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J. Nonlinear Sci. Appl. 8 (2015), 997–1004
Research Article
On certain Euler difference sequence spaces of
fractional order and related dual properties
Ugur Kadaka,∗, P. Baliarsinghb
a
Department of Mathematics, Bozok University, 66100 Yozgat , Turkey.
b
Department of Mathematics, School of Applied Sciences, KIIT University, India.
Abstract
The purpose of this paper is to generalize the Euler sequences of nonabsolute type by introducing a generalized Euler mean difference operator E r (∆(α̃) ) of order α. We investigate their topological structures as
well as some interesting results concerning the operator E r (∆(α̃) ) for a proper fraction α̃. Also we obtain
the α-, β- and γ-duals of these sets.
Keywords: Euler sequence spaces of nonabsolute type, linear operator, matrix transformations, α-, β- and
γ-duals.
2010 MSC: 46A45, 46A35, 46B45.
1. Introduction
By Γ(α̃), we denote the Euler gamma function of a real number α̃. Using the definition, Γ(α̃) with
α̃ ∈
/ {0, −1, −2, −3 . . . } can be expressed as an improper integral as follows:
Z ∞
Γ(α̃) =
e−t tα̃−1 dt.
(1.1)
0
Also, the Euler gamma function is known as the generalized factorial function. Let w be the set of all
sequences of real numbers and `∞ , c and c0 respectively be the Banach spaces of bounded, convergent and
null sequences x = (xk ) with the usual norm kxk = supk |xk | . By bs, cs, `1 and `p we denote the spaces of
all bounded, convergent, absolutely and p−absolutely convergent series, respectively.
∗
Corresponding author
Email addresses: ugurkadak@gmail.com (Ugur Kadak), pb.math10@gmail.com (P. Baliarsingh)
Received 2015-4-27
U. Kadak, P. Baliarsingh, J. Nonlinear Sci. Appl. 8 (2015), 997–1004
998
For a positive proper fraction α̃, Baliarsingh and Dutta [6, 7] ( also, see [8, 13]) have defined the
generalized fractional difference operator ∆(α̃) as
∆
(α̃)
(xk ) =
∞
X
(−1)i
i=0
Γ(α̃ + 1)
xk−i .
i!Γ(α̃ − i + 1)
(1.2)
Throughout the text it is assumed that the series defined in (1.2) is convergent for x ∈ w. More specifically,
it is convenient to express the difference operator ∆(α̃) as a triangle i.e.,
(
Γ(α̃+1)
(−1)(n−k) (n−k)!Γ(
α̃−n+k+1) , (0 ≤ k ≤ n)
(∆(α̃) )nk =
0,
(k > n)
In fact, this difference matrix includes several difference matrices introduced by Ahmad and Mursaleen [1],
Malkowsky et al. [18] and many others (see[5, 10, 14, 15, 16, 17, 19, 20, 23]).
The well known Euler mean matrix E r = (ernk ) of order r, (0 < r < 1) is defined by the matrix
  n (1 − r)n−k rk , (k ≤ n)
r
k
enk =

0,
(k > n).
Equivalently, we may write




Er = 


1
0
0
0
1−r
r
0
0
(1 − r)2 2(1 − r)r
r2
0
3
2
2
(1 − r) 3(1 − r) r 3(1 − r)r r3
..
..
..
..
.
.
.
.
...
...
...
...
..
.




.


Combining the Euler mean matrix of order r and the difference matrix of order α̃, we define the product
matrix E r (∆(α̃) ) as
 n
X

Γ(α̃ + 1)
n

i−k
(−1)
ri (1 − r)n−i , (0 ≤ k ≤ n)
r
(α̃)
n − i (i − k)!Γ(α̃ − i + k + 1)
(E (∆ ))nk =

 i=k
0,
(k > n)
Moreover, (E r (∆(α̃) ))nk can be written as follows:

(E r (∆(α̃) ))nk
1

(1 − r) − α̃r

= (1 − r)2 − 2α̃(1 − r)r +

..
.
α̃(α̃−1) 2
r
2!
0
0 0 ···
r
0 0 ···
2
2(1 − r)r − α̃r r2 0 · · ·
..
.. .. . .
.
.
. .



.

