difference sequence spaces

Meng and Song Journal of Inequalities and Applications (2017) 2017:194 DOI 10.1186/s13660-017-1470-4

RESEARCH

Open Access

On some binomial B(m)-difference sequence spaces Jian Meng* and Meimei Song *

Correspondence: [email protected] Department of Mathematics, Tianjin University of Technology, Tianjin, 300000, P.R. China

Abstract In this paper, we introduce the binomial sequence spaces b0a,b (B(m) ), bca,b (B(m) ) and a,b (m) (B ) by combining the binomial transformation and difference operator. We b∞ 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 bca,b (B(m) ). Keywords: sequence space; matrix domain; Schauder basis; α -, β - and γ -duals

1 Introduction and preliminaries Let w denote the space of all sequences. By p , ∞ , c and c , we denote the spaces of absolutely p-summable, bounded, convergent and null sequences, respectively, where  ≤ p < ∞. A Banach sequence space Z is called a BK -space [] provided each of the maps pn : Z → C defined by pn (x) = xn is continuous for all n ∈ N, which is of great importance in the characterization of matrix transformations between sequence spaces. One can prove that the sequence spaces ∞ , c and c are BK -spaces with their usual sup-norm. Let Z be a sequence space, then Kizmaz [] introduced the following difference sequence spaces: Z() = (xk ) ∈ w : (xk ) ∈ Z for Z ∈ {∞ , c, c }, where xk = xk – xk+ for each k ∈ N. Et and Colak [] defined the generalization of the difference sequence spaces Z m = (xk ) ∈ w : m xk ∈ Z for Z ∈ {∞ , c, c }, where m ∈ N,  xk = xk and m xk = m– xk – m– xk+ for each k ∈ N, i m which is equivalent to the binomial representation m xk = m i= (–) i xk+i . Since then, many authors have studied further generalization of the difference sequence spaces [–]. Moreover, Altay and Polat [], Başarir and Kara [–], Başarir, Kara and Konca [], Kara [], Kara and İlkhan [, ], Polat and Başar [], Song and Meng [] and many others have studied new sequence spaces from matrix point of view that represent difference operators. For an infinite matrix A = (an,k ) and x = (xk ) ∈ w, the A-transform of x is defined by Ax = {(Ax)n } and is supposed to be convergent for all n ∈ N, where (Ax)n = ∞ k= an,k xk . © The Author(s) 2017. 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.

Meng and Song Journal of Inequalities and Applications (2017) 2017:194

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For two sequence spaces X and Y and an infinite matrix A = (an,k ), the sequence space XA is defined by XA = x = (xk ) ∈ w : Ax ∈ 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 ⊆ YA . The Euler means Er of order r is defined by the matrix Er = (ern,k ), where  < r <  and ⎧ ⎨ n ( – r)n–k rk

if  ≤ k ≤ n,

⎩

if k > n.

k

ern,k =

The Euler sequence spaces er , erc and er∞ were defined by Altay and Başar [] and Altay, Başar and Mursaleen [] as follows: n n n–k k ( – r) r xk =  , = x = (xk ) ∈ w : lim n→∞ k k=

n n r n–k k ( – r) r xk exists , ec = x = (xk ) ∈ w : lim n→∞ k k=

er

and n n n–k k ( – r) r xk < ∞ . = x = (xk ) ∈ w : sup n∈N k= k

er∞

Altay and Polat [] defined further generalization of the Euler sequence spaces er (∇), erc (∇) and er∞ (∇) by Z(∇) = x = (xk ) ∈ w : (∇xk ) ∈ Z for Z ∈ {er , erc , er∞ }, where ∇xk = xk – xk– for each k ∈ N. Here any term with negative subscript is equal to naught. Polat and Başar [] employed the matrix domain technique of the triangle limitation method for obtaining the following sequence spaces: Z ∇ (m) = x = (xk ) ∈ w : ∇ (m) xk ∈ Z (m) for Z ∈ {er , erc , er∞ }, where ∇ (m) = (δn,k ) is a triangle matrix defined by

