Some normed binomial difference sequence spaces related to the ${\ell }_p$ spaces

Abstract

The aim of this paper is to introduce the normed binomial sequence spaces $b_p^{r,s}(\mathrm{\nabla })$ by combining the binomial transformation and difference operator, where $1\le p\le \mathrm{\infty }$. We prove that these spaces are linearly isomorphic to the spaces ${\ell }_p$ and ${\ell }_{\mathrm{\infty }}$, respectively. Furthermore, we compute Schauder bases and the α-, β- and γ-duals of these sequence spaces.

Introduction and preliminaries

Let w denote the space of all sequences. By ${\ell }_p$, ${\ell }_{\mathrm{\infty }}$, c and $c_0$, we denote the spaces of p-absolutely summable, bounded, convergent and null sequences, respectively, where $1\le p<\mathrm{\infty }$. Let Z be a sequence space, then Kizmaz [1] introduced the following difference sequence spaces:

$$Z(\mathrm{\Delta })=\{(x_k)\in w:(\mathrm{\Delta }{x}_k)\in Z\}$$

for $Z={\ell }_{\mathrm{\infty }},c,c_0$, where $\mathrm{\Delta }{x}_k=x_k-x_{k+1}$ for each $k\in \mathbb{N}=\{1,2,3,\dots \}$, the set of positive integers. Since then, many authors have studied further generalization of the difference sequence spaces [2–6]. Moreover, Altay and Polat [7], Başarir and Kara [8–12], Kara [13], Kara and İlkhan [14], Polat and Başar [15], and many others have studied new sequence spaces from a 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 ${(Ax)}_n={\sum }_{k=0}^{\mathrm{\infty }}{a}_{n,k}{x}_k$. For two sequence spaces X, Y and an infinite matrix $A=(a_{n,k})$, the sequence space $X_A$ is defined by $X_A=\{x=(x_k)\in w:Ax\in X\}$, 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

$$e_{n,k}^r=\{\begin{array}{cc}\left(\begin{array}{c}n\\ k\end{array}\right){(1-r)}^{n-k}{r}^k\hfill & \text{if }0\le k\le n,\hfill \\ 0\hfill & \text{if }k>n.\hfill \end{array}$$

The Euler sequence spaces $e_p^r$ and $e_{\mathrm{\infty }}^r$ were defined by Altay, Başar and Mursaleen [16] as follows:

$$e_p^r=\{x=(x_k)\in w:\sum _n{\left|\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right){(1-r)}^{n-k}{r}^k{x}_k\right|}^p<\mathrm{\infty }\}$$

and

$$e_{\mathrm{\infty }}^r=\{x=(x_k)\in w:\underset{n\in \mathbb{N}}{sup}\left|\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right){(1-r)}^{n-k}{r}^k{x}_k\right|<\mathrm{\infty }\}.$$

Altay and Polat [7] defined further generalization of the Euler sequence spaces $e_0^r(\mathrm{\nabla })$, $e_c^r(\mathrm{\nabla })$ and $e_{\mathrm{\infty }}^r(\mathrm{\nabla })$ by

$$\begin{array}{rl}& e_0^r(\mathrm{\nabla })=\{x=(x_k)\in w:(\mathrm{\nabla }{x}_k)\in e_0^r\},\\ & e_c^r(\mathrm{\nabla })=\{x=(x_k)\in w:(\mathrm{\nabla }{x}_k)\in e_c^r\}\end{array}$$

and

$$e_{\mathrm{\infty }}^r(\mathrm{\nabla })=\{x=(x_k)\in w:(\mathrm{\nabla }{x}_k)\in e_{\mathrm{\infty }}^r\},$$

where $\mathrm{\nabla }{x}_k=x_k-x_{k-1}$ for each $k\in \mathbb{N}$. Here any term with negative subscript is equal to naught. Many authors have used especially the Euler matrix for defining new sequence spaces, for instance, Kara and Başarir [17], Karakaya and Polat [18] and Polat and Başar [15].