Let A = (ank ) be an infinite matrix of real numbers ank , where n, k ∈ N0 , the set of all natural numbers
including zero. For the sequence spaces X and Y , we write a matrix mapping A : X → Y defined by
X
(Ax)n =
ank xk , (n ∈ N0 ).
(1.3)
k
P
For every x = (xk ) ∈ X, we call Ax the A−transform of x if the series k ank xk converges for each n ∈ N0 .
By (X, Y ), we denote the class of all infinite matrices A such that A : X → Y . Thus, A ∈ (X, Y ) if and
only if the series in (1.3) converges for each n ∈ N0 .
U. Kadak, P. Baliarsingh, J. Nonlinear Sci. Appl. 8 (2015), 997–1004
999
The matrix domain λA of an infinite matrix A in a sequence space λ is defined by
λA = {x = (xk ) ∈ w : Ax ∈ λ},
In fact, for most cases the new sequence space λA generated by the limitation matrix A can be expressed
as either an expansion or contraction of the original space λ. In some cases they may overlap each others.
It is known that the duality is an important concept in the theory of sequence spaces, especially for
studying of topological structures of sequence spaces. Let X, Y ⊂ w and define the set S(X, Y ) by
S(X, Y ) = {z = (zk ) : xz = (xk zk ) ∈ Y for all x ∈ X}.
(1.4)
With the notation of (1.4), we redefine the α−, β− and γ−duals of a sequence space X respectively as
follows:
X α = S(X, `1 ), X β = S(X, cs) and X γ = S(X, bs).
Now, we give the following results involving the inverse of the matrices ∆(α̃) , E r and E r (∆(α̃) )
Lemma 1.1 ([9, 12]). The inverse of the difference matrix ∆(α̃) is given by the triangle
(
Γ(−α̃+1)
(−1)(n−k) (n−k)!Γ(−
(−α̃)
α̃−n+k+1) , (0 ≤ k ≤ n),
(∆
)nk =
0,
(k > n).
Lemma 1.2. The inverse of the Euler mean matrix E r is given by the triangle
(−1)(n−k) nk (1 − r)n−k r−n , (0 ≤ k ≤ n),
1/r
(E )nk =
0,
(k > n).
Proof. Proof is straightforward (see [12]).
Lemma 1.3. The inverse of the Euler mean difference matrix E r (∆(α̃) ) is given by a triangle (bnk ), where
 n
X

j Γ(−α̃ + 1)(1 − r)j−k r−j

(n−k)

, (0 < k ≤ n),
(−1)


k (n − j)!Γ(−α̃ − n + j + 1)
 j=k
bnk =
1
(k = n),

rn ,





0,
(k > n).
Proof. Proof follows from Lemma 1.1 and Lemma 1.2.
2. New Euler difference sequence spaces
In this section, we define certain sequence spaces of non absolute type erp (∆(α̃) ), er0 (∆(α̃) ),erc (∆(α̃) ) and
er∞ (∆(α̃) ) by combining the Euler mean operator E r and the fractional difference operator ∆(α̃) . We also
investigate their certain topological properties. In addition, the α−, β− and γ−duals of these spaces have
been determined.
For a positive real number α̃ and 0 < r < 1, we define certain classes of Euler fractional difference
sequence spaces as

p

n n


X X
X
n
Γ(α̃ + 1)
i−j
i
n−i (−1)
erp (∆(α̃) ) = x = (xk ) :
<
∞
r
(1
−
r)
x
j


n − i (i − j)!Γ(α̃ − i + j + 1)
n j=0 i=j


n X
n


X
n
Γ(α̃ + 1)
r
(α̃)
i−j
e0 (∆ ) = x = (xk ) : lim
(−1)
ri (1 − r)n−i xj = 0
n→∞


n − i (i − j)!Γ(α̃ − i + j + 1)
j=0 i=j


n X
n


X
n
Γ(α̃
+
1)
erc (∆(α̃) ) = x = (xk ) : lim
(−1)i−j
ri (1 − r)n−i xj exists
n→∞


n − i (i − j)!Γ(α̃ − i + j + 1)
j=0 i=j
U. Kadak, P. Baliarsingh, J. Nonlinear Sci. Appl. 8 (2015), 997–1004
1000