(m) δn,k

=

⎧ ⎨(–)n–k ⎩

m n–k

if max{, n – m} ≤ k ≤ n, if  ≤ k < max{, n – m} or k > n,

for all k, n, m ∈ N. Also, Başarir and Kayikçi [] defined the matrix B(m) = (b(m) n,k ) by b(m) n,k

⎧ ⎨ =

m m–n+k n–k s n–k r

⎩

if max{, n – m} ≤ k ≤ n, if  ≤ k < max{, n – m} or k > n,


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which is reduced to the matrix ∇ (m) in the case r = , s = –. Kara and Başarir [] introduced the spaces er (B(m) ), erc (B(m) ) and er∞ (B(m) ) of B(m) -difference sequences. Recently Bişgin [, ] defined another generalization of the Euler sequence spaces and a,b introduced the binomial sequence spaces ba,b , bca,b , b∞ and bpa,b . Let a, b ∈ R and a, b = . a,b ) is defined by Then the binomial matrix Ba,b = (bn,k

a,b bn,k

⎧ ⎨ =

 (a+b)n

⎩

n n–k k b k a

if  ≤ k ≤ n, if k > n,

for all k, n ∈ N. For ab >  we have (i) Ba,b < ∞, a,b (ii) limn→∞ bn,k =  for each k ∈ N, a,b (iii) limn→∞ k bn,k = . Thus, the binomial matrix Ba,b is regular for ab > . Unless stated otherwise, we assume that ab > . If we take a + b = , we obtain the Euler matrix Er , so the binomial matrix generalizes the Euler matrix. Bişgin defined the following binomial sequence spaces:

n  n n–k k = x = (xk ) ∈ w : lim a b xk =  , n→∞ (a + b)n k k=

n  n n–k k a,b a b xk exists , bc = x = (xk ) ∈ w : lim n→∞ (a + b)n k k=

ba,b

and n  n n–k k = x = (xk ) ∈ w : sup a b xk < ∞ . n n∈N (a + b) k= k

a,b b∞

The purpose of the present paper is to study the binomial difference spaces ba,b (B(m) ), a,b (m) bca,b (B(m) ) and b∞ (B ) whose Ba,b (B(m) )-transforms are in the spaces c , c and ∞ , respectively. These new sequence spaces are the generalization of the sequence spaces defined in [, ] and []. 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 a,b (m) In this section, we introduce the spaces ba,b (B(m) ), bca,b (B(m) ) and b∞ (B ) and prove the BK -property and inclusion relations. We first define the binomial difference sequence spaces ba,b (B(m) ), bca,b (B(m) ) and a,b (m) b∞ (B ) by Z B(m) = x = (xk ) ∈ w : B(m) xk ∈ Z a,b }. By using the notion of (.), the sequence spaces ba,b (B(m) ), bca,b (B(m) ) for Z ∈ {ba,b , bca,b , b∞ a,b (m) (B ) can be redefined by and b∞

ba,b B(m) = ba,b B(m) ,

bca,b B(m) = bca,b B(m) ,

(m) a,b a,b B = b∞ B(m) . b∞

(.)


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a,b (m) It is obvious that the sequence spaces ba,b (B(m) ), bca,b (B(m) ) and b∞ (B ) may be reduced to some sequence spaces in the special cases of a, b, s, r and m ∈ N. For instance, if we take a + b = , then we obtain the spaces er (B(m) ), erc (B(m) ) and er∞ (B(m) ), defined by Kara and Başarir []. If we take a + b = , r =  and s = –, then we obtain the spaces er (∇ (m) ), erc (∇ (m) ) and er∞ (∇ (m) ), defined by Polat and Başar []. Especially, taking r =  and s = –, we obtain the new binomial difference sequence spaces ba,b (∇ (m) ), bca,b (∇ (m) ) a,b (∇ (m) ). and b∞ Let us define the sequence y = (yn ) as the Ba,b (B(m) )-transform of a sequence x = (xk ), that is,

