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

R E S E A R C H

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 spacesba,b₀ (B(m)_),ba,b

c (B(m)) and ba,b

∞(B(m)) by combining the binomial transformation and difference operator. We

prove theBK-property and some inclusion relations. Furthermore, we obtain Schauder bases and compute the

α

β

γ

c (B(m)). Keywords: sequence space; matrix domain; Schauder basis;

α

β

γ

1 Introduction and preliminaries

Letwdenote the space of all sequences. Byp, ∞, candc, we denote the spaces of

absolutely p-summable, bounded, convergent and null sequences, respectively, where ≤p<∞. A Banach sequence spaceZis called aBK-space [] provided each of the maps

pn:Z→Cdefined bypn(x) =xnis continuous for alln∈N, which is of great importance in

the characterization of matrix transformations between sequence spaces. One can prove 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=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=xkandmxk=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 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 matrixA= (an,k) andx= (xk)∈w, theA-transform ofx is defined by

Ax={(Ax)n}and is supposed to be convergent for alln∈N, where (Ax)n=

_∞

For two sequence spacesXandYand an infinite matrixA= (an,k), the sequence spaceXA

is defined by

XA=

x= (xk)∈w:Ax∈X

, (.)

which is called the domain of matrixAin the spaceX. By (X:Y), we denote the class of all matrices such thatX⊆YA.

The Euler meansEr_{of orderris defined by the matrixE}r_{= (e}r

n,k), where  <r<  and

er_n,k=

⎧ ⎨ ⎩

n k

( –r)n–k_rk _{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= 

,

er_c=

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_,er_c,er_∞}, where∇xk=xk–xk–for eachk∈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)_x k

∈Z

forZ∈ {er_,er

c,er∞}, where∇(m)= (δ (m)

n,k) is a triangle matrix defined by

δ(m)_n,k =

⎧ ⎨ ⎩

(–)n–k m n–k

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

for allk,n,m∈N. Also, Başarir and Kayikçi [] defined the matrixB(m)_{= (b}(m) n,k) by

b(m)_n,k =

⎧ ⎨ ⎩

m n–k

rm–n+k_sn–k _{ifmax{,n–m} ≤k≤n,}

which is reduced to the matrix∇(m)_{in the caser= ,s= –. Kara and Başarir []}

intro-duced the spaceser

(B(m)),erc(B(m)) ander∞(B(m)) ofB(m)-difference sequences.

Recently Bişgin [, ] defined another generalization of the Euler sequence spaces and introduced the binomial sequence spacesba,b_ ,ba,b_c ,ba,b_∞ andba,b_p . Leta,b∈Randa,b= . Then the binomial matrixBa,b_{= (b}a,b

n,k) is defined by

ba,b_n,k=

⎧ ⎨ ⎩

 (a+b)n

n k

an–k_bk _{if ≤k≤n,}

 ifk>n, for allk,n∈N. Forab>  we have

(i) Ba,b _<∞,

(ii) limn→∞ba,bn,k= for eachk∈N, (iii) lim_n→∞kba,b_n,k= .

Thus, the binomial matrixBa,b is regular forab> . Unless stated otherwise, we assume thatab> . If we takea+b= , we obtain the Euler matrixEr_{, so the binomial matrix}

generalizes the Euler matrix. Bişgin defined the following binomial sequence spaces:

ba,b_ =

x= (xk)∈w: lim n→∞

 (a+b)n

n

k=

n k

an–kbkxk= 

,

ba,b_c =

x= (xk)∈w: lim n→∞

 (a+b)n

n

k=

n k

an–kbkxkexists

,

and

ba,b_∞ =

x= (xk)∈w:sup n∈N

 (a+b)n

n

k=

n k

an–kbkxk

<∞

.

