Binomial difference sequence spaces of fractional order

R E S E A R C H

Open Access

Binomial difference sequence spaces of

fractional order

Jian Meng

1

_{and Liquan Mei}

1*

*_{Correspondence:}

[email protected]

1_{School of Mathematics and}

Statistics, Xi’an Jiaotong University, Xi’an, P.R. China

Abstract

In this paper, we introduce the sequence spacesbr,s₀(∇(α)),br,s

c (∇(α)), andbr,s∞(∇(α)). We investigate some functional properties, inclusion relations, and the

α

-,

β

-,

γ

-, and continuous duals of these sets.

MSC: 46A45; 46B45; 46B50

Keywords: Sequence space; Matrix transformation; Fractional difference operator;

α

-,

β

-, and

γ

-duals

1 Introduction

Letw,p,∞,c, andc0denote the spaces of all,p-absolutely summable, bounded,

conver-gent, and null sequencesx= (xk) with complex termsxk, respectively, where 1≤p<∞ andk∈N={0, 1, 2, . . .}. A sequence spaceXis called aBK-space if it is a Banach space with continuous coordinatespk:X→Cdefined bypk(x) =xkforx= (xk)∈Xandk∈N. The most important result of the theory ofBK-spaces is that matrix mappings between

BK-spaces are continuous [13]. The sequence spaces_∞,c, and c0 with theirsup-norm

areBK-spaces.

The concept of a difference sequence space was firstly introduced by Kizmaz [22] by defining the setZ() ={x= (xk) : (xk)∈Z} forZ∈ {_∞,c,c0}, wherexk=xk–xk+1

fork∈N. The idea of a difference sequence was generalized by Et and Çolak [14–16] by defining the spacesZ(m_{) ={x= (xk) : (}m_{xk)∈Z}forZ∈ {}

∞,c,c0}, wherem∈N,

mxk=m–1xk–m–1xk+1 fork∈N. For a positive proper fractionα, Baliarsingh and

Dutta [4,5] defined the fractional difference operatorαby

αxk= ∞

i=0

(–1)i (α+ 1)

i!(α–i+ 1)xk+i

fork∈N, where the Euler gamma function(p) of a real numberpwithp∈ {/ 0, –1, –2, –3, . . .}can be expressed as an improper integral(p) =₀∞e–ttp–1dt. It is observed that

(i) (p+ 1) =p!forp∈N,

(ii) (p+ 1) =p(p)forp∈R\ {0, –1, –2, –3, . . .}.

Some definitions of fractional derivatives have been generalized by using a set of new difference sequence spaces of fractional order [3]. Application of fractional derivatives

becomes more apparent in diffusion processes, modeling mechanical systems, and many other fields.

LetX,Y be two sequence spaces, and letA= (an,k) be an infinite matrix with complex

numbers an,k,n,k∈N. Let the sequence spaceXA defined by XA={x= (xk) :Ax∈X}

denote the domain of matrixAin the spaceX, whereAx={(Ax)n}, theA-transform ofx,

is defined by (Ax)n=

∞

k=0an,kxk,n∈N. Let (X:Y) denote the class of all matrices such

thatX⊆YA. The matrix domain approach has been employed by Başarir and Kara [6–

10], Kara and İlkhan [19–21], Polat and Başar [26], Song and Meng [23–25,27], and many others to introduce new sequence spaces.

The Euler sequence spaceser₀,er_c, ander_∞were defined by Altay and Başar [1] and Altay, Başar, and Mursaleen [2]. Moreover, Kadak and Baliarsingh [18] introduced a generalized Euler mean difference operatorEr_(∇(α)_{) of fractional order, where the matrix∇}(α)_{= (∇}(α)

n,k)

is defined by

∇(α)

n,k=

⎧ ⎨ ⎩

(–1)n–k (α+1)

(n–k)!(α–n+k+1) if 0≤k≤n,

0 ifk>n.

Letr,s∈Randr+s= 0. Then the binomial matrixBr,s_{= (b}r,s

n,k) is defined by

br_n,_,s_k=

⎧ ⎨ ⎩

1 (s+r)n

n

k sn–krk if 0≤k≤n,

0 ifk>n,

for allk,n∈N. Ifs+r= 1, then we obtain the Euler matrixEr. Bişgin [11,12] defined the binomial sequence spaces

br₀,s=

x= (xk) : lim n→∞

1 (s+r)n

n

k=0

n k

sn–krkxk= 0

,

br_c,s=

x= (xk) : lim

n→∞

1 (s+r)n

n

k=0

n k

sn–krkxkexists

,

and

br_∞,s=

x= (xk) :sup

n∈N

1 (s+r)n

n

k=0

n k

sn–krkxk

<∞

.

