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
- andγ
-duals of these sequence spaces.Keywords: sequence space; matrix domain; Schauder basis;
α
-,β
- andγ
-duals1 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∞,candcareBK-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=xk–xk+ 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–xk–m–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
spacesX,Yand an infinite matrixA= (an,k), the sequence spaceXAis defined by
XA=
x= (xk)∈w:Ax∈X
, (.)
which is called the domain of matrixA. By (X:Y), we denote the class of all matrices such thatX⊆YA.
The Euler meansErof orderris defined by the matrixEr= (er
n,k), where <r< and
ern,k=
⎧ ⎨ ⎩ n
k
( –r)n–krk if ≤k≤n,
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)n–krkxk=
,
erc=
x= (xk)∈w: lim n→∞
n
k=
n k
( –r)n–krkxkexists
,
and
er∞=
x= (xk)∈w:sup n∈N
n
k=
n k
( –r)n–krkxk
<∞
.
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=xk–xk–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)=
⎧ ⎨ ⎩
(–)n–km n–k
if max{,n–m} ≤k≤n, if ≤k<max{,n–m}ork>n,
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)),b∞r,s(∇(m)) and prove theBK
-property and inclusion relations.
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
sn–krk∇(m)xk
(.)
for eachn∈N, where
∇(m)x
k= m
i=
(–)i
m i
xk–i= m
i=max{,k–m}
(–)k–i
m k–i
xi.
Then the binomial difference sequence spacesbr,s(∇(m)),br,s
c (∇(m)) andb∞r,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 b∞r,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)∈cand define the sequencex= (xk) by
xk= k
i=
(s+r)i
k
j=i
m+k–j–
k–j
j i
r–j(–s)j–iyi (.)
for eachk∈N. Then we have
lim n→∞
Br,s∇(m)xk
n=nlim→∞
(s+r)n
n
k=
n k
sn–krk∇(m)xk
= lim n→∞yn= ,
which implies thatx∈br,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
Theorem . The inclusions br,s(∇(m)) ⊆ br,s
(∇(m+)), brc,s(∇(m)) ⊆ brc,s(∇(m+)) and br,s
∞(∇(m))⊆b∞r,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 thatx∈br,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+k–j–
k–j
–
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 havex∈br,s(∇(m))\er(∇(m)). This shows that the inclusioner(∇(m))⊆
br,s(∇(m)) strictly holds.
3 The Schauder basis and
α
-,β
- andγ
-dualsFor a normed space (X, · ), a sequence{xk:xk∈X}k∈N is called aSchauder basis[] if
for everyx∈X, 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+i–j–
i–j
j
k
r–j(–s)j–k ifi≥k,
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), (.)
Proof Obviously,Br,s(∇(m)g(k)
i (r,s)) =ek∈c, 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.
Forx∈br,s(∇(m)) andn∈N, 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)xi–x(in)
k=
⎧ ⎨ ⎩
if ≤k<n, [Br,s(∇(m)x
i)]k ifk≥n,
for eachk∈N.
For everyε> , there is a positive integernsuch that
Br,s∇(m)xi
k< ε
for allk≥n. Then we have
x–x(n)br,s
(∇(m))=supk≥n
Br,s∇(m)xi
k≤sup k≥n
Br,s∇(m)xi
k< ε
<ε,
which impliesx∈br,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+n–j–
n–j
j k
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 x∈bcr,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)) =ek∈c⊆candg∈bcr,s(∇(m)). Forx∈brc,s(∇(m)), we put y=x–lgand we havey∈br,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)) andb∞r,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
Xα=M(X,), Xβ=M(X,cs) and Xγ=M(X,bs),
respectively.
Let us give the following properties:
sup K∈
n
k∈K 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.
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
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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 4 May 2017 Accepted: 20 July 2017
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