The purpose of the present paper is to study the binomial difference spacesba,b_ (B(m)_),

ba,b

c (B(m)) andba,b∞(B(m)) whoseBa,b(B(m))-transforms are in the spacesc,cand∞,

respec-tively. 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 spacesba,b_ (B(m)),ba,b_c (B(m)) andba,b_∞(B(m)) and prove the

BK-property and inclusion relations.

We first define the binomial difference sequence spaces ba,b_ (B(m)_{), b}a,b

c (B(m)) and

ba,b

∞(B(m)) by

ZB(m)=x= (xk)∈w:

B(m)xk

∈Z

forZ∈ {ba,b_ ,ba,b

c ,ba,b∞}. By using the notion of (.), the sequence spacesb a,b

 (B(m)),ba,bc (B(m))

andba,b

∞(B(m)) can be redefined by ba,b_ B(m)=ba,b_{ B}(m), ba,bc

B(m)=ba,b_{c B}(m), ba,b∞

It is obvious that the sequence spacesba,b_ (B(m)_{), b}a,b

c (B(m)) andba,b∞(B(m)) may be

re-duced to some sequence spaces in the special cases ofa,b,s,randm∈N. For instance, if we takea+b= , then we obtain the spaceser

(B(m)),erc(B(m)) ander∞(B(m)), defined by

Kara and Başarir []. If we takea+b= , r=  ands= –, then we obtain the spaces

er

(∇(m)),erc(∇(m)) ander∞(∇(m)), defined by Polat and Başar []. Especially, takingr= 

ands= –, we obtain the new binomial difference sequence spacesba,b_ (∇(m)_),ba,b c (∇(m))

andba,b_∞(∇(m)).

Let us define the sequencey= (yn) as theBa,b(B(m))-transform of a sequencex= (xk),

that is,

yn=

Ba,bB(m)xk

n=

 (a+b)n

n

k= n

i=k

m i–k

n i

an–ibirm+k–isi–kxk (.)

for eachn∈N. Then the sequence spacesba,b_ (B(m)_),ba,b

c (B(m)) andba,b∞(B(m)) can be

rede-fined by all sequences whoseBa,b_(B(m)_{)-transforms are in the spacesc}

,cand∞.

Theorem . The sequence spaces ba,b_ (B(m)),ba,b_c (B(m))and ba,b_∞(B(m))are BK -spaces with their sup-norm defined by

x _ba,b

 (B(m))= x bac,b(B(m))= x ba∞,b(B(m)₎=sup

n∈N

Ba,bB(m)xk

n.

Proof The sequence spacesba,b_ ,ba,b

c andba,b∞ areBK-spaces with theirsup-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

ba,b (B(m)),ba,bc (B(m)) andba,b∞(B(m)) areBK-spaces.

Theorem . The sequence spaces ba,b_ (B(m)_),ba,b

c (B(m))and ba,b∞(B(m))are linearly

isomor-phic to the spaces c,c and∞,respectively.

Proof Similarly, we prove the theorem only for the spaceba,b_ (B(m)_{). To proveb}a,b  (B(m))∼=

c, we must show the existence of a linear bijection between the spacesba,b (B(m)) andc.

Considerba,b_ (B(m)_)→c

byT(x) =Ba,b(B(m)xk). The linearity ofT is obvious andx= 

wheneverT(x) = . Therefore,Tis injective.

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

xk= k

i=

(a+b)i

k

j=i

m+k–j– 

k–j j i

(–s)k–j

rm+k–j(–a) j–i_b–j_y

i (.)

for eachk∈N. Then we have

lim n→∞

Ba,bB(m)xk

n=_nlim_→∞

 (a+b)n

n

k=

n k

an–kbkB(m)xk

= lim n→∞yn= ,

which implies thatx∈ba,b_ (B(m)_{) andT(x) =y. Consequently,Tis surjective and is norm}

preserving. Thus,ba,b_ (B(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

Theorem . The inclusion ba,b_ (B(m)_)⊆ba,b

c (B(m))⊆ba,b∞(B(m))holds.