Recently Bişgin [19, 20] defined another type of generalization of the Euler sequence spaces and introduced the binomial sequence spaces $b_0^{r,s}$, $b_c^{r,s}$, $b_{\mathrm{\infty }}^{r,s}$ and $b_p^{r,s}$. Let $r,s\in \mathbb{R}$ and $r+s\ne 0$. Then the binomial matrix $B^{r,s}=(b_{n,k}^{r,s})$ is defined by

$$b_{n,k}^{r,s}=\{\begin{array}{cc}\frac{1}{{(s+r)}^n}\left(\begin{array}{c}n\\ k\end{array}\right)s^{n-k}{r}^k\hfill & \text{if }0\le k\le n,\hfill \\ 0\hfill & \text{if }k>n,\hfill \end{array}$$

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

• (i) $\parallel B^{r,s}\parallel <\mathrm{\infty }$,
• (ii) ${lim}_{n\to \mathrm{\infty }}{b}_{n,k}^{r,s}=0$ for each $k\in \mathbb{N}$,
• (iii) ${lim}_{n\to \mathrm{\infty }}{\sum }_k{b}_{n,k}^{r,s}=1$.

Thus, the binomial matrix $B^{r,s}$ is regular for $sr>0$. Unless stated otherwise, we assume that $sr>0$. If we take $s+r=1$, we obtain the Euler matrix $E^r$. So the binomial matrix generalizes the Euler matrix. Bişgin [20] defined the following spaces of binomial sequences:

$$b_p^{r,s}=\{x=(x_k)\in w:\sum _n{\left|\frac{1}{{(s+r)}^n}\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)s^{n-k}{r}^k{x}_k\right|}^p<\mathrm{\infty }\}$$

and

$$b_{\mathrm{\infty }}^{r,s}=\{x=(x_k)\in w:\underset{n\in \mathbb{N}}{sup}\left|\frac{1}{{(s+r)}^n}\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)s^{n-k}{r}^k{x}_k\right|<\mathrm{\infty }\}.$$

The main purpose of the present paper is to study the normed difference spaces $b_p^{r,s}(\mathrm{\nabla })$ and $b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })$ of the binomial sequence whose $B^{r,s}(\mathrm{\nabla })$-transforms are in the spaces ${\ell }_p$ and ${\ell }_{\mathrm{\infty }}$, respectively. These new sequence spaces are the generalization of the sequence spaces defined in [7] and [20]. Also, we 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 $b_p^{r,s}(\mathrm{\nabla })$ and $b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })$ and prove that these sequence spaces are linearly isomorphic to the spaces ${\ell }_p$ and ${\ell }_{\mathrm{\infty }}$, respectively.

We first define the binomial difference sequence spaces $b_p^{r,s}(\mathrm{\nabla })$ and $b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })$ by

$$b_p^{r,s}(\mathrm{\nabla })=\{x=(x_k)\in w:(\mathrm{\nabla }{x}_k)\in b_p^{r,s}\}$$

and

$$b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })=\{x=(x_k)\in w:(\mathrm{\nabla }{x}_k)\in b_{\mathrm{\infty }}^{r,s}\}.$$

Let us define the sequence $y=(y_n)$ as the $B^{r,s}(\mathrm{\nabla })$-transform of a sequence $x=(x_k)$, that is,

$$y_n={[B^{r,s}(\mathrm{\nabla }{x}_k)]}_n=\frac{1}{{(s+r)}^n}\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)s^{n-k}{r}^k(\mathrm{\nabla }{x}_k)$$ 2.1

for each $n\in \mathbb{N}$. Then the binomial difference sequence spaces $b_p^{r,s}(\mathrm{\nabla })$ or $b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })$ can be redefined by all sequences whose $B^{r,s}(\mathrm{\nabla })$-transforms are in the space ${\ell }_p$ or ${\ell }_{\mathrm{\infty }}$.