n n a,b (m)  m n n–i i m+k–i i–k a br s xk (.) yn = B B xk n = n (a + b) i–k i k= i=k a,b (m) for each n ∈ N. Then the sequence spaces ba,b (B(m) ), bca,b (B(m) ) and b∞ (B ) can be redea,b (m) fined by all sequences whose B (B )-transforms are in the spaces c , c and ∞ . a,b (m) Theorem . The sequence spaces ba,b (B(m) ), bca,b (B(m) ) and b∞ (B ) are BK -spaces with their sup-norm defined by

x ba,b (B(m) ) = x ba,b (B(m) ) = x ba,b (B(m) ) = sup Ba,b B(m) xk n . c



∞

n∈N

a,b Proof The sequence spaces ba,b , bca,b and b∞ are BK -spaces with their sup-norm (see [], Theorem . and [], Theorem .). Moreover, B(m) is a triangle matrix and (.) holds. By using Theorem .. of Wilansky [], we deduce that the binomial sequence spaces a,b (m) (B ) are BK -spaces. ba,b (B(m) ), bca,b (B(m) ) and b∞ a,b (m) Theorem . The sequence spaces ba,b (B(m) ), bca,b (B(m) ) and b∞ (B ) are linearly isomorphic to the spaces c , c and ∞ , respectively.

Proof Similarly, we prove the theorem only for the space ba,b (B(m) ). To prove ba,b (B(m) ) ∼ = c , we must show the existence of a linear bijection between the spaces ba,b (B(m) ) and c . Consider ba,b (B(m) ) → c by T(x) = Ba,b (B(m) xk ). The linearity of T is obvious and x =  whenever T(x) = . Therefore, T is injective. Let y = (yn ) ∈ c and define the sequence x = (xk ) by

k k m+k–j– j (–s)k–j i xk = (a + b) (–a)j–i b–j yi (.) m+k–j r k – j i i= j=i for each k ∈ N. Then we have

lim B

n→∞

a,b

B

(m)

xk

n  n n–k k (m) a b B xk = lim yn = , = lim n n n→∞ (a + b) n→∞ k k=

which implies that x ∈ ba,b (B(m) ) and T(x) = y. Consequently, T is surjective and is norm preserving. Thus, ba,b (B(m) ) ∼ = c . The following theorems give some inclusion relations for this class of sequence spaces. We have the well known inclusion c ⊆ c ⊆ ∞ , then the corresponding extended versions also preserve this inclusion.


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a,b (m) Theorem . The inclusion ba,b (B(m) ) ⊆ bca,b (B(m) ) ⊆ b∞ (B ) holds.

Theorem . The inclusions ea (B(m) ) ⊆ ba,b (B(m) ), eac (B(m) ) ⊆ bca,b (B(m) ) and ea∞ (B(m) ) ⊆ a,b (m) (B ) strictly hold. b∞ Proof Similarly, we only prove the inclusion ea (B(m) ) ⊆ ba,b (B(m) ). If a + b = , we have Ea = Ba,b . So ea (B(m) ) ⊆ ba,b (B(m) ) holds. Let  < a <  and b = . We define a sequence x = (xk ) by xk = (– a )k for each k ∈ N. It is clear that a

E B

(m)

xk

m m i m–i a i n = sr (– – a) ∈/ c –  i i=

and B

a,b

B

(m)

xk

n m  m i m–i a i – = sr ∈ c .  +a i i=

So, we have x ∈ ba,b (B(m) ) \ ea (B(m) ). This shows that the inclusion ea (B(m) ) ⊆ ba,b (B(m) ) strictly holds.

3 The Schauder basis and α -, β - and γ -duals For a normed space (X, · ), a sequence {xk : xk ∈ X}k∈N is called a Schauder basis [] if for every x ∈ X, there is a unique scalar sequence (λk ) such that x – nk= λk xk →  as n → ∞. We shall construct Schauder bases for the sequence spaces ba,b (B(m) ) and bca,b (B(m) ). We define the sequence g (k) (a, b) = {gi(k) (a, b)}i∈N by ⎧ ⎨ gi(k) (a, b) = ⎩(a + b)k i m+i–j– j (–s)i–j b–j (–a)j–k j=k

i–j

k rm+i–j

if  ≤ i < k, if i ≥ k,

for each k ∈ N. Theorem . The sequence (g (k) (a, b))k∈N is a Schauder basis for the binomial sequence space ba,b (B(m) ) and every x = (xi ) ∈ ba,b (B(m) ) has a unique representation by x=