Theorem . The inclusions ea

(B(m))⊆b a,b

 (B(m)), eac(B(m))⊆ba,bc (B(m))and ea∞(B(m))⊆ ba,b

∞(B(m))strictly hold.

Proof Similarly, we only prove the inclusionea

(B(m))⊆ba,b(B(m)). Ifa+b= , we have

Ea=Ba,b. Soea_(B(m))⊆ba,b_ (B(m)) holds. Let  <a<  andb= . We define a sequence

x= (xk) byxk= (–_a)kfor eachk∈N. It is clear that

EaB(m)xk

=

_m

i=

m

i

sirm–i

–a 

i

(– –a)n

/

∈c

and

Ba,bB(m)xk

=

_m

i=

m

i

sirm–i

–a 

i

  +a

n

∈c.

So, we have x∈ba,b_ (B(m))\ea_(B(m)). This shows that the inclusionea_(B(m))⊆ba,b_ (B(m))

strictly holds.

3 The Schauder basis and

α

β

γ

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

for everyx∈X, there is a unique scalar sequence (λk) such that x–

n

k=λkxk →

as n→ ∞. We shall construct Schauder bases for the sequence spaces ba,b_ (B(m)) and

ba,b c (B(m)).

We define the sequenceg(k)_{(a,b) ={g}(k)

i (a,b)}i∈Nby

g_i(k)(a,b) =

⎧ ⎨ ⎩

 if ≤i<k, (a+b)ki

j=k

m+i–j– i–j

_j

k

_(–s)i–j rm+i–jb

–j_(–a)j–k _ifi≥k,

for eachk∈N.

Theorem . The sequence(g(k)_(a,b))

k∈Nis a Schauder basis for the binomial sequence space ba,b_ (B(m)_{)and every x= (x}

i)∈ba,b (B(m))has a unique representation by

x=

k

λk(a,b)g(k)(a,b), (.)

whereλk(a,b) = [Ba,b(B(m)xi)]kfor each k∈N.

Proof Obviously,Ba,b_(B(m)_g(k)

i (a,b)) =ek∈c, whereek is the sequence with  in thekth

place and zeros elsewhere for eachk∈N. This implies thatg(k)(a,b)∈ba,b_ (B(m)) for each

k∈N.

Forx∈ba,b_ (B(m)_{) andn∈N, we put}

x(n)=

n

k=

By the linearity ofBa,b_(B(m)_{), we have}

Ba,bB(m)x(n)_i =

n

k=

λk(a,b)Ba,b

B(m)g_i(k)(a,b)=

n

k=

λk(a,b)ek

and

Ba,bB(m)xi–x(n)i

k=

⎧ ⎨ ⎩

 if ≤k<n, [Ba,b(B(m)xi)]k ifk≥n,

for eachk∈N.

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

Ba,bB(m)xi

k<

ε

 for allk≥n. Then we have

x–x(n)_ba,b

 (B(m))=sup_k≥n

Ba,bB(m)xi

k≤sup k≥n

Ba,bB(m)xi

k<

ε

<ε,

which implies thatx∈ba,b_ (B(m)_{) is represented as in (.).}

To show the uniqueness of this representation, we assume that

x=

k

μk(a,b)g(k)(a,b).

Then we have

Ba,bB(m)xi

k=

k

μk(a,b)

Ba,bB(m)g_i(k)(a,b)_k=

k

μk(a,b)(ek)k=μk(a,b),

which is a contradiction with the assumption thatλk(a,b) = [Ba,b(B(m)xi)]kfor eachk∈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 b}a,b

c (B(m)) and every x∈ba,bc (B(m)) has a

unique representation by

x=lg+

k

λk(a,b) –l

g(k)(a,b). (.)