The purpose of this paper is to generalize the sequence spaceser0(∇(α)),erc(∇(α)), and er

∞(∇(α)) and introduce the binomial difference sequence spacesbr0,s(∇(α)),brc,s(∇(α)), and br,s

∞(∇(α)) of fractional order whose∇(α)-transforms are in the spacesb0r,s, brc,s, and br∞,s.

These new sequence spaces are generalizations of the spaces defined in [11,12,18,23,25,

26].

2 Difference sequence spaces of fractional order

In this chapter, we introduce the binomial difference sequence spacesbr₀,s(∇(α)_),br,s c (∇(α)),

andbr_∞,s(∇(α)_{) of fractional order and investigate some functional properties and inclusion}

Define the spacesbr₀,s(∇(α)_),br,s

c (∇(α)), andb∞r,s(∇(α)) by

Z∇(α) _{=x= (x}

k) :∇(α)(xk)∈Z

forZ∈ {br₀,s,br,s

c ,br∞,s}. By taking the∇(α)-transform ofxin the spacesb r,s

0 ,brc,s, andbr∞,sthe

spacesbr₀,s(∇(α)_),br,s

c (∇(α)), andb∞r,s(∇(α)) can be redefined by

br₀,s∇(α) _=br,s

0 ∇(α), brc,s

∇(α) _=br,s

c ∇(α), br_∞,s

∇(α) _=br,s

∞ ∇(α). (2.1)

The sequence spacesbr₀,s(∇(α)_),br,s

c (∇(α)), andb∞r,s(∇(α)) include some particular cases in

certain cases ofs,r, andα.

(i) Forα= 0, these sequence spaces generalize the spacesbr₀,s,br,s

c , andbr∞,sdefined by

Bişgin [11,12].

(ii) Fors+r= 1, these sequence spaces generalize the spaceser

0(∇(α)),erc(∇(α)), and er

∞(∇(α))defined by Kadak and Baliarsingh [18].

(iii) Forα=m∈N, these sequence spaces generalize the spacesb₀r,s(∇(m)_),br,s c (∇(m)),

andbr,s

∞(∇(m))defined by Meng and Song [23].

(iv) Fors+r= 1andα=m∈N, these sequence spaces generalize the spaceser

0(∇(m)),

er_c(∇(m)), ander_∞(∇(m))defined by Polat and Başar [26].

Define the sequencey= (yn) by theBr,s(∇(α)_{)-transform of a sequencex= (xk), that is,}

yn=Br,s∇(α) (xk)_n

= 1

(s+r)n n

k=0

n

i=k

(–1)i–k

n i

(α+ 1) (i–k)!(α–i+k+ 1)s

n–i_ri_{xk (2.2)}

for eachn∈N.

Theorem 2.1 Let Z∈ {br₀,s,br,s

c ,br∞,s}.Then Z(∇(α))are BK -spaces with the norm x Z(∇(α)₎= ∇(α)_(xk)

Z.

Proof Theorem2.1of Bişgin [11,12] and Theorem 4.3.12 of Wilansky [29] imply that the

spacesZ(∇(α)_{) areBK-spaces.}

Theorem 2.2 The inclusion br₀,s(∇(α)_)⊆br,s

c (∇(α))⊆b∞r,s(∇(α))is strict.

Proof Proof follows from Lemma 2.3 of Et and Nuray [17].

Theorem 2.3 The inclusions er

0(∇(α))⊆b0r,s(∇(α)),erc(∇(α))⊆brc,s(∇(α)), and er∞(∇(α))⊆

br,s

∞(∇(α))are strict.

Proof We only give the proof of the inclusion er₀(∇(α)_)⊆br,s

0 (∇(α)). The others can be

proved similarly. It is clear thater

0(∇(α))⊆b

r,s

0 (∇(α)). Further, to show that this inclusion is strict, let 0 <

r< 1 ands= 4 and define the sequencex= (xk) by

xk=

k

j=0

(–1)k–j (–α+ 1) (k–j)!(–α–k+j+ 1)

–3

r

fork∈N. We have [Er_(∇(α)_)(xk)]

n= ((–2 –r)n) /∈c0 and [Br,s(∇(α))(xk)]n= ((₄₊1_r)n)∈c0.

Therefore, the inclusioner₀(∇(α)_)⊆br,s

0 (∇(α)) is strict.

Theorem 2.4 The spaces br₀,s(∇(α)_),br,s

c (∇(α)),and b∞r,s(∇(α))are linearly isomorphic to the

spaces c0,c,and∞,respectively.

Proof We prove the theorem only for the spacebr0,s(∇(α)). To proveb0r,s(∇(α))∼=c0, we will

show the existence of a linear bijection between the spacesbr₀,s(∇(α)_{) andc} 0.

