Alexandria Engineering Journal (2017) xxx, xxx–xxx
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Alexandria Engineering Journal
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SHORT COMMUNICATION
A note on fractional difference operators
P. Baliarsingh, L. Nayak *
Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar 751024, India
Received 4 January 2017; revised 9 February 2017; accepted 13 February 2017
KEYWORDS
Fractional order difference
operator;
Sequence spaces;
Fractional derivatives and
integrals
Abstract In the present article, following on very recent and new approach of fractional difference
operator by Baliarsingh (2016), we establish some new ideas involving the exponent rules of this
operator. In fact, the new results are sufficient enough to describe the results obtained by Baliarsingh and also provide certain improvements on them.
Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an
open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction, preliminaries and definitions
Let w be the space of all real valued sequences and N be the set
of all nonnegative integers. Then any subspace of w is called a
sequence space and by ‘1 ; c and c0 , we denote the spaces of all
bounded, convergent and null sequences, respectively, normed
by kxk1 ¼ supk jxk j.
Kizmaz [2] has introduced the idea of difference sequence
spaces associated with basic sequences ‘1 ; c and c0 by defining
the difference operator D of order one, where
ðDxÞk ¼ xk xkþ1 ;
ðk 2 NÞ:
Later on, these spaces have been generalized to the case of
integral order m by Et and Colak [3] using operator Dm and
m
X
m
ð1Þi
xkþi ; ðk 2 NÞ:
ðDm xÞk ¼
i
i¼0
Baliarsingh [4] generalized the above difference operator by
introducing fractional difference operator Da , where
* Corresponding author.
E-mail address: laxmipriyamath@gmail.com (L. Nayak).
Peer review under responsibility of Faculty of Engineering, Alexandria
University.
ðDa xÞk ¼
1
X
ð1Þi
i¼0
Cða þ 1Þ
xkþi ;
i!Cða i þ 1Þ
ðk 2 NÞ;
where CðaÞ denotes the Euler gamma function of a real number a and a R f0; 1; 2; . . .g.
Very recently, Baliarsingh [1] has introduced a generalized
fractional difference operator as
ðDha;b;c xÞk ¼
1
X
ðaÞi ðbÞi
x ;
aþbc ki
i!ðcÞ
i¼0
ih
ðk 2 NÞ;
ð1:1Þ
where h is a positive constant and a; b; c are real numbers such
that c R N. Also, ðaÞk denotes the Pochhammer symbol or
shifted factorial of a real number a which is being defined using
familiar Euler gamma function as
(
1;
ða ¼ 0 or k ¼ 0Þ
ðaÞk ¼ CðaþkÞ
¼
aða
þ
1Þða
þ
2Þ.
.
.ða
þ
k
1Þ;
ðk 2 NÞ
CðaÞ
It is noted that the series defined in (1.1) converges for all
c R N and c > a þ b (see [5]). In different suitable choice of
generalizes following special
a; b; c and h, the operator Da;b;c
h
cases:
(i) Let a ¼ 1; b ¼ c and h ¼ 1, the operator Dha;b;c reduces to
difference operator Dð1Þ of order one (see [6]).
http://dx.doi.org/10.1016/j.aej.2017.02.022
1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: P. Baliarsingh, L. Nayak, A note on fractional difference operators, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.
aej.2017.02.022
2
P. Baliarsingh, L. Nayak
(ii) Let a ¼ m 2 N; b ¼ c and h ¼ 1, the operator Dha;b;c
reduces to difference operator DðmÞ of integral order m
(see [3]).
(iii) Let a ¼ a 2 R; b ¼ c and h ¼ 1, the operator Dha;b;c
reduces to difference operator DðaÞ of order a (see [9]).
(iv) Let b ¼ 1; c R N and h ¼ 1, then the difference
sequence ðDha;1;c xÞk ¼ xk ac xk1 ; ðk 2 NÞ.
(v) Let b ¼ 2; c R N and h ¼ 1, then the difference
x ; ðk 2 NÞ.
sequence ðDha;2;c xÞk ¼ xk 2ac xk1 þ aða1Þ
cðc1Þ k2
In the above article, author introduced new difference
Þ for
operator and related sequence spaces XðDa;b;c
h
x 2 fc; c0 ; ‘1 g, which generalize most of the difference
sequence spaces such as spaces defined in [6–13] and many
others. It is shown that these spaces are complete linear
spaces with the norm
kxkDa;b;c ¼
h
½a
X
jxk j þ supDha;b;c ðxk Þ:
i¼0
ð1:2Þ
k
where ½a indicates the integral part of a. It is noted that under
limiting condition of h, i.e., h ! 0 and for a variable x, the
operator Da;b;c
generates a fractional difference operator Da;b;c
h
h;x
which describes the notion of fractional derivatives and integrals of a function fðxÞ. The linearity and exponent properties
of this operator (see [1]) are as follows:
Theorem 1.1. If a; b > 0, then
(i) The difference operator Da;b;c
is a linear operator over R.
h;x
b;b;b
a;b;b
a;b;b P1 ðbÞi
(ii) Dh;x Dh;x f ðxÞ ¼ Dh;x
i¼0 i!ha f ðx ihÞ ¼
aþb;b;b
Dh;x
f ðxÞ.
a;b;b
(iii) Dh;x
Da;b;b
f ðxÞ ¼ Da;b;b
Da;b;b
h;x
h;x
h;x f ðxÞ ¼ f ðxÞ.