Proof Obviously,Ba,b_(B(m)_gk

i(a,b)) =ek∈candg∈ba,bc (B(m)). Forx∈ba,bc (B(m)), we put

y=x–lgand we havey∈ba,b_ (B(m)_{). Hence, we deduce thatyhas a unique representation}

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

proof.

Corollary . The sequence spaces ba,b_ (B(m)_{)and b}a,b

Köthe and Toeplitz [] first computed the dual whose elements can be represented as sequences and defined theα-dual (or Köthe-Toeplitz dual). Next, we compute theα-,β -andγ-duals of the sequence spacesba,b_ (B(m)_),ba,b

c (B(m)) andba,b∞(B(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,c) and Xγ =M(X,∞),

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 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. Theorem . Theα-dual of the spaces ba,b_ (B(m)_),ba,b

c (B(m))and ba,b∞(B(m))is the set

U_a,b=

u= (uk)∈w:sup K∈

k

i∈K

(a+b)i

k

j=i

m+k–j– 

k–j j i

×(–s)k–j

rm+k–jb

–j_(–a)j–i_u k

<∞

.

Proof Letu= (uk)∈wandx= (xk) be defined by (.), then we have

ukxk= k

i=

(a+b)i

k

j=i

m+k–j– 

k–j j i

(–s)k–j

rm+k–jb

–j_(–a)j–i_u kyi=

for eachk∈N, whereGa,b_{= (g}a,b

k,i) is defined by

g_k,ia,b=

⎧ ⎨ ⎩

(a+b)ik j=i

m+k–j– k–j

_j

i

_(–s)k–j

rm+k–jb–j(–a)j–iuk if ≤i≤k,

 ifi>k.

Therefore, we deduce thatux= (ukxk)∈wheneverx∈ba,b (B(m)),ba,bc (B(m)) orba,b∞(B(m)),

if and only ifGa,b_y∈

, whenevery∈c,cor∞. This implies thatu= (uk)∈[ba,b (B(m))]α,

[ba,b

c (B(m))]αor [ba,b∞(B(m))]αif and only ifGa,b∈(c:),Ga,b∈(c:) orGa,b∈(∞:) by

Parts (i) of Lemma .. So we obtain

u= (uk)∈

ba,b_ B(m)α=ba,b_c B(m)α=ba,b_∞B(m)α

if and only if

sup K∈

k

i∈K

(a+b)i

k

j=i

m+k–j– 

k–j j i

(–s)k–j

rm+k–jb

–j_(–a)j–i_u k

<∞.

Thus, we have [ba,b_ (B(m))]α_{= [b}a,b

c (B(m))]α= [ba,b∞(B(m))]α=U a,b

 .

Now, we define the setsU_a,b,U_a,b,U_a,bandU_a,bby

U_a,b=

u= (uk)∈w:sup n∈N

k

|un,k|<∞

,

U_a,b=

u= (uk)∈w: lim

n→∞un,kexists for eachk∈N

,

U_a,b=

u= (uk)∈w: lim n→∞

k

|un,k|=

k

lim

n→∞un,k

,

and

U_a,b=

u= (uk)∈w: lim n→∞

k

un,kexists

,

where

un,k= (a+b)k n

i=k i

j=k

m+i–j– 

i–j

j k

(–s)i–j

rm+i–j(–a) j–k_(b)–j_u

i.

Theorem . We have the following relations:

(i) [ba,b_ (B(m)_)]β_=Ua,b

 ∩U

a,b  , (ii) [ba,b

c (B(m))]β=U a,b

 ∩U

a,b

 ∩U

a,b  , (iii) [ba,b

Proof Letu= (uk)∈wandx= (xk) be defined by (.), then we consider the following

equation:

n

k=

ukxk= n

k=

uk

⎡ ⎣k

i=

(a+b)i k

j=i

m+k–j– 

k–j j i

(–s)k–j

rm+k–j(–a) j–i_b–j_y

i

⎤ ⎦

=

n

k=

⎡ ⎣(a+b)k

n

i=k i

j=k

m+i–j– 

i–j

j k

(–s)i–j

rm+i–j(–a) j–k_b–j_u

i

⎤ ⎦yk

=Ua,by_n,

whereUa,b_{= (u}a,b

n,k) is defined by

ua,b_n,k=

⎧ ⎨ ⎩

(a+b)kn i=k

i j=k

m+i–j– i–j

_j

k

_(–s)i–j rm+i–j(–a)

j–k_(b)–j_u

i if ≤k≤n,

 ifk>n.

