, 169-176 (1981) MathSciNetView ArticleMATHGoogle Scholar
  3. Et, M, Colak, R: On generalized difference sequence spaces. Soochow J. Math. 21, 377-386 (1995) MathSciNetMATHGoogle Scholar
  4. Bektaş, C, Et, M, Çolak, R: Generalized difference sequence spaces and their dual spaces. J. Math. Anal. Appl. 292, 423-432 (2004) MathSciNetView ArticleMATHGoogle Scholar
  5. Dutta, H: Characterization of certain matrix classes involving generalized difference summability spaces. Appl. Sci. 11, 60-67 (2009) MathSciNetMATHGoogle Scholar
  6. Reddy, BS: On some generalized difference sequence spaces. Soochow J. Math. 26, 377-386 (2010) MathSciNetGoogle Scholar
  7. Tripathy, BC, Esi, A: A new type of difference sequence spaces. Int. J. Sci. Technol. 1, 147-155 (2006) Google Scholar
  8. Altay, B, Polat, H: On some new Euler difference sequence spaces. Southeast Asian Bull. Math. 30, 209-220 (2006) MathSciNetMATHGoogle Scholar
  9. Başarir, M, Kara, EE: On compact operators on the Riesz \({B}^{m}\)-difference sequence spaces. Iran. J. Sci. Technol. 35, 279-285 (2011) MathSciNetMATHGoogle Scholar
  10. Başarir, M, Kara, EE: On some difference sequence spaces of weighted means and compact operators. Ann. Funct. Anal. 2, 114-129 (2011) MathSciNetView ArticleMATHGoogle Scholar
  11. Başarir, M, Kara, EE: On compact operators on the Riesz \({B}^{m}\)-difference sequence spaces II. Iran. J. Sci. Technol. 33, 371-376 (2012) MathSciNetMATHGoogle Scholar
  12. 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) MathSciNetView ArticleMATHGoogle Scholar
  13. 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) View ArticleMATHGoogle Scholar
  14. Başarir, M, Kara, EE, Konca, Ş: On some new weighted Euler sequence spaces and compact operators. Math. Inequal. Appl. 17, 649-664 (2014) MathSciNetMATHGoogle Scholar
  15. Kara, EE: Some topological and geometrical properties of new Banach sequence spaces. J. Inequal. Appl. 2013, 38 (2013) MathSciNetView ArticleMATHGoogle Scholar
  16. Kara, EE, İlkhan, M: On some Banach sequence spaces derived by a new band matrix. Br. J. Math. Comput. Sci. 9, 141-159 (2015) View ArticleGoogle Scholar
  17. Kara, EE, İlkhan, M: Some properties of generalized Fibonacci sequence spaces. Linear Multilinear Algebra 64, 2208-2223 (2016) MathSciNetView ArticleMATHGoogle Scholar
  18. Polat, H, Başar, F: Some Euler spaces of difference sequences of order m. Acta Math. Sci. 27, 254-266 (2007) MathSciNetView ArticleMATHGoogle Scholar
  19. Song, MM, Meng, J: Some normed binomial difference sequence spaces related to the \(\ell_{p}\) spaces. J. Inequal. Appl. 2017, 128 (2017) MathSciNetView ArticleMATHGoogle Scholar
  20. Altay, B, Başar, F: On some Euler sequence spaces of nonabsolute type. Ukr. Math. J. 57, 1-17 (2005) MathSciNetView ArticleMATHGoogle Scholar
  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) MathSciNetView ArticleMATHGoogle Scholar
  22. Başarir, M, Kayikçi, M: On the generalized \({B}^{m}\)-Riesz difference sequence space and β-property. J. Inequal. Appl. 2009(1), 18 (2009) MathSciNetMATHGoogle Scholar
  23. Kara, EE, Başarir, M: On compact operators and some Euler \({B}^{(m)}\)-difference sequence spaces. J. Math. Anal. Appl. 379, 499-511 (2011) MathSciNetView ArticleMATHGoogle Scholar
  24. Bişgin, MC: The binomial sequence spaces of nonabsolute type. J. Inequal. Appl. 2016, 309 (2016) MathSciNetView ArticleMATHGoogle Scholar
  25. Bişgin, MC: The binomial sequence spaces which include the spaces \(\ell _{p}\) and \(\ell_{\infty}\) and geometric properties. J. Inequal. Appl. 2016, 304 (2016) MathSciNetView ArticleMATHGoogle Scholar
  26. Wilansky, A: Summability Through Functional Analysis. North-Holland Mathematics Studies, vol. 85. Elsevier, Amsterdam (1984) MATHGoogle Scholar
  27. Köthe, G, Toeplitz, O: Linear Raume mit unendlichvielen Koordinaten and Ringe unenlicher Matrizen. J. Reine Angew. Math. 171, 193-226 (1934) MathSciNetMATHGoogle Scholar
  28. Stieglitz, M, Tietz, H: Matrixtransformationen von Folgenräumen Eine Ergebnisubersict. Math. Z. 154, 1-16 (1977) MathSciNetView ArticleMATHGoogle Scholar
  29. Başar, F: Summability Theory and Its Applications. Bentham Science, Istanbul (2012). ISBN:978-1-60805-420-6 MATHGoogle Scholar


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