λk (a, b)g (k) (a, b),

(.)

k

where λk (a, b) = [Ba,b (B(m) xi )]k for each k ∈ N. Proof Obviously, Ba,b (B(m) gi(k) (a, b)) = ek ∈ c , where ek is the sequence with  in the kth place and zeros elsewhere for each k ∈ N. This implies that g (k) (a, b) ∈ ba,b (B(m) ) for each k ∈ N. For x ∈ ba,b (B(m) ) and n ∈ N, we put x(n) =

n k=

λk (a, b)g (k) (a, b).


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By the linearity of Ba,b (B(m) ), we have n n (m) (k) a,b = B λ (a, b)B g (a, b) = λk (a, b)ek Ba,b B(m) x(n) k i i k=

k=

and ⎧ ⎨ a,b (m) = B B xi – x(n) i k ⎩[Ba,b (B(m) xi )]k

if  ≤ k < n, if k ≥ n,

for each k ∈ N. For every ε > , there is a positive integer n such that a,b (m) ε B B xi < k  for all k ≥ n . Then we have x – x(n) a,b (m) = sup Ba,b B(m) xi ≤ sup Ba,b B(m) xi < ε < ε, b (B ) k k  k≥n k≥n which implies that x ∈ ba,b (B(m) ) is represented as in (.). To show the uniqueness of this representation, we assume that x=

μk (a, b)g (k) (a, b).

k

Then we have

Ba,b B(m) xi k = μk (a, b) Ba,b B(m) gi(k) (a, b) k = μk (a, b)(ek )k = μk (a, b), k

k

which is a contradiction with the assumption that λk (a, b) = [Ba,b (B(m) xi )]k for each k ∈ N. This shows the uniqueness of this representation. Theorem . Let g = (, , , , . . .) and limk→∞ λk (a, b) = l. The set {g, g () (a, b), g () (a, b), . . . , g (k) (a, b), . . .} is a Schauder basis for the space bca,b (B(m) ) and every x ∈ bca,b (B(m) ) has a unique representation by x = lg +

λk (a, b) – l g (k) (a, b).

(.)

k

Proof Obviously, Ba,b (B(m) gik (a, b)) = ek ∈ c and g ∈ bca,b (B(m) ). For x ∈ bca,b (B(m) ), we put y = x – lg and we have y ∈ ba,b (B(m) ). Hence, we deduce that y has a unique representation by (.), which implies that x has a unique representation by (.). Thus, we complete the proof. Corollary . The sequence spaces ba,b (B(m) ) and bca,b (B(m) ) are separable.


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Köthe and Toeplitz [] 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-,βa,b (m) and γ -duals of the sequence spaces ba,b (B(m) ), bca,b (B(m) ) and b∞ (B ). For the sequence spaces X and Y , define multiplier space M(X, Y ) by M(X, Y ) = u = (uk ) ∈ w : ux = (uk xk ) ∈ Y for all x = (xk ) ∈ X . Then the α-, β- and γ -duals of a sequence space X are defined by X α = M(X,  ),

X β = M(X, c)

and X γ = M(X, ∞ ),

respectively. Let us give the following properties: an,k < ∞, sup

K∈ n

sup

(.)

k∈K

|an,k | < ∞,

(.)

n∈N k

lim an,k = ak

n→∞

lim

n→∞

lim

n→∞

(.)

an,k = a,

k

for each k ∈ N,

|an,k | =

k

(.)

lim an,k , k

(.)