It is observed that for any arbitrary a and b, these proper, but they may not
ties are fully satisfied with the operator Da;b;c
h
a;b;c
be satisfied with the operator Dh;x . This happens only due to
dynamic nature of fractional derivatives. While calculating
fractional derivatives of a constant function, one may get
two different values (zero and nonzero) depending on the
order of the derivatives. For instance, 1.5th derivative of the
function fðxÞ ¼ c (a constant) is zero (asymptotically) whereas
0.5th derivative of fðxÞ is nonzero. As an application of this
operator, the fractional derivatives of the function fðxÞ ¼ xb
has been given as follows:
Theorem 1.2. Let a 2 R and 0 – b 2 R, then
a;b;b b
x ¼ xba
Dh;x
Cðb þ 1Þ
:
Cðb þ 1 aÞ
It is remarked that the well-known Euler gamma function is
restricted in the set of negative integers including zero, the
above result is not valid for negative integer b 6 1. As a
result, above theorem needs further stronger conditions on a
and b.
2. Main results
The main objective of the present section is to improve the
results in Theorems 1.1 and 1.2 by adding certain new conditions. Some counter examples are provided in support to the
each point of modifications in the above theorems.
Theorem 2.1. Let a 2 R and b R f1; 2; . . .g, then
a;b;b b
Dh;x
x ¼ xba
Cðb þ 1Þ
:
Cðb þ 1 aÞ
Proof. Consider the function as fðxÞ ¼ x1 ; x – 0, then it is
clearly observed that b ¼ 1 – 0, and hence the condition
for Theorem 1.2 is satisfied. But, 0.5th derivative of fðxÞ cannot be determined. This is due to the fact that the Euler gamma
function has singularities in the set of all negative integers
including zero. However, for any negative fractional value of
b
b; Da;b;b
h;x x exists. For instance, if b ¼ 3=2 and a ¼ 1=2, then
Cð3=2 þ 1Þ
:
Cð3=2 þ 1 1=2Þ
Cð1=2Þ
¼ x2
¼ 0:
Cð1Þ
1=2;b;b 3=2
Dh;x
x
¼ x22
3
1
b
It is also noticed that for b ¼ 0, the derivative Da;b;b
h;x x has different values as per the nature of a. Therefore, we have the following corollary. h
Corollary 2.2. Let a 2 R and fðxÞ ¼ c be a constant function,
then
8
>
< 0;
c
Da;b;b
fðxÞ
¼
h;x
>
: xa c ;
Cð1aÞ
ða ¼ 1; 2; 3; . . .Þ
ða ¼ 0Þ
:
ðotherwiseÞ
Proof. For any nonnegative integer a, the proof is obvious. If
a is a fraction, then using Theorem 2.1 the fractional derivative
of the constant function fðxÞ ¼ c can be determined as
a;b;b
a;b;b 0
fðxÞ ¼ Da;b;b
Dh;x
h;x c ¼ cDh;x x ¼ c
xa
:
Cð1 aÞ
If we analyze the behavior of the fractional derivatives of a
constant function geometrically, it provides very interesting
and dynamic nature of fractional calculus. In this regard, we
consider a constant function fðxÞ ¼ 2 for all x, then its ath
derivative with respect to x is given by
a;b;b
Dh;x
fðxÞ ¼ 2
xa
:
Cð1 aÞ
Clearly, from the above expression, it is observed that for
a;b;b
fðxÞ takes always posany a < 1 the fractional derivative Dh;x
itive values whereas for a ¼ 1; 2; 3; . . ., it takes only the value 0.
For any nonintegral value a > 1, Da;b;b
h;x fðxÞ ! 0 as x ! 1 and
takes positive and negative values as per the nature of ð1Þ½a .
This follows immediately from Fig. 1.
Please cite this article in press as: P. Baliarsingh, L. Nayak, A note on fractional difference operators, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.
aej.2017.02.022
Fractional difference operators
3
Proof. It is well known that Theorem 1.1 is valid only for any
nonnegative integral values of a and b, and not substantial
arbitrarily. In order to prove the present theorem, it is sufficient to illustrate some suitable counter examples which contradict the results presented in Theorem 1.1.
10
Blue color:−0.5≤ α< 0
Green color:α=0 and 1
Red color:0.1≤ α≤ 1.5
8
α,b,b
Δ h,x (2)
6
(i) Let us consider a function f ðxÞ ¼ xp , for any p > 0 and
take a ¼ p and b ¼ p þ 1, then using Theorem 2.1, we
obtain
4
2
pþ1;b;b p
Db;b;b
x ¼ x1
h;x fðxÞ ¼ Dh;x
0
−2
0
1
2
3
4
5
6
7
8
9
10
x
Figure 1
fðxÞ ¼ 2.
Now, for the left hand side of (i), we have
p;b;b
a;b;b
Db;b;b
Dpþ1;b;b
xp ¼ Dp;b;b
Dh;x
h;x
h;x ð0Þ
h;x fðxÞ ¼ Dh;x
Fractional derivatives of the constant function
In this figure, we have plotted the curves of the fractional
derivatives of order a of the function fðxÞ ¼ 2 with different
values of a ranging from 0:5 to 1:5. Actually, the curves
indicating blue color in the figure signify fractional derivatives
of fðxÞ ¼ 2 of negative orders 0:5; 0:4; 0:3; 0:2 and 0:1
(from top to bottom), whereas the curves indicating red color
signify the derivatives of the function of different positive
orders 0:1; 0:2; 0:3; 0:4; 0:5; 0:6; 0:7; 0:8; 0:9; 1:1; 1:2; 1:3; 1:4 and
1:5 (from top to bottom). Two green horizontal straight lines
represent the derivatives of the function of nonnegative
integral orders 0 and 1 (from top to bottom). From the figure
it is noticed that the curves involving higher positive order
fractional derivatives either exactly overlap or asymptotically
meet the line fðxÞ ¼ 0. It is also observed that the curves
involving fractional derivatives of order a < 1, are lying above
to the line fðxÞ ¼ 0, hence the respective derivatives are always
positive. For 1 < a < 2, the corresponding curves are lying
below to the line fðxÞ ¼ 0, hence subsequent derivatives are
negative, but if 2 < a < 3, the derivatives of order a are
positive. For instance, if a ¼ 5=2 and fðxÞ ¼ 2, then
5=2;b;b
Dh;x
fðxÞ ¼ 2
x5=2
3
¼ pffiffiffi x5=2 :
Cð3=2Þ 2 p
From the above equation, it is concluded that
positive as the sign of ð1Þ
the proof. h
½5=2
fðxÞ
D5=2;b;b
h;x
is
is positive. This step concludes
Since fractional derivative of a constant function need not
be zero, we often encounter problems while discussing several
properties of fractional calculus. This idea suggests the nonuniform or deviation behavior of the difference operator Da;b;c
h;x .
The most common and remarkable deviation of this operator
are being found out in linearity involving exponent rules. As a
part of this, now, we quote following results involving dynamic
nature of this operator in exponent rules:
Theorem 2.3. Let a and b be two arbitrary real numbers, then
b;b;b
aþb;b;b
a;b;b
Db;b;b
Da;b;b
f ðxÞ.
(i) Dh;x
h;x f ðxÞ – Dh;x
h;x f ðxÞ – Dh;x
a;b;b
a;b;b
a;b;b
a;b;b
(ii) Dh;x Dh;x f ðxÞ – Dh;x
Dh;x f ðxÞ – f ðxÞ.
Cðp þ 1Þ
¼ 0:
Cð0Þ
¼ 0:
ð2:1Þ
For reverse direction, we first calculate
a;b;b
p
pp
fðxÞ ¼ Dp;b;b
Dh;x
h;x x ¼ x
Cðp þ 1Þ
¼ pCðpÞ:
Cð1Þ
And secondly, we deduce that
pþ1;b;b
p
Db;b;b
Da;b;b
Dp;b;b
h;x fðxÞ ¼ Dh;x
h;x x
h;x
¼ Dpþ1;b;b
ðpCðpÞÞ
h;x
¼ xp1
pCðpÞ
:
CðpÞ
ð2:2Þ
Finally, for the right hand side of (i), we may write
Daþb;b;b
fðxÞ ¼ D2pþ1;b;b
xp ¼ xp2p1
h;x
h;x
¼ xp1
pCðpÞ
:
CðpÞ
Cðp þ 1Þ
CðpÞ
ð2:3Þ
Now, replacing a and b as p þ 1 and p; respectivelly , it
is observed that Eq. (2.3) is unaltered but
Db;b;b
Da;b;b
ð2:4Þ
h;x fðxÞ ¼ 0:
h;x
Combining Eqs. (2.1)–(2.4), we conclude the proof.
(ii) Using similar techniques as discussed in (i), we may
complete the proof by taking f ðxÞ ¼ x1=2 ; a ¼ 1=2
and b ¼ 1=2. h
3. Conclusion
In this note, we provide certain interesting ideas on fractional
derivatives which suggest the adequate improvement of the
results obtained in [1]. As discussed earlier, the nature of fractional derivative of a function is nonuniform and dynamic
which is completely different to that of the derivatives in classical sense. For example:
Fractional derivative of a nonconstant function may be
constant even zero also.
Fractional derivative of a constant function may not be
zero.
Please cite this article in press as: P. Baliarsingh, L. Nayak, A note on fractional difference operators, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.
aej.2017.02.022
4
Fractional derivative of a periodic function with period P
may not be periodic with the same period even not periodic
also.
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Please cite this article in press as: P. Baliarsingh, L. Nayak, A note on fractional difference operators, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.
aej.2017.02.022