n n


XX
n
Γ(α̃
+
1)
er∞ (∆(α̃) ) = x = (xk ) : sup (−1)i−j
ri (1 − r)n−i xj < ∞


n − i (i − j)!Γ(α̃ − i + j + 1)
n j=0 i=j
It is noted that the spaces erp (∆(α̃) ), er0 (∆(α̃) ), erc (∆(α̃) ) and er∞ (∆(α̃) ) can be derived by taking E r (∆(α̃) )−
transform of x in the spaces `p , c0 , c and `∞ , respectively i.e.,
erp (∆(α̃) ) = (`p )E r (∆(α̃) ) ; cr0 (∆(α̃) ) = (c0 )E r (∆(α̃) ) ; cr (∆(α̃) ) = (c)E r (∆(α̃) ) ; `∞ (∆(α̃) ) = (`∞ )E r (∆(α̃) )
Keeping the above new sets in mind, we define the sequence y = (yk ), which is used as the E r (∆(α̃) )−transform
of a sequence x = (xk ) i.e.,
yk =
k X
k
X
j=0 i=j
(−1)i−j
k
Γ(α̃ + 1)
ri (1 − r)k−i xj ,
k − i (i − j)!Γ(α̃ − i + j + 1)
(k ∈ N0 ).
(2.1)
In particular, the above new spaces include the following special cases:
(i) For α̃ = 0, above classes reduce to the classes defined by Altay and Başar [2, 3].
(ii) For α̃ = 1, above classes reduce to the classes defined by Altay and Polat [4].
(iii) For α̃ = m ∈ N, above classes reduce to the classes defined by Polat and Başar [21].
We will now give some interesting results of these spaces concerning their topological structures, bases
and duals.
Theorem 2.1. For a positive proper fraction α̃, the sequence space erp (∆(α̃) ) is a complete normed linear
space with co-ordinate wise addition and scalar multiplication which is a BK-space with the norm
kxkerp (∆(α̃) ) = kE r (∆(α̃) )xkp (1 ≤ p < ∞).
Also, the sequence spaces er0 (∆(α̃) ), erc (∆(α̃) ) and er∞ (∆(α̃) ) are complete normed linear spaces with co-ordinate
wise addition and scalar multiplication, moreover BK-space with the norm
kxkerc (∆(α̃) ) = kxker∞ (∆(α̃) ) = kE r (∆(α̃) )xk∞ .
Proof. The proof is straightforward, hence omitted.
Theorem 2.2. For a positive proper fraction α̃, the sequence spaces erp (∆(α̃) ), er0 (∆(α̃) ), erc (∆(α̃) ) and
er∞ (∆(α̃) ) are linearly isomorphic to the classical spaces `p , c0 , c and `∞ , respectively.
Proof. We prove the theorem for the space er∞ (∆(α̃) ). We show that there exists a linear bijection between the
spaces er∞ (∆(α̃) ) and `∞ . With the notation (2.1) we define a mapping T : er∞ (∆(α̃) ) → `∞ by x 7→ y = T x.
Clearly, T is linear. If T x = θ = (0, 0, 0, . . . ), then x = θ; therefore, T is injective. Let y ∈ `∞ and using
Lemma 1.3, define a sequence x = (xk ) via yk as
k X
k
X
Γ(−α̃ + 1)(1 − r)j−i r−j
(k−i) j
yi , (k ∈ N0 ).
(2.2)
xk =
(−1)
i (k − j)!Γ(−α̃ − k + j + 1)
i=0 j=i
Then, we have
X
n
n X
n
Γ(α̃ + 1)
i
n−i i−j
sup (−1)
r (1 − r) xj = sup |yn | = kyk∞ < ∞.
n − i (i − j)!Γ(α̃ − i + j + 1)
n n
j=0 i=j
Thus, we obtain that x ∈ er∞ (∆(α̃) ) and consequently T is surjective. This completes the proof.
U. Kadak, P. Baliarsingh, J. Nonlinear Sci. Appl. 8 (2015), 997–1004
1001
Theorem 2.3. Let λk = (E r (∆(α̃) )x)k for all k ∈ N0 . Now for fixed k ∈ N0 define the sequence
(k)
(k)
bn = {bn }n∈N0 by
 1
(k = n),

rn ,


n
j−k
−j
 X
j Γ(−α̃ + 1)(1 − r) r
(−1)(n−k)
, (0 ≤ k ≤ n),
b(k)
n =
k (n − j)!Γ(−α̃ − n + j + 1)