n→∞

where is the collection of all finite subsets of N. Lemma . ([]) Let A = (an,k ) be an infinite matrix. Then the following statements hold: (i) A ∈ (c :  ) = (c :  ) = (∞ :  ) if and only if (.) holds. (ii) A ∈ (c : c) if and only if (.) and (.) hold. (iii) A ∈ (c : c) if and only if (.), (.) and (.) hold. (iv) A ∈ (∞ : c) if and only if (.) and (.) hold. (v) A ∈ (c : ∞ ) = (c : ∞ ) = (∞ : ∞ ) if and only if (.) holds. a,b (m) Theorem . The α-dual of the spaces ba,b (B(m) ), bca,b (B(m) ) and b∞ (B ) is the set

k m+k–j– j i = u = (uk ) ∈ w : sup (a + b) k – j i K∈

Ua,b

k

i∈K

(–s)k–j –j j–i × m+k–j b (–a) uk < ∞ . r

j=i

Proof Let u = (uk ) ∈ w and x = (xk ) be defined by (.), then we have

k m+k–j– j (–s)k–j –j (a + b) b (–a)j–i uk yi = Ga,b y k uk xk = m+k–j k–j i r i= j=i k

i


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a,b for each k ∈ N, where Ga,b = (gk,i ) is defined by

a,b gk,i =

⎧ ⎨(a + b)i k m+k–j– j (–s)k–j b–j (–a)j–i u

if  ≤ i ≤ k,

⎩

if i > k.

j=i

k

i rm+k–j

k–j

a,b (m) Therefore, we deduce that ux = (uk xk ) ∈  whenever x ∈ ba,b (B(m) ), bca,b (B(m) ) or b∞ (B ),

if and only if Ga,b y ∈  , whenever y ∈ c , c or ∞ . This implies that u = (uk ) ∈ [ba,b (B(m) )]α , a,b (m) α [bca,b (B(m) )]α or [b∞ (B )] if and only if Ga,b ∈ (c :  ), Ga,b ∈ (c :  ) or Ga,b ∈ (∞ :  ) by

Parts (i) of Lemma .. So we obtain α a,b (m) α a,b (m) α = bc B = b∞ B u = (uk ) ∈ ba,b B(m) if and only if

k k–j m+k–j– j (–s) –j j–i (a + b)i b (–a) u sup k < ∞. m+k–j k–j i r K∈

j=i k i∈K a,b (m) α Thus, we have [ba,b (B(m) )]α = [bca,b (B(m) )]α = [b∞ (B )] = Ua,b .

Now, we define the sets Ua,b , Ua,b , Ua,b and Ua,b by Ua,b

= u = (uk ) ∈ w : sup |un,k | < ∞ , n∈N k

Ua,b = u = (uk ) ∈ w : lim un,k exists for each k ∈ N , n→∞ Ua,b = u = (uk ) ∈ w : lim lim |un,k | = u n,k , n→∞

k

k

n→∞

and un,k exists , Ua,b = u = (uk ) ∈ w : lim n→∞

k

where

i n m+i–j– j (–s)i–j (–a)j–k (b)–j ui . un,k = (a + b) rm+i–j i – j k i=k j=k k

Theorem . We have the following relations: (i) [ba,b (B(m) )]β = Ua,b ∩ Ua,b , (ii) [bca,b (B(m) )]β = Ua,b ∩ Ua,b ∩ Ua,b , a,b (m) β (iii) [b∞ (B )] = Ua,b ∩ Ua,b .


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Proof Let u = (uk ) ∈ w and x = (xk ) be defined by (.), then we consider the following equation: ⎡ ⎤

k k k–j (–s) m + k – j –  j uk xk = uk ⎣ (a + b)i (–a)j–i b–j yi ⎦ m+k–j r k – j i i= j=i k= k=

n

n

⎤

i n i–j (–s) m + i – j –  j ⎣(a + b)k (–a)j–k b–j ui ⎦ yk = rm+i–j i – j k k= i=k j=k n

⎡

= U a,b y n , a,b where U a,b = (un,k ) is defined by

a,b = un,k

⎧ ⎨(a + b)k n i m+i–j– j (–s)i–j (–a)j–k (b)–j u

if  ≤ k ≤ n,

⎩

if k > n.

i=k

j=k

i–j

k rm+i–j

i

Therefore, we deduce that ux = (uk xk ) ∈ c whenever x ∈ ba,b (B(m) ) if and only if U a,b y ∈ c whenever y ∈ c , which implies that u = (uk ) ∈ [ba,b (B(m) )]β if and only if U a,b ∈ (c : c) by Part (ii) of Lemma .. So we obtain [ba,b (B(m) )]β = Ua,b ∩ Ua,b . Using Parts (iii), (iv) instead of Part (ii) of Lemma ., the proof can be completed in a similar way. Similarly, we give the following theorem without proof. a,b (m) Theorem . The γ -dual of the spaces ba,b (B(m) ), bca,b (B(m) ) and b∞ (B ) is the set Ua,b .