Theorem 2.1

The sequence spaces $b_p^{r,s}(\mathrm{\nabla })$ and $b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })$ are complete linear metric spaces with the norm defined by

$$f_{b_p^{r,s}(\mathrm{\nabla })}(x)={\parallel y\parallel }_p={(\sum _{n=1}^{\mathrm{\infty }}|y_n{|}^p)}^{\frac{1}{p}}$$

and

$$f_{b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })}(x)={\parallel y\parallel }_{\mathrm{\infty }}=\underset{n\in \mathbb{N}}{sup}|y_n|,$$

where $1\le p<\mathrm{\infty }$ and the sequence $y=(y_n)$ is defined by the $B^{r,s}(\mathrm{\nabla })$-transform of x.

Proof

The proof of the linearity is a routine verification. It is obvious that $f_{b_p^{r,s}}(\alpha x)=|\alpha |f_{b_p^{r,s}}(x)$ and $f_{b_p^{r,s}}(x)=0$ if and only if $x=\theta$ for all $x\in b_p^{r,s}(\mathrm{\nabla })$, where θ is the zero element in $b_p^{r,s}$ and $\alpha \in \mathbb{R}$. We consider $x,z\in b_p^{r,s}(\mathrm{\nabla })$, then we have

$$\begin{array}{rl}{f}_{b_p^{r,s}(\mathrm{\nabla })}(x+z)& ={\left(\sum _n\right|{\left(B^{r,s}[\mathrm{\nabla }(x_k+z_k)]\right)}_n{|}^p)}^{\frac{1}{p}}\\ & \le {\left(\sum _n\right|{[B^{r,s}(\mathrm{\nabla }{x}_k)]}_n{|}^p)}^{\frac{1}{p}}+{\left(\sum _n\right|{[B^{r,s}(\mathrm{\nabla }{z}_k)]}_n{|}^p)}^{\frac{1}{p}}=f_{b_p^{r,s}(\mathrm{\nabla })}(x)+f_{b_p^{r,s}(\mathrm{\nabla })}(z).\end{array}$$

Hence $f_{b_p^{r,s}(\mathrm{\nabla })}$ is a norm on the space $b_p^{r,s}(\mathrm{\nabla })$.

Let $(x_m)$ be a Cauchy sequence in $b_p^{r,s}(\mathrm{\nabla })$, where $x_m={(x_{m_k})}_{k=1}^{\mathrm{\infty }}\in b_p^{r,s}(\mathrm{\nabla })$ for each $m\in \mathbb{N}$. For every $\epsilon >0$, there is a positive integer $m_0$ such that $f_{b_p^{r,s}(\mathrm{\nabla })}(x_m-x_l)<\epsilon$ $\text{for }m,l\ge m_0$. Then we get

$$\left|{\left(B^{r,s}[\mathrm{\nabla }(x_{m_k}-x_{l_k})]\right)}_n\right|\le {\left(\sum _n\right|{\left(B^{r,s}[\mathrm{\nabla }(x_{m_k}-x_{l_k})]\right)}_n{|}^p)}^{\frac{1}{p}}<\epsilon$$

for $m,l\ge m_0$ and each $k\in \mathbb{N}$. So ${(B^{r,s}(\mathrm{\nabla }{x}_{m_k}))}_{m=1}^{\mathrm{\infty }}$ is a Cauchy sequence in the set of real numbers $\mathbb{R}$. Since $\mathbb{R}$ is complete, we have ${lim}_{m\to \mathrm{\infty }}{B}^{r,s}(\mathrm{\nabla }{x}_{m_k})=B^{r,s}(\mathrm{\nabla }{x}_k)$ for each $k\in \mathbb{N}$. We compute

$$\sum _{n=0}^i\left|{\left(B^{r,s}[\mathrm{\nabla }(x_{m_k}-x_{l_k})]\right)}_n\right|\le f_{b_p^{r,s}(\mathrm{\nabla })}(x_m-x_l)<\epsilon$$ 2.2

for $m>m_0$. We take i and l →∞, then the inequality (2.2) implies that

$$f_{b_p^{r,s}(\mathrm{\nabla })}(x_m-x)\to 0.$$

We have

$$f_{b_p^{r,s}(\mathrm{\nabla })}(x)\le f_{b_p^{r,s}(\mathrm{\nabla })}(x_m-x)+f_{b_p^{r,s}(\mathrm{\nabla })}(x_m)<\mathrm{\infty },$$

that is, $x\in b_p^{r,s}(\mathrm{\nabla })$. Thus, the space $b_p^{r,s}(\mathrm{\nabla })$ is complete. For the space $b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })$, the proof can be completed in a similar way. So, we omit the detail. □

