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R E S E A R C H

Open Access

Binomial difference sequence spaces of

order

m

Jian Meng and Meimei Song

*

*Correspondence: ms153106305@126.com

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

Abstract

In this paper, we introduce the binomial sequence spacesbr,s0(∇(m)),br,sc (∇(m)) and br,s

∞(∇(m)) by combining the binomial transformation andmth order difference operator. We prove theBK-property and some inclusion relations. Also, we obtain the Schauder bases and compute the

α

-,

β

- and

γ

-duals of these sequence spaces.

Keywords: sequence space; matrix domain; Schauder basis;

α

-,

β

- and

γ

-duals

1 Introduction and preliminaries

Letwdenote the space of all sequences. By,candc, we denote the spaces of bounded,

convergent and null sequences, respectively. We writebs,csandp for the spaces of all

bounded, convergent andp-absolutely summable series, respectively; ≤p<∞. A Ba-nach sequence spaceZ is called aBK-space [] provided each of the mapspn:Z→C

defined bypn(x) =xnis continuous for alln∈N, which is of great importance in the

char-acterization of matrix transformations between sequence spaces. It is well known that the sequence spaces∞,candcareBK-spaces with their usualsup-norm.

LetZbe a sequence space, then Kizmaz [] introduced the following difference sequence spaces:

Z() =(xk)∈w: (xk)∈Z

forZ∈ {,c,c}, wherexk=xkxk+ for eachk∈N. Et and Colak [] defined the

generalization of the difference sequence spaces

Zm=(xk)∈w:

mxk

Z

forZ∈ {∞,c,c}, wherem∈N,xk=xk,mxk=m–xkm–xk+ for eachk∈N,

which is equivalent to the binomial representationmxk=

m i=(–)i

m

i

xk+i. Since then,

many authors have studied further generalization of the difference sequence spaces [–]. Moreover, Altay and Polat [], Başarir [], Başarir, Kara and Konca [], Başarir and Kara [–], Başarir, Öztürk and Kara [], Polat and Başarir [] and many others have studied new sequence spaces from matrix point of view that represent difference operators.

For an infinite matrixA= (an,k) andx= (xk)∈w, theA-transform ofx is defined by

(Ax)n=

k=an,kxk and is supposed to be convergent for all n∈N. For two sequence

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spacesX,Yand an infinite matrixA= (an,k), the sequence spaceXAis defined by

XA=

x= (xk)∈w:AxX

, (.)

which is called the domain of matrixA. By (X:Y), we denote the class of all matrices such thatXYA.

The Euler meansErof orderris defined by the matrixEr= (er

n,k), where  <r<  and

ern,k=

⎧ ⎨ ⎩ n

k

( –r)nkrk if kn,

 ifk>n.

The Euler sequence spaceser

,ercander∞were defined by Altay and Başar [] and Altay,

Başar and Mursaleen [] as follows:

er=

x= (xk)∈w: lim n→∞

n

k=

n k

( –r)nkrkxk= 

,

erc=

x= (xk)∈w: lim n→∞

n

k=

n k

( –r)nkrkxkexists

,

and

er=

x= (xk)∈w:sup n∈N

n

k=

n k

( –r)nkrkxk

<∞

.

Altay and Polat [] defined further generalization of the Euler sequence spaceser

(∇),erc(∇)

ander(∇) by

Z(∇) =x= (xk)∈w: (∇xk)∈Z

forZ∈ {er,erc,er∞}, where∇xk=xkxk–for eachk∈N. Here any term with negative

subscript is equal to naught.

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

Z∇(m)=x= (xk)∈w:

∇(m)x

k

Z

forZ∈ {er,er

c,er∞}, where∇(m)= (δ

(m)

n,k) is a triangle matrix defined by

δ(nm,k)=

⎧ ⎨ ⎩

(–)nkm nk

if max{,nm} ≤kn,  if ≤k<max{,nm}ork>n,

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Recently Bişgin [, ] defined another generalization of the Euler sequence spaces and

Thus, the binomial matrixBr,sis regular forsr> . Unless stated otherwise, we assume that sr> . If we takes+r= , we obtain the Euler matrixEr. So the binomial matrix generalizes

the Euler matrix. Bişgin defined the following spaces of binomial sequences:

br,s=

cand∞, 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 andα-,β- andγ-duals of these sequence spaces.

2 The binomial difference sequence spaces

In this section, we introduce the spacesbr,s(∇(m)),br,s

c (∇(m)),br,s(∇(m)) and prove theBK

-property and inclusion relations.

(4)

then obtain the spacesbr,s,br,s

c andbr∞,sdefined by Bişgin [, ]. On takings+r= , we

obtain the spaceser

(∇(m)),erc(∇(m)) ander∞(∇(m)) defined by Polat and Başar [].