Let us denote the transformationT:br₀,s(∇(α)_)→c

0byT(x) =Br,s(∇(α))(xk). The linearity

ofTis clear, andx= 0 wheneverT(x) = 0. HenceTis injective. Lety= (yn)∈c0and define the sequencex= (xk) by

xk=

k

i=0

(s+r)i

k

j=i

(–1)k–j

j i

(–α+ 1) (k–j)!(–α–k+j+ 1)r

–j_(–s)j–i_{yi (2.3)}

fork∈N. Then we have

lim

n→∞

Br,s∇(α) (xk)_n= lim

n→∞

1 (s+r)n

n

k=0

n k

sn–krk∇(α) (xk) = lim

n→∞yn= 0,

which implies thatx∈br₀,s(∇(α)_{). Therefore, we obtain thatTis surjective and norm}

pre-serving. This completes the proof.

We shall construct the Schauder bases for the sequence spacesbr₀,s(∇(α)_{) andb}r,s c (∇(α)).

Because the isomorphismTbetweenbr₀,s(∇(α)_{) andc}

0(or betweenbrc,s(∇(α)) andc) is onto,

the inverse image of the basis of the spacec0(orc) is the basis of the spaceb0r,s(∇(α)) (or

br,s

c (∇(α))). Fork∈N, define the sequenceg(k)(r,s) ={g

(k)

i (r,s)}i∈Nby

g_i(k)(r,s) =

⎧ ⎨ ⎩

0 if 0≤i<k,

(s+r)ki j=k(–1)i–j

_j

k

(–α+1) (i–j)!(–α–i+j+1)r

–j_(–s)j–k _ifi≥k.

Theorem 2.5 The sequence(g(k)(r,s))k∈Nis the Schauder basis for the space br0,s(∇(α)),and

every x in br₀,s(∇(α)_{)has a unique representation by}

x=

k

λk(r,s)g(k)(r,s), (2.4)

whereλk(r,s) = [Br,s_(∇(α)_)(xi)]

kfor k∈N.

Theorem 2.6 Define the sequence g= (gn)by

gn=

n

k=0

(s+r)k

n

j=k

(–1)n–j

j k

(–α+ 1) (n–j)!(–α–n+j+ 1)r

–j_(–s)j–k

for n∈Nandlim_k→∞λk(r,s) =l.The set{g,g(0)(r,s),g(1)(r,s),g(2)(r,s), . . .}is the Schauder basis for the space br,s

c (∇(α)),and every x in bcr,s(∇(α))has a unique representation by

x=lg+

k

λk(r,s) –l

3 The

α

-,

β

-,

γ

-, and continuous duals

In this section, we determine theα-,β-,γ-, and continuous duals of the spacesb₀r,s(∇(α)_),

br_c,s(∇(α)_{), andb}r,s

∞(∇(α)).

For two sequence spacesXandY, the setM(X,Y) is defined by

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

Letbsandcsdenote the sequence spaces of all bounded and convergent series, respec-tively. In particular,

Xα=M(X,1), Xβ=M(X,cs), and Xγ =M(X,bs)

are called theα-, β-, andγ-duals of the sequence spaceX, respectively. The space of all bounded linear functionals onXdenoted byX∗ is called the continuous dual of the spaceX.

Let us give the following properties needed in Lemma3.1:

sup

K∈

n

k∈K an,k

<∞, (3.1)

sup

n∈N

k

|an,k|<∞, (3.2)

lim

n→∞an,k=ak fork∈N, (3.3)

lim

n→∞

k

an,k=a, (3.4)

lim

n→∞

k

|an,k|=

k

lim

n→∞an,k

, (3.5)

whereis the collection of all finite subsets ofN.

Lemma 3.1([28]) Let A= (an,k)be an infinite matrix.Then

(i) A∈(c0:1) = (c:1) = (∞:1)if and only if(3.1)holds.

(ii) A∈(c0:c)if and only if(3.2)and(3.3)hold.

(iii) A∈(c:c)if and only if(3.2), (3.3),and(3.4)hold. (iv) A∈(_∞:c)if and only if(3.3)and(3.5)hold.

(v) A∈(c0:∞) = (c:∞) = (∞:∞)if and only if(3.2)holds.

Theorem 3.2 We have[br₀,s(∇(α)_)]α_{= [b}r,s

c (∇(α))]α= [b∞r,s(∇(α))]α=U r,s

1 ,where

U₁r,s=

u= (uk) :sup

K∈

k

i∈K

(s+r)i

k

j=i

(–1)k–j

j i

× (–α+ 1) (k–j)!(–α–k+j+ 1)r

–j_(–s)j–i_u k

<∞

Proof We immediately derive by (2.3) that

ukxk= k

i=0

(s+r)i

k

j=i

(–1)k–j

j i

(–α+ 1) (k–j)!(–α–k+j+ 1)r

–j_(–s)j–i_u kyi=

Gr,sy_k

fork∈N, whereGr,s_{= (g}r,s

k,i) is defined by

g_kr,_,s_i=

⎧ ⎨ ⎩

(s+r)ik j=i(–1)k–j

_j

i

(–α+1)

(k–j)!(–α–k+j+1)r–j(–s)j–iuk if 0≤i≤k,

0 ifi>k.