Theorem 2.2

The sequence spaces $b_p^{r,s}(\mathrm{\nabla })$ and $b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })$ are linearly isomorphic to the spaces ${\ell }_p$ and ${\ell }_{\mathrm{\infty }}$, respectively, where $1\le p<\mathrm{\infty }$.

Proof

Similarly, we only prove the theorem for the space $b_p^{r,s}(\mathrm{\nabla })$. To prove $b_p^{r,s}(\mathrm{\nabla })\cong {\ell }_p$, we must show the existence of a linear bijection between the spaces $b_p^{r,s}(\mathrm{\nabla })$ and ${\ell }_p$.

Consider $T:b_p^{r,s}(\mathrm{\nabla })\to {\ell }_p$ by $T(x)=B^{r,s}(\mathrm{\nabla }{x}_k)$. The linearity of T is obvious and $x=\theta$ whenever $T(x)=\theta$. Therefore, T is injective.

Let $y=(y_n)\in {\ell }_p$ and define the sequence $x=(x_k)$ by

$$x_k=\sum _{i=0}^k{(s+r)}^i\sum _{j=i}^k\left(\begin{array}{c}j\\ i\end{array}\right)r^{-j}{(-s)}^{j-i}{y}_i$$ 2.3

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

$$\begin{array}{rcl}{f}_{b_p^{r,s}(\mathrm{\nabla })}(x)& =& {\parallel {[B^{r,s}(\mathrm{\nabla }{x}_k)]}_n\parallel }_p\\ & =& {\left(\sum _{n=1}^{\mathrm{\infty }}{|\frac{1}{{(s+r)}^n}\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)s^{n-k}{r}^k(\mathrm{\nabla }{x}_k)|}^p\right)}^{\frac{1}{p}}\\ & =& {(\sum _{n=1}^{\mathrm{\infty }}|y_n{|}^p)}^{\frac{1}{p}}={\parallel y\parallel }_p<\mathrm{\infty },\end{array}$$

which implies that $x\in b_p^{r,s}(\mathrm{\nabla })$ and $T(x)=y$. Consequently, T is surjective and is norm preserving. Thus, $b_p^{r,s}(\mathrm{\nabla })\cong {\ell }_p$. □

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

For a normed space $(X,\parallel \cdot \parallel )$, a sequence ${\{x_k:x_k\in X\}}_{k\in \mathbb{N}}$ is called a Schauder basis [21] if for every $x\in X$, there is a unique scalar sequence $({\lambda }_k)$ such that $\parallel x-{\sum }_{k=0}^n{\lambda }_k{x}_k\parallel \to 0\text{ as }n\to \mathrm{\infty }$. Next, we shall give a Schauder basis for the sequence space $b_p^{r,s}(\mathrm{\nabla })$.

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

$$g_i^{(k)}(r,s)=\{\begin{array}{cc}0\hfill & \text{if }0\le i<k,\hfill \\ {(s+r)}^k{\sum }_{j=k}^i\left(\begin{array}{c}j\\ k\end{array}\right)r^{-j}{(-s)}^{j-k}\hfill & \text{if }i\ge k,\hfill \end{array}$$

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 spaces $b_p^{r,s}(\mathrm{\nabla })$ and every $x=(x_i)\in b_p^{r,s}(\mathrm{\nabla })$ has a unique representation by

$$x=\sum _k{\lambda }_k(r,s)g^{(k)}(r,s),$$ 3.1

where $1\le p<\mathrm{\infty }$ and ${\lambda }_k(r,s)={[B^{r,s}(\mathrm{\nabla }{x}_i)]}_k$ for each $k\in \mathbb{N}$.

