Skip to main content

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

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

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

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

which is called the domain of matrix A in the space X. By \((X : Y)\), we denote the class of all matrices such that \(X \subseteq Y_{A}\).

The Euler means \(E^{r}\) of order r is defined by the matrix \(E^{r}=(e_{n,k}^{r})\), where \(0< r<1\) and

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

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

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

and

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

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

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

for \(Z\in\{e_{0}^{r}, e_{c}^{r}, e_{\infty}^{r}\}\), where \(\nabla x_{k}=x_{k}-x_{k-1}\) for each \(k\in\mathbb{N}\). Here any term with negative subscript is equal to naught.

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

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

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

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

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

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

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

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

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

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

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

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

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

and

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

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

The binomial difference sequence spaces

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

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

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

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

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

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

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

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

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

Theorem 2.1

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

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

Proof

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

Theorem 2.2

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

Proof

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

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

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

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

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

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

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

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

Theorem 2.3

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

Theorem 2.4

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

Proof

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

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

and

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

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

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

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

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

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

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

Theorem 3.1

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

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

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

Proof

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

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

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

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

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

and

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

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

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

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

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

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

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

To show the uniqueness of this representation, we assume that

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

Then we have

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

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

Theorem 3.2

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

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

Proof

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

Corollary 3.3

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

Köthe and Toeplitz [27] first computed the dual whose elements can be represented as sequences and defined the α-dual (or Köthe-Toeplitz dual). Next, we compute the α-,β- and γ-duals of the sequence spaces \(b_{0}^{a,b}(B^{(m)})\), \(b_{c}^{a,b}(B^{(m)})\) and \(b_{\infty }^{a,b}(B^{(m)})\).

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

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

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

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

respectively.

Let us give the following properties:

$$\begin{aligned}& \sup_{K\in\Gamma} \sum_{n} \biggl\vert \sum_{k\in K} a_{n,k}\biggr\vert < \infty, \end{aligned}$$
(3.3)
$$\begin{aligned}& \sup_{n\in\mathbb{N}} \sum_{k} \vert a_{n,k}\vert < \infty, \end{aligned}$$
(3.4)
$$\begin{aligned}& \lim_{n\rightarrow\infty}a_{n,k}=a_{k} \quad \text{for each } k\in \mathbb{N}, \end{aligned}$$
(3.5)
$$\begin{aligned}& \lim_{n\rightarrow\infty}\sum_{k}a_{n,k}=a, \end{aligned}$$
(3.6)
$$\begin{aligned}& \lim_{n\rightarrow\infty}\sum_{k} \vert a_{n,k}\vert =\sum_{k}\Bigl\vert \lim_{n\rightarrow\infty}a_{n,k}\Bigr\vert , \end{aligned}$$
(3.7)

where Γ is the collection of all finite subsets of \(\mathbb{N}\).

Lemma 3.4

[28]

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

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (iv)

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

  5. (v)

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

Theorem 3.5

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

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

Proof

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

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

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

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

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

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

if and only if

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

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

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

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

and

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

where

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

Theorem 3.6

We have the following relations:

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

Proof

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

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

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

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

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

Similarly, we give the following theorem without proof.

Theorem 3.7

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

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

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

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

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

Lemma 4.1

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

Proof

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

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

Lemma 4.2

[28]

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

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

For brevity of notation, we write

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

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

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

Theorem 4.3

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

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

Theorem 4.4

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

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

Theorem 4.5

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

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

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.

References

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

    MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    Article  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  15. Kara, EE: Some topological and geometrical properties of new Banach sequence spaces. J. Inequal. Appl. 2013, 38 (2013)

    MathSciNet  Article  MATH  Google 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)

    Article  Google Scholar 

  17. Kara, EE, İlkhan, M: Some properties of generalized Fibonacci sequence spaces. Linear Multilinear Algebra 64, 2208-2223 (2016)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  19. Song, MM, Meng, J: Some normed binomial difference sequence spaces related to the \(\ell_{p}\) spaces. J. Inequal. Appl. 2017, 128 (2017)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  27. Köthe, G, Toeplitz, O: Linear Raume mit unendlichvielen Koordinaten and Ringe unenlicher Matrizen. J. Reine Angew. Math. 171, 193-226 (1934)

    MathSciNet  MATH  Google Scholar 

  28. Stieglitz, M, Tietz, H: Matrixtransformationen von Folgenräumen Eine Ergebnisubersict. Math. Z. 154, 1-16 (1977)

    MathSciNet  Article  MATH  Google Scholar 

  29. Başar, F: Summability Theory and Its Applications. Bentham Science, Istanbul (2012). ISBN:978-1-60805-420-6

    MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jian Meng.

Additional information

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

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), 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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Meng, J., Song, M. On some binomial \(B^{(m)}\)-difference sequence spaces. J Inequal Appl 2017, 194 (2017). https://doi.org/10.1186/s13660-017-1470-4

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

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