(m) 4 Certain matrix mappings on the space ba,b c (B ) In this section, we characterize matrix transformations from bca,b (B(m) ) into p , ∞ and c. Let us define the matrix = (θn,k ) via an infinite matrix = (λn,k ) by = (Ba,b (B(m) ))– , that is,

∞ m+n–j– j (–s)n–j θn,k = (a + b) (–a)j–k b–j λn,j , rm+n–j n – j k j=k k

(.)

where (Ba,b (B(m) ))– is the inverse of the Ba,b (B(m) )-transform. We now give the following lemmas. Lemma . Let Z be any given sequence space and the entries of the matrices = (λn,k ) and = (θn,k ) are connected with equation (.). If (λn,k )k ∈ [bca,b (B(m) )]β for all n ∈ N, then

∈ (bca,b (B(m) ) : Z) if and only if ∈ (c : Z). Proof Let ∈ (bca,b (B(m) ) : Z) and y = (yn ) ∈ c. For every x = (xk ) ∈ bca,b (B(m) ), we have xk = [(Ba,b (B(m) ))– yn ]k . Since (λn,k )k ∈ [bca,b (B(m) )]β for all n ∈ N, this implies the existence of the

-transform of x, i.e. x exists. So we obtain x = (Ba,b (B(m) ))– y = y, which implies that ∈ (c : Z). Conversely, let ∈ (c : Z) and x ∈ bca,b (B(m) ). For every y ∈ c, we have yn = [Ba,b (B(m) xk )]n . Since (λn,k )k ∈ [bca,b (B(m) )]β for all n ∈ N, this implies that y exists, which can be proved


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in a similar way to the proof of Theorem .. So we have y = Ba,b (B(m) )x = x, which shows that ∈ (bca,b (B(m) ) : Z). Lemma . ([]) Let A = (an,k ) be an infinite matrix. Then the following statement holds: A ∈ (c : p ) if and only if sup

p a n,k < ∞,

K∈ n

 ≤ p < ∞.

(.)

k∈K

For brevity of notation, we write

n i m + i – j –  j (–s)i–j (–a)j–k (b)–j an,j , tn,k = (a + b)k rm+i–j i – j k i=k j=k

i l m + i – j –  j (–s)i–j l = (a + b)k (–a)j–k (b)–j an,j tn,k rm+i–j i – j k i=k j=k for all n, k ∈ N. l By using Lemma ., there are some immediate consequences with tn,k or tn,k in place of an,k in Lemma . and Lemma .. Theorem . A ∈ (bca,b (B(m) ) : p ) if and only if sup

p t n,k < ∞,

K∈ n

(.)

k∈K

tn,k exists for each k, n ∈ N, tn,k converges for each n ∈ N,

(.) (.)

k

sup

l l t < ∞, n,k

n ∈ N.

(.)

l∈N k=

Theorem . A ∈ (bca,b (B(m) ) : ∞ ) if and only if (.) and (.) hold, and sup

|tn,k | < ∞.

(.)

n∈N k

Theorem . A ∈ (bca,b (B(m) ) : c) if and only if (.), (.) and (.) hold, and lim tn,k

n→∞

lim

n→∞

exists for each k ∈ N, tn,k

exists.

(.) (.)

k

5 Conclusion a,b By considering the definitions of the binomial matrix Ba,b = (bn,k ) and the difference opa,b (m) a,b (m) a,b (m) erator, we introduce the sequence spaces b (B ), bc (B ) and b∞ (B ). These spaces are the natural continuation of [, –]. Our results are the generalization of the matrix


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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 [] is added to the list of references.

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

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