Proof

Obviously, $B^{r,s}(\mathrm{\nabla }{g}_i^{(k)}(r,s))=e_k\in {\ell }_p$, where $e_k$ is the sequence with 1 in the kth place and zeros elsewhere for each $k\in \mathbb{N}$. This implies that $g^{(k)}(r,s)\in b_p^{r,s}(\mathrm{\nabla })$ for each $k\in \mathbb{N}$.

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

$$x^{(m)}=\sum _{k=0}^m{\lambda }_k(r,s)g^{(k)}(r,s).$$

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

$$B^{r,s}\left(\mathrm{\nabla }{x}_i^{(m)}\right)=\sum _{k=0}^m{\lambda }_k(r,s)B^{r,s}(\mathrm{\nabla }{g}_i^{(k)}(r,s))=\sum _{k=0}^m{\lambda }_k(r,s)e_k$$

and

$${\left[B^{r,s}\left(\mathrm{\nabla }(x_i-x_i^{(m)})\right)\right]}_k=\{\begin{array}{cc}0\hfill & \text{if }0\le k\le m,\hfill \\ {[B^{r,s}(\mathrm{\nabla }{x}_i)]}_k\hfill & \text{if }k>m,\hfill \end{array}$$

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

For any given $\epsilon >0$, there is a positive integer $m_0$ such that

$$\sum _{k=m_0+1}^{\mathrm{\infty }}|{[B^{r,s}(\mathrm{\nabla }{x}_i)]}_k{|}^p<{\left(\frac{\epsilon }{2}\right)}^p$$

for all $k\ge m_0$. Then we have

$$\begin{array}{rcl}{f}_{b_p^{r,s}(\mathrm{\nabla })}(x-x^{(m)})& =& {\left(\sum _{k=m+1}^{\mathrm{\infty }}\right|{[B^{r,s}(\mathrm{\nabla }{x}_i)]}_k{|}^p)}^{\frac{1}{p}}\\ & \le & {\left(\sum _{k=m_0+1}^{\mathrm{\infty }}\right|{[B^{r,s}(\mathrm{\nabla }{x}_i)]}_k{|}^p)}^{\frac{1}{p}}\\ & <& \frac{\epsilon }{2}<\epsilon ,\end{array}$$

which implies that $x\in b_p^{r,s}(\mathrm{\nabla })$ is represented as (3.1).

To prove the uniqueness of this representation, we assume that

$$x=\sum _k{\mu }_k(r,s)g^{(k)}(r,s).$$

Then we have

$${[B^{r,s}(\mathrm{\nabla }{x}_i)]}_k=\sum _k{\mu }_k(r,s){\left[B^{r,s}(\mathrm{\nabla }{g}_i^{(k)}(r,s))\right]}_k=\sum _k{\mu }_k(r,s){(e_k)}_k={\mu }_k(r,s),$$

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

Corollary 3.2

The sequence space $b_p^{r,s}(\mathrm{\nabla })$ is separable, where $1\le p<\mathrm{\infty }$.

For the duality theory, the study of sequence spaces is more useful when we investigate them equipped with linear topologies. Köthe and Toeplitz [22] first computed duals whose elements can be represented as sequences and defined the α-dual (or Köthe-Toeplitz dual).

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

$$M(X,Y)=\{u=(u_k)\in w:ux=(u_k{x}_k)\in Y\text{ for all }x=(x_k)\in X\}.$$

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

$$X^{\alpha }=M(X,{\ell }_1),X^{\beta }=M(X,c)\text{and}{X}^{\gamma }=M(X,{\ell }_{\mathrm{\infty }}),$$

respectively.