j=k


0,
(k > n),
for all n, k ∈ N0 . Then
(k)
(i) The sequence {bn }n∈N0 is a basis for the space er0 (∆(α̃) ) and any x ∈ er0 (∆(α̃) ) has a unique representation in the form
X
x=
λk b(k) .
k
(ii) The set {z, β (k) } is a basis for the space erc (∆(α̃) ) and any x ∈ erc (∆(α̃) ) has a unique representation in
the form
X
x = lz +
(λk − l)b(k) ,
k
where l = lim λk and z = (zk ), defined by
k→∞
zk =
k X
k
X
i=0 j=i
(k−i)
(−1)
j Γ(−α̃ + 1)(1 − r)j−i r−j
.
i (k − j)!Γ(−α̃ − k + j + 1)
3. Dual properties
In this section, we formulate and prove theorems determining the α-, β-, and γ-duals of the Euler
sequence spaces of nonabsolute type.
It is well known that {`∞ }β = `1 and {`p }β = `q where 1 ≤ p < ∞ and p−1 + q −1 = 1. We shall
throughout denote the collection of all finite subsets of N by F. We begin with quoting lemmas, due to
Stieglitz and Tietz [22], that are needed in proving the next theorems.
Lemma 3.1. (i) A = (ank ) ∈ (c0 : `1 ) = (c : `1 ) if and only if
X X
sup
ank < ∞.
K∈F n
(3.1)
k∈K
(ii) A = (ank ) ∈ (c0 : c) if and only if
lim ank = `k for all k, and
n→∞
sup
X
|ank | < ∞.
n∈N k
(iii) A = (ank ) ∈ (c0 : `∞ ) if and only if (3.3) holds.
(3.2)
(3.3)
U. Kadak, P. Baliarsingh, J. Nonlinear Sci. Appl. 8 (2015), 997–1004
1002
Theorem 3.2. Define the sets hα̃∞ (r), hα̃ (r) and hα̃p (r) as follows:


n


n−k
X X
j
(−1)
Γ(1 − α̃)
j−k −j ,
(1
−
r)
r
x
<
∞
hα̃∞ (r) := x = (xk ) ∈ ω : sup
j


k
k∈N n j=k (n − j)! Γ(1 − α̃ − n + j)


n


n−k
X X X
(−1)
Γ(1 − α̃)
j
j−k −j hα̃ (r) := x = (xk ) ∈ ω : sup
<
∞
,
(1
−
r)
r
x
j


k
K∈F n k∈K j=k (n − j)! Γ(1 − α̃ − n + j)


p
n


n−k
X X X
(−1)
Γ(1 − α̃)
j
j−k −j hα̃p (r) := x = (xk ) ∈ ω : sup
<
∞
.
(1
−
r)
r
x
j


k
K∈F k n∈K j=k (n − j)! Γ(1 − α̃ − n + j)
Then {er1 (∆(α̃) )}α = hα̃∞ (r), {er0 (∆(α̃) )}α = {erc (∆(α̃) )}α = hα̃ (r) and {erp (∆(α̃) )}α = hα̃p (r).
Proof. Since the proof for the spaces er1 (∆(α̃) ) and erp (∆(α̃) ) is obtained by analogy, we consider only the
spaces er0 (∆(α̃) ) and erc (∆(α̃) ).
Let x = (xn ) ∈ ω consider the matrix Brα̃ = (bα̃nk (r)) defined by
 P
(−1)n−k Γ(1−α̃)
n
j
j−k r −j , (k < n),


j=k (n−j)! Γ(1−α̃−n+j) k (1 − r)
α̃
n
bnk (r) =
(3.4)
1/r
, (k = n),


0
, (k > n).
Bearing in mind the relation (2.2), we easily obtain that


n
n
n−k
X
X
(−1)
Γ(1 − α̃)
j

x n wn =
(1 − r)j−k r−j xj  yk = (Brα̃ y)n , n ∈ N.
(n − j)! Γ(1 − α̃ − n + j) k
k=0
(3.5)
j=k
We, therefore, observe by (3.5) that xw = (xn wn ) ∈ `1 whenever w ∈ er0 (∆(α̃) ) or erc (∆(α̃) ) if and only if
Brα̃ y ∈ `1 whenever y ∈ c0 or c. Then using Lemma 3.1(i) we derive that
n
n−k
X X X
j
(−1)
Γ(1 − α̃)
j−k −j (1
−
r)
r
x
sup
j
< ∞.
k
K∈F n k∈K j=k (n − j)! Γ(1 − α̃ − n + j)
This yields that {er0 (∆(α̃) )}α = {erc (∆(α̃) )}α = hα̃ (r).
Theorem 3.3. Define the sets dα̃1 (r), dα̃2 (r) and dα̃3 (r) by


n X
n


n−k
X
(−1)
Γ(1 − α̃)
j
j−k −j (xk ) ∈ ω : sup
(1
−
r)
r
x
<
∞
,
dα̃1 (r) :=
j


k
n∈N k=0 j=k (n − j)! Γ(1 − α̃ − n + j)


n


n−k
X
(−1)
Γ(1 − α̃)
j
dα̃2 (r) :=
(xk ) ∈ ω : lim
(1 − r)j−k r−j xj exists ,
n→∞


(n − j)! Γ(1 − α̃ − n + j) k
j=k


n X
n


n−k
X
(−1)
Γ(1 − α̃)
j
dα̃3 (r) :=
(xk ) ∈ ω : lim
(1 − r)j−k r−j xj exists .
n→∞