Thereforeux= (ukxk)∈1 wheneverx∈br0,s(∇(α)),brc,s(∇(α)) or br∞,s(∇(α)) if and only if

Gr,s_y∈

1whenevery∈c0,cor∞. This implies thatu= (uk)∈[b0r,s(∇(α))]α, [bcr,s(∇(α))]α,

or [br,s

∞(∇(α))]α if and only ifGr,s∈(c0:1) = (c:1) = (∞:1). We derive by part (i) of

Lemma3.1thatu= (uk)∈[b₀r,s(∇(α)_)]α_{= [b}r,s

c (∇(α))]α= [b∞r,s(∇(α))]αif and only if

sup K∈ k

i∈K

(s+r)i

k

j=i

(–1)k–j

j i

(–α+ 1) (k–j)!(–α–k+j+ 1)r

–j_(–s)j–i_uk

<∞,

which yields that [br₀,s(∇(α)_)]α_{= [b}r,s

c (∇(α))]α= [br∞,s(∇(α))]α=U1r,s.

To determine theβ- andγ-duals of the spacesbr₀,s(∇(α)_),br,s

c (∇(α)), andb∞r,s(∇(α)), we define

the following sets:

U₂r,s=

u= (uk) :sup

n∈N

k

|un,k|<∞

,

U₃r,s=

u= (uk) : lim

n→∞un,kexists for eachk∈N

,

U₄r,s=

u= (uk) : lim

n→∞

k

|un,k|=

k

lim

n→∞un,k

,

and

U₅r,s=

u= (uk) : lim

n→∞

k

un,kexists

,

where

un,k= (s+r)k n

i=k i

j=k

(–1)i–j

j k

(–α+ 1) (i–j)!(–α–i+j+ 1)r

–j_(–s)j–k_u i.

Theorem 3.3

(i) [br0,s(∇(α))]β=U2r,s∩U3r,s,

(ii) [br,s

c (∇(α))]β=U r,s

2 ∩U

r,s

3 ∩U

r,s

5 ,

(iii) [br,s

∞(∇(α))]β=U3r,s∩U4r,s,

(iv) [br₀,s(∇(α)_)]γ _{= [b}r,s

c (∇(α))]γ= [b∞r,s(∇(α))]γ =U r,s

Proof We consider the equality

n

k=0

ukxk= n

k=0

uk

_k

i=0

(s+r)i

k

j=i

(–1)k–j

j i

(–α+ 1) (k–j)!(–α–k+j+ 1)r

–j_(–s)j–i_y i

=

n

k=0

(s+r)k

n

i=k i

j=k

(–1)i–j

j k

(–α+ 1) (i–j)!(–α–i+j+ 1)r

–j_(–s)j–k_u i

yk

=Ur,sy_n,

whereUr,s= (ur_n,_,s_k) is defined by

ur_n,_,s_k=

⎧ ⎨ ⎩

(s+r)kn i=k

i

j=k(–1)i–j

_j

k

(–α+1)

(i–j)!(–α–i+j+1)r–j(–s)j–kui if 0≤k≤n,

0 ifk>n.

Thereforeux= (ukxk)∈cswheneverx∈br0,s(∇(α)) if and only ifUr,sy∈cwhenevery∈c0.

This implies thatu= (uk)∈[br0,s(∇(α))]βif and only ifUr,s∈(c0:c). We obtain by part (ii)

of Lemma3.1that [br₀,s(∇(α)_)]β_=Ur,s

2 ∩U

r,s

3 . The proof can be completed in a similar way

by parts (iii), (iv), (v) instead of Part (ii) of Lemma3.1, so we omit the details.

Theorem 3.4 The spaces[br₀,s(∇(α)_)]∗_and[br,s

c (∇(α))]∗are equivalent to1.

Proof Proof follows from Theorem 3.6 of Bişgin [11] and the fact that ifZ is a Banach

space, then [Z(∇(α)_)]∗_{is equivalent toX}∗_[17].

4 Conclusion

In this paper, we have discussed some results obtained from the matrix domain of the binomial matrix and the difference matrix of fractional order. Our main aim is to general-ize the results on the matrix domain of the Euler matrix. It is immediate that our results reduce to the sequence spaces defined in [11,12,18,23,25,26].

Acknowledgements

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

Funding

This work is supported by NSF of China (11371289).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the writing of 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: 6 July 2018 Accepted: 1 October 2018

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