We give the following properties:

$$\underset{n\in \mathbb{N}}{sup}\sum _k|a_{n,k}{|}^q<\mathrm{\infty },$$ 3.2

$$\underset{k\in \mathbb{N}}{sup}\sum _n|a_{n,k}|<\mathrm{\infty },$$ 3.3

$$\underset{n,k\in \mathbb{N}}{sup}|a_{n,k}|<\mathrm{\infty },$$ 3.4

$$\underset{n\to \mathrm{\infty }}{lim}{a}_{n,k}=a_k\text{for each }k\in \mathbb{N},$$ 3.5

$$\underset{K\in \mathrm{\Gamma }}{sup}\sum _k|\sum _{n\in K}{a}_{n,k}{|}^q<\mathrm{\infty },$$ 3.6

$$\underset{n\to \mathrm{\infty }}{lim}\sum _k|a_{n,k}|=\sum _k\left|\underset{n\to \mathrm{\infty }}{lim}{a}_{n,k}\right|,$$ 3.7

where Γ is the collection of all finite subsets of $\mathbb{N}$, $\frac{1}{p}+\frac{1}{q}=1$ and $1<p\le \mathrm{\infty }$.

Lemma 3.3

[23]

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

• (i) $A\in ({\ell }_1:{\ell }_1)$ if and only if (3.3) holds.
• (ii) $A\in ({\ell }_1:c)$ if and only if (3.4) and (3.5) hold.
• (iii) $A\in ({\ell }_1:{\ell }_{\mathrm{\infty }})$ if and only if (3.4) holds.
• (iv) $A\in ({\ell }_p:{\ell }_1)$ if and only if (3.6) holds with $\frac{1}{p}+\frac{1}{q}=1$ and $1<p\le \mathrm{\infty }$.
• (v) $A\in ({\ell }_p:c)$ if and only if (3.2) and (3.5) hold with $\frac{1}{p}+\frac{1}{q}=1$ and $1<p<\mathrm{\infty }$.
• (vi) $A\in ({\ell }_p:{\ell }_{\mathrm{\infty }})$ if and only if (3.2) holds with $\frac{1}{p}+\frac{1}{q}=1$ and $1<p<\mathrm{\infty }$.
• (vii) $A\in ({\ell }_{\mathrm{\infty }}:c)$ if and only if (3.5) and (3.7) hold with $\frac{1}{p}+\frac{1}{q}=1$ and $1<p<\mathrm{\infty }$.
• (viii) $A\in ({\ell }_{\mathrm{\infty }}:{\ell }_{\mathrm{\infty }})$ if and only if (3.2) holds with $q=1$.

Theorem 3.4

We define the set $U_1^{r,s}$ and $U_2^{r,s}$ by

$$U_1^{r,s}=\{u=(u_k)\in w:\underset{i\in \mathbb{N}}{sup}\sum _k\left|{(s+r)}^i\sum _{j=i}^k\left(\begin{array}{c}j\\ i\end{array}\right)r^{-j}{(-s)}^{j-i}{u}_k\right|<\mathrm{\infty }\}$$

and

$$U_2^{r,s}=\{u=(u_k)\in w:\underset{K\in \mathrm{\Gamma }}{sup}\sum _i{\left|\sum _{k\in K}{(s+r)}^i\sum _{j=i}^k\left(\begin{array}{c}j\\ i\end{array}\right)r^{-j}{(-s)}^{j-i}{u}_k\right|}^q<\mathrm{\infty }\}.$$

Then ${[b_1^{r,s}(\mathrm{\nabla })]}^{\alpha }=U_1^{r,s}$ and ${[b_p^{r,s}(\mathrm{\nabla })]}^{\alpha }=U_2^{r,s}$, where $1<p\le \mathrm{\infty }$.

Proof

Let $u=(u_k)\in w$ and $x=(x_k)$ be defined by (2.3), then we have

$$u_k{x}_k=\sum _{i=0}^k{(s+r)}^i\sum _{j=i}^k\left(\begin{array}{c}j\\ i\end{array}\right)r^{-j}{(-s)}^{j-i}{u}_k{y}_i={\left(G^{r,s}y\right)}_k$$

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

$$g_{k,i}^{r,s}=\{\begin{array}{cc}{(s+r)}^i{\sum }_{j=i}^k\left(\begin{array}{c}j\\ i\end{array}\right)r^{-j}{(-s)}^{j-i}{u}_k\hfill & \text{if }0\le i\le k,\hfill \\ 0\hfill & \text{if }i>k.\hfill \end{array}$$

Therefore, we deduce that $ux=(u_k{x}_k)\in {\ell }_1$ whenever $x\in b_1^{r,s}(\mathrm{\nabla })$ or $b_p^{r,s}(\mathrm{\nabla })$ if and only if $G^{r,s}y\in {\ell }_1$ whenever $y\in {\ell }_1$ or ${\ell }_p$, which implies that $u=(u_k)\in {[b_1^{r,s}(\mathrm{\nabla })]}^{\alpha }\text{ or }{[b_p^{r,s}(\mathrm{\nabla })]}^{\alpha }$ if and only if $G^{r,s}\in ({\ell }_1:{\ell }_1)$ and $G^{r,s}\in ({\ell }_p:{\ell }_1)$ by parts (i) and (iv) of Lemma 3.3, we obtain $u=(u_k)\in {[b_1^{r,s}(\mathrm{\nabla })]}^{\alpha }$ if and only if

$$\underset{i\in \mathbb{N}}{sup}\sum _k\left|{(s+r)}^i\sum _{j=i}^k\left(\begin{array}{c}j\\ i\end{array}\right)r^{-j}{(-s)}^{j-i}{u}_k\right|<\mathrm{\infty }$$

and $u=(u_k)\in {[b_p^{r,s}(\mathrm{\nabla })]}^{\alpha }$ if and only if

$$\underset{K\in \mathrm{\Gamma }}{sup}\sum _i{\left|\sum _{k\in K}{(s+r)}^i\sum _{j=i}^k\left(\begin{array}{c}j\\ i\end{array}\right)r^{-j}{(-s)}^{j-i}{u}_k\right|}^q<\mathrm{\infty }.$$

Thus, we have ${[b_1^{r,s}(\mathrm{\nabla })]}^{\alpha }=U_1^{r,s}$ and ${[b_p^{r,s}(\mathrm{\nabla })]}^{\alpha }=U_2^{r,s}$, where $1<p\le \mathrm{\infty }$. □

Now, we define the sets $U_3^{r,s}$, $U_4^{r,s}$, $U_5^{r,s}$, $U_6^{r,s}$ and $U_7^{r,s}$ by

$$\begin{array}{rl}& U_3^{r,s}=\{u=(u_k)\in w:\underset{n\to \mathrm{\infty }}{lim}{(s+r)}^k\sum _{i=k}^n\sum _{j=k}^i\left(\begin{array}{c}j\\ k\end{array}\right)r^{-j}{(-s)}^{j-k}{u}_i\text{ exists for each }k\in \mathbb{N}\},\\ & U_4^{r,s}=\{u=(u_k)\in w:\underset{n,k\in \mathbb{N}}{sup}\left|{(s+r)}^k\sum _{i=k}^n\sum _{j=k}^i\left(\begin{array}{c}j\\ k\end{array}\right)r^{-j}{(-s)}^{j-k}{u}_i\right|<\mathrm{\infty }\},\\ & U_5^{r,s}=\{u=(u_k)\in w:\underset{n\to \mathrm{\infty }}{lim}\sum _k\left|{(s+r)}^k\sum _{i=k}^n\sum _{j=k}^i\left(\begin{array}{c}j\\ k\end{array}\right)r^{-j}{(-s)}^{j-k}{u}_i\right|\\ & \phantom{U_5^{r,s}=}=\sum _k\left|\underset{n\to \mathrm{\infty }}{lim}{(s+r)}^k\sum _{i=k}^n\sum _{j=k}^i\left(\begin{array}{c}j\\ k\end{array}\right)r^{-j}{(-s)}^{j-k}{u}_i\right|\},\\ & U_6^{r,s}=\{u=(u_k)\in w:\underset{n\in \mathbb{N}}{sup}\sum _{k=0}^n{\left|{(s+r)}^k\sum _{i=k}^n\sum _{j=k}^i\left(\begin{array}{c}j\\ k\end{array}\right)r^{-j}{(-s)}^{j-k}{u}_i\right|}^q<\mathrm{\infty }\},1<q<\mathrm{\infty },\end{array}$$