(n − j)! Γ(1 − α̃ − n + j) k
k=0 j=k
Then {er0 (∆(α̃) )}β = dα̃1 (r) ∩ dα̃2 (r) and {erc (∆(α̃) )}β = dα̃1 (r) ∩ dα̃2 (r) ∩ dα̃3 (r).
U. Kadak, P. Baliarsingh, J. Nonlinear Sci. Appl. 8 (2015), 997–1004
1003
Proof. Since the proof can also be obtained for the space erc (∆(α̃) ) in a similar way, we omit it and give only
the proof for the space er0 (∆(α̃) ). Consider the equality


n
n
k
n−k
X
X
X
k
(−1)
Γ(1 − α̃)

x k wk =
(r − 1)k−j r−k yj  xk
(n − j)! Γ(1 − α̃ − n + j) j
k=0
k=0 j=0


n
n
n−k
X
X
j
(−1)
Γ(1 − α̃)

(1 − r)j−k r−j xj  yk
=
(n − j)! Γ(1 − α̃ − n + j) k
k=0
=
j=k
(Brα̃ y)n
(3.6)
where Brα̃ = (bα̃nk (r)) is defined in (3.4). Thus, we deduce from Lemma 3.1(ii) with (3.6) that (xk wk ) ∈ cs
whenever w ∈ er0 (∆(α̃) ) if and only if Brα̃ y ∈ c whenever y = (yk ) ∈ c0 . Therefore, we derive from (3.2) and
(3.3) that
lim bα̃nk (r) exists for each k ∈ N and sup
n→∞
n
X
n∈N k=0
|bα̃nk (r)| < ∞,
which shows that {er0 (∆(α̃) )}β = dα̃1 (r) ∩ dα̃2 (r).
Theorem 3.4. The γ-duals of the spaces er0 (∆(α̃) ), erc (∆(α̃) ) and er∞ (∆(α̃) ) is dα̃1 (r).
Proof. It is natural that the present theorem may be proved by the technique used in the proofs of Theorem
3.2 and 3.3. But, we prefer to follow the classical way and give only the proof for the space eα̃0 (r).
Let x = (xk ) ∈ dα̃1 (r) and w = (wk ) ∈ er0 (∆(α̃) ). Consider the equality
n
n k
X
n−k
X
X
(−1)
Γ(1 − α̃)
k
xk wk = (r − 1)k−j r−k yj xk k=0 j=0 (n − j)! Γ(1 − α̃ − n + j) j
k=0
n
X
α̃
= bnk (r)yk k=0
≤
n
X
|bα̃nk (r)| |yk |
k=0
which gives us by taking supremum over n ∈ N that
n
n
n
X
X
X
sup xk wk ≤ sup
|bα̃nk (r)| |yk | ≤ kyk∞ sup
|bα̃nk (r)| ≤ ∞.
n∈N n∈N
n∈N
k=0
k=0
k=0
This means that x = (xk ) ∈ {er0 (∆(α̃) )}γ . Hence, we have
dα̃1 (r) ⊂ {er0 (∆(α̃) )}γ .
(3.7)
r
(α̃) γ
r
(α̃)
let
the sequence
x = (xk ) ∈ {e0 (∆ )} and w ∈ e0 (∆ ). Then, one can conclude that
PConversely,
n
α̃
α̃ = (bα̃ (r)) is in
b
(r)y
∈
`
whenever
(x
w
)
∈
bs.
This
implies
that
the
triangle
matrix
B
∞
k
k
k
r
k=0 nk
nk
n∈N
the class (c0 : `∞ ). Hence, the condition
sup
n
X
n∈N k=0
|bα̃nk (r)| < ∞
is satisfied, which implies that x = (xk ) ∈ dα̃1 (r). In other words,
{er0 (∆(α̃) )}γ ⊂ dα̃1 (r).
(3.8)
Therefore, by combining inclusions (3.7) and (3.8), we establish that the γ-dual of the space er0 (∆(α̃) ) is
dα̃1 (r), which completes the proof.
U. Kadak, P. Baliarsingh, J. Nonlinear Sci. Appl. 8 (2015), 997–1004
1004
Conclusion
In this article, certain results on some Euler spaces of difference sequences of order m, (m ∈ N), have
been extended to the sequence spaces of positive fractional order α̃. The results presented in this article
not only generalize the earlier works done by several authors [2, 3, 4, 21] but also give a new perspective
concerning the development of the Euler spaces of difference sequences. As future work we will study certain
matrix transformations of Euler spaces of fractional order and Riesz mean difference sequence of fractional
order.
Acknowledgements:
We express our sincere thanks to the referees for their valuable comments and suggestions on this
manuscript.
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