Therefore, we deduce thatux= (ukxk)∈cwheneverx∈ba,b (B(m)) if and only ifUa,by∈c

whenevery∈c, which implies thatu= (uk)∈[ba,b(B(m))]βif and only ifUa,b∈(c:c) by

Part (ii) of Lemma .. So we obtain [ba,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.

Theorem . Theγ-dual of the spaces ba,b_ (B(m)_),ba,b

c (B(m))and ba,b∞(B(m))is the set U a,b  .

4 Certain matrix mappings on the spaceba,b c (B(m))

In this section, we characterize matrix transformations fromba,b

c (B(m)) intop,∞andc.

Let us define the matrix= (θn,k) via an infinite matrix= (λn,k) by=(Ba,b(B(m)))–,

that is,

θn,k= (a+b)k

∞

j=k

m+n–j– 

n–j

j k

(–s)n–j

rm+n–j(–a) j–k_b–j_λ

n,j, (.)

where (Ba,b_(B(m)₎₎–_{is the inverse of theB}a,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∈[ba,bc (B(m))]βfor all n∈N,then

∈(ba,b

c (B(m)) :Z)if and only if∈(c:Z).

Proof Let∈(ba,b

c (B(m)) :Z) andy= (yn)∈c. For everyx= (xk)∈ba,bc (B(m)), we havexk=

[(Ba,b(B(m)))–yn]k. Since (λn,k)k∈[ba,bc (B(m))]βfor alln∈N, this implies the existence of the

-transform ofx,i.e.xexists. So we obtainx=(Ba,b_(B(m)₎₎–_{y=y, which implies}

that∈(c:Z).

Conversely, let∈(c:Z) andx∈ba,b_c (B(m)). For everyy∈c, we haveyn= [Ba,b(B(m)xk)]n.

in a similar way to the proof of Theorem .. So we havey=Ba,b_(B(m)_{)x=x, which}

shows that∈(ba,b

c (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 K∈

n

k∈K

an,k

p<∞, ≤p<∞. (.)

For brevity of notation, we write

tn,k= (a+b)k n

i=k i

j=k

m+i–j– 

i–j

j k

(–s)i–j

rm+i–j(–a) j–k_(b)–j_a

n,j,

t_n,kl = (a+b)k

l

i=k i

j=k

m+i–j– 

i–j

j k

(–s)i–j

rm+i–j(–a) j–k_(b)–j_a

n,j

for alln,k∈N.

By using Lemma ., there are some immediate consequences withtn,k ortln,kin place

ofan,kin Lemma . and Lemma ..

Theorem . A∈(ba,b

c (B(m)) :p)if and only if sup

K∈

n

k∈K

tn,k

p<∞, (.)

tn,k exists for each k,n∈N, (.)

k

tn,k converges for each n∈N, (.)

sup l∈N

l

k=

tl_n,k<∞, n∈N. (.)

Theorem . A∈(ba,b

c (B(m)) :∞)if and only if(.)and(.)hold,and sup

n∈N

k

|tn,k|<∞. (.)

Theorem . A∈(ba,b

c (B(m)) :c)if and only if(.), (.)and(.)hold,and lim

n→∞tn,k exists for each k∈N, (.) lim

n→∞

k

tn,k exists. (.)

5 Conclusion

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.

Publisher’s Note

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

Received: 15 May 2017 Accepted: 8 August 2017

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