and

$$U_7^{r,s}=\{u=(u_k)\in w:\underset{n\in \mathbb{N}}{sup}\sum _{k=0}^n\left|{(s+r)}^k\sum _{i=k}^n\sum _{j=k}^i\left(\begin{array}{c}j\\ k\end{array}\right)r^{-j}{(-s)}^{j-k}{u}_i\right|<\mathrm{\infty }\}.$$

Theorem 3.5

We have the following relations:

• (i) ${[b_1^{r,s}(\mathrm{\nabla })]}^{\beta }=U_3^{r,s}\cap U_4^{r,s}$,
• (ii) ${[b_p^{r,s}(\mathrm{\nabla })]}^{\beta }=U_3^{r,s}\cap U_6^{r,s}$, where $1<p<\mathrm{\infty }$,
• (iii) ${[b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })]}^{\beta }=U_3^{r,s}\cap U_5^{r,s}$,
• (iv) ${[b_1^{r,s}(\mathrm{\nabla })]}^{\gamma }=U_4^{r,s}$,
• (v) ${[b_p^{r,s}(\mathrm{\nabla })]}^{\gamma }=U_6^{r,s}$, where $1<p<\mathrm{\infty }$,
• (vi) ${[b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })]}^{\gamma }=U_7^{r,s}$.

Proof

Let $u=(u_k)\in w$ and $x=(x_k)$ be defined by (2.3), then we consider the following equation:

$$\begin{array}{rl}\sum _{k=0}^n{u}_k{x}_k& =\sum _{k=0}^n{u}_k\left[\sum _{i=0}^k{(s+r)}^i\sum _{j=i}^k\left(\begin{array}{c}j\\ i\end{array}\right)r^{-j}{(-s)}^{j-i}{y}_i\right]\\ & =\sum _{k=0}^n\left[{(s+r)}^k\sum _{i=k}^n\sum _{j=k}^i\left(\begin{array}{c}j\\ k\end{array}\right)r^{-j}{(-s)}^{j-k}{u}_i\right]y_k={\left(U^{r,s}y\right)}_n,\end{array}$$

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

$$u_{n,k}=\{\begin{array}{cc}{(s+r)}^k{\sum }_{i=k}^n{\sum }_{j=k}^i\left(\begin{array}{c}j\\ k\end{array}\right)r^{-j}{(-s)}^{j-k}{u}_i\hfill & \text{if }0\le k\le n,\hfill \\ 0\hfill & \text{if }k>n.\hfill \end{array}$$

Therefore, we deduce that $ux=(u_k{x}_k)\in c$ whenever $x\in b_1^{r,s}(\mathrm{\nabla })$ if and only if $U^{r,s}y\in c$ whenever $y\in {\ell }_1$, which implies that $u=(u_k)\in {[b_1^{r,s}(\mathrm{\nabla })]}^{\beta }$ if and only if $U^{r,s}\in ({\ell }_1:c)$. By Lemma 3.3(ii), we obtain ${[b_1^{r,s}(\mathrm{\nabla })]}^{\beta }=U_3^{r,s}\cap U_4^{r,s}$. Using Lemma 3.3(i) and (iii)-(viii) instead of (ii), the proof can be completed in a similar way. So, we omit the details. □

Conclusion

By considering the definitions of the binomial matrix $B^{r,s}=(b_{n,k}^{r,s})$ and the difference operator, we introduce the sequence spaces $b_p^{r,s}(\mathrm{\nabla })$ and $b_{\mathrm{\infty }}^{r,s}(\mathrm{\nabla })$. These spaces are the natural continuations of [1, 7, 20]. Our results are the generalizations of the matrix domain of the Euler matrix of order r. In order to give fully inform the reader on related topics with applications and a possible line of further investigation, the e-book [24] is added to the list of references.