Let us define the sequencey= (yn) as theBr,s(∇(m))-transform of a sequencex= (xk) by

yn=

Br,s∇(m)xk

n=

 (s+r)n

n

k=

n k

snkrk∇(m)xk

(.)

for eachn∈N, where

∇(m)x

k= m

i=

(–)i

m i

xki= m

i=max{,km}

(–)ki

m ki

xi.

Then the binomial difference sequence spacesbr,s(∇(m)),br,s

c (∇(m)) andbr,s(∇(m)) can be

redefined by all sequences whoseBr,s((m))-transforms are in the spacesc

,cand∞.

Theorem . Let Z∈ {br,s,bcr,s,br,s}.Then Z(∇(m))is a BK -space with the norm x Z(∇(m))=

(∇(m)xk) Z.

Proof The sequence spacesbr,s,br,s

c andbr∞,sareBK-spaces (see [], Theorem . and [],

Theorem .). Moreover,∇(m)is a triangle matrix and (.) holds. By using Theorem ..

of Wilansky [], we deduce that the binomial sequence spacesbr,s(∇(m)),br,s

c (∇(m)) and br,s

∞(∇(m)) areBK-spaces. Theorem . The sequence spaces br,s(∇(m)),br,s

c (∇(m))and br,s(∇(m))are linearly isomor-phic to the spaces c,c and∞,respectively.

Proof Similarly, we prove the theorem only for the spacebr,s(∇(m)). To provebr,s(∇(m))∼=

c, we must show the existence of a linear bijection between the spacesbr,s(∇(m)) andc.

ConsiderT :br,s(∇(m))→c byT(x) =Br,s(∇(m)xk). The linearity ofT is obvious and x=  wheneverT(x) = . Therefore,T is injective.

Lety= (yn)∈cand define the sequencex= (xk) by

xk= k

i=

(s+r)i

k

j=i

m+kj– 

kj

j i

rj(–s)jiyi (.)

for eachk∈N. Then we have

lim n→∞

Br,s∇(m)xk

n=nlim→∞

 (s+r)n

n

k=

n k

snkrk∇(m)xk

= lim n→∞yn= ,

which implies thatxbr,s(∇(m)) andT(x) =y. Consequently,T is surjective and is norm

preserving. Thus,br,s(∇(m))=c

.

The following theorems give some inclusion relations for this class of sequence spaces. We have the well-known inclusionc⊆c∞, then the corresponding extended versions

also preserve this inclusion.

Theorem . The inclusion br,s(∇(m))br,s

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Theorem . The inclusions br,s(∇(m)) br,s

(∇(m+)), brc,s(∇(m)) ⊆ brc,s(∇(m+)) and br,s

∞(∇(m))⊆br,s(∇(m+))hold.

Proof Letx= (xk)∈br,s(∇(m)), then the inequality

Br,s∇(m+)xk

n=B

r,s(m)(x

k)

n

=Br,s∇(m)xk

n

Br,s∇(m)xk

n–

Br,s∇(m)xk

n+B

r,s(m)x

k

n–

holds and tends to  asn→ ∞, which implies thatxbr,s(∇(m+)).

Theorem . The inclusions er(∇(m))⊆br,s(∇(m)),erc(∇(m))⊆brc,s(∇(m))and er∞(∇(m))⊆ br,s

∞(∇(m))strictly hold.

Proof Similarly, we only prove the inclusioner(∇(m))br,s

(∇(m)). Ifr+s= , we haveEr=

Br,s. Soer

(∇(m))⊆br,s(∇(m)) holds. Take  <r<  ands= . We define a sequencex= (xk)

by

xk= k

j=

m+kj– 

kj

–

r

j

for all m,k∈N. It is clear that [Er((m)x

k)]n = ((– –r)n) /∈c and [Br,s(∇(m)xk)]n=

((+r)n)c

. So, we havexbr,s(∇(m))\er(∇(m)). This shows that the inclusioner(∇(m))⊆

br,s(∇(m)) strictly holds.

3 The Schauder basis and

α

-,

β

- and

γ

-duals

For a normed space (X, · ), a sequence{xk:xkX}k∈N is called aSchauder basis[] if

for everyxX, there is an unique scalar sequence (λk) such that x–

n

k=λkxk →

asn→ ∞. We shall construct the Schauder bases for the sequence spacesbr,s(∇(m)) and

br,s c (∇(m)).

We define the sequenceg(k)(r,s) ={gi(k)(r,s)}i∈Nby

gi(k)(r,s) =

⎧ ⎨ ⎩

 if ≤i<k, (s+r)ki

j=k

m+ij–

ij

j

k

rj(–s)jk ifik,

for eachk∈N.

Theorem . The sequence(g(k)(r,s))

k∈N is a Schauder basis for the binomial sequence space br,s(∇(m))and every x= (x

i)∈br,s(∇(m))has an unique representation by

x=

k

λk(r,s)g(k)(r,s), (.)

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Proof Obviously,Br,s((m)g(k)

i (r,s)) =ekc, whereekis the sequence with  in thekth

place and zeros elsewhere for eachk∈N. This implies thatg(k)(r,s)br,s

(∇(m)) for each

k∈N.

Forxbr,s(∇(m)) andnN, we put

x(n)=

n

k=

λk(r,s)g(k)(r,s).

By the linearity ofBr,s((m)), we have

Br,s∇(m)x(in)=

n

k=

λk(r,s)Br,s

∇(m)g(k)

i (r,s)

=

n

k=

λk(r,s)ek

and

Br,s∇(m)xix(in)

k=

⎧ ⎨ ⎩

 if ≤k<n, [Br,s((m)x

i)]k ifkn,

for eachk∈N.

For everyε> , there is a positive integernsuch that

Br,s∇(m)xi

k< ε

 for allkn. Then we have

xx(n)br,s

(∇(m))=supkn

Br,s∇(m)xi

k≤sup kn

Br,s∇(m)xi

k< ε

<ε,

which impliesxbr,s(∇(m)) is represented as in (.).

To show the uniqueness of this representation, we assume that

x=

k

μk(r,s)g(k)(r,s).

Then we have

Br,s∇(m)xi

k=

k

μk(r,s)

Br,s∇(m)gi(k)(r,s)k=

k

μk(r,s)(ek)k=μk(r,s),

which is a contradiction with the assumption thatλk(r,s) = [Br,s(∇(m)xi)]kfor eachk∈N.

This shows the uniqueness of this representation.

Theorem . We define g= (gn)by

gn= n

k=

(s+r)k

n

j=k

m+nj– 

nj

j k

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for all n∈ N and limk→∞λk(r,s) =l. The set {g,g()(r,s),g()(r,s), . . . ,g(k)(r,s), . . .} is a Schauder basis for the space br,s

c (∇(m))and every xbcr,s(∇(m))has an unique represen-tation by

x=lg+

k

λk(r,s) –l

g(k)(r,s). (.)

Proof Obviously,Br,s((m)gk

i(r,s)) =ekc⊆candgbcr,s(∇(m)). Forxbrc,s(∇(m)), we put y=xlgand we haveybr,s(∇(m)). Hence, we deduce thatyhas an unique representation

by (.), which implies thatxhas an unique representation by (.). Thus, we complete the

proof.

From Theorem ., we know thatbr,s(∇(m)) andbrc,s(∇(m)) are Banach spaces. By com-bining this fact with Theorem . and Theorem ., we can give the following corollary.

Corollary . The sequence spaces br,s(∇(m))and br,s

c (∇(m))are separable.

Köthe and Toeplitz [] first computed the dual whose elements can be represented as sequences and defined theα-dual (or Köthe-Toeplitz dual). Chandra and Tripathy [] generalized the notion of Köthe-Toeplitz dual of sequence spaces. Next, we compute the

α-,β- andγ-duals of the sequence spacesbr,s(∇(m)),br,s

c (∇(m)) andbr,s(∇(m)).

For the sequence spacesXandY, define multiplier spaceM(X,Y) by

M(X,Y) =u= (uk)∈w:ux= (ukxk)∈Yfor allx= (xk)∈X

. Then theα-,β- andγ-duals of a sequence spaceXare defined by

=M(X,), =M(X,cs) and =M(X,bs),

respectively.

Let us give the following properties:

sup K

n

kK an,k

<∞, (.)

sup n∈N

k

|an,k|<∞, (.)

lim

n→∞an,k=ak for eachk∈N, (.) lim

n→∞

k

an,k=a, (.)

lim n→∞

k

|an,k|=

k

lim

n→∞an,k

, (.)

whereis the collection of all finite subsets ofN.

Lemma .([]) Let A= (an,k)be an infinite matrix,then:

(i) A∈(c:) = (c:) = (∞:)if and only if(.)holds.

(8)
(9)

where

Theorem . The following equations hold:

(i) [br,s(∇(m))]β=Ur,s

Proof Since the proof may be obtained in the same way for (ii) and (iii), we only prove (i). Letu= (uk)∈wandx= (xk) be defined by (.), then we consider the following equation:

Proof Using Lemma .(v) instead of (ii), the proof can be given in a similar way. So, we

omit the details.

4 Conclusion

By considering the definitions of the binomial matrixBr,s= (brn,,sk) andmth order differ-ence operator, we introduce the sequdiffer-ence spacesbr,s(∇(m)),brc,s(∇(m)) andbr,s(∇(m)). These spaces are the natural continuation of [, , , ]. Our results are the generalization of the matrix domain of the Euler matrix of orderr.

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

(10)

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 4 May 2017 Accepted: 20 July 2017

References

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References

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