research article fractional sums and differences with...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013, Article ID 104173, 6 pageshttp://dx.doi.org/10.1155/2013/104173

Research ArticleFractional Sums and Differences with Binomial Coefficients

Thabet Abdeljawad,1 Dumitru Baleanu,1,2,3 Fahd Jarad,1 and Ravi P. Agarwal4

1 Department of Mathematics, Faculty of Art and Sciencs, Cankaya University, Balgat, 06530 Ankara, Turkey2 Institute of Space Sciences, P.O. BOX MG-23, 76900 Magurele-Bucharest, Romania3 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University,P.O. Box 80204, Jeddah 21589, Saudi Arabia

4Department of Mathematics, Texas A & M University, 700 University Boulevard, Kingsville, TX, USA

Correspondence should be addressed toThabet Abdeljawad; [email protected]

Received 31 March 2013; Accepted 3 May 2013

Academic Editor: Shurong Sun

Copyright © 2013 Thabet Abdeljawad et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral andthen defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The secondapproach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtainGrunwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and rightfractional integrals and derivatives representing the second approach.Then, we use the discrete version of the Q-operator and somediscrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemannones describing the first approach.

1. Introduction and Preliminaries

Fractional calculus (FC) is developing very fast in both theo-retical and applied aspects. As a result, FC is used intensivelyand successfully in the last few decades to describe the ano-malous processes which appear in complex systems [1–6].Very recently, important results in the field of fractionalcalculus and its applications were reported (see e.g., [7–10]and the references therein). The complexity of the real worldphenomena is a great source of inspiration for the researchersto invent new fractional tools which will be able to dig muchdipper into the mysteries of the mother nature. Historicallythe FC passed through different periods of evolutions, andit started to face very recently a new provocation: how toformulate properly its discrete counterpart [11–24]. At thisstage, we have to stress on the fact that in the classicaldiscrete equations their roots are based on the functionaldifference equations, therefore, the natural question is to findthe generalization of these equations to the fractional case. Inother words, we will end up with generalizations of the basicoperators occurring in standard difference equations. As itwas expected, there were several attempts to do this gener-alization as well as to apply this new techniques to investigate

the dynamics of some complex processes. In recent years,the discrete counterpart of the fractional Riemann-Liouville,Caputo, was investigated mainly thinking how to apply tech-niques from the time scales calculus to the expressions of thefractional operators. Despite of the beauty of the obtainedresults, one simple question arises: can we obtain the sameresults from a new point of view which is more simplerand more intuitive? Having all above mentioned thinks inmind we are going to use the binomial theorem in orderto get Grunwald-Letnikov fractional derivatives. After that,we proved that the results obtained coincide with the onesobtained by the discretization of the Riemann-Liouville oper-ator. In this manner, we believe that it becomes more clearwhat the fractional difference equations bring new in descrip-tion of the related complex phenomena described.

For a natural number 𝑛, the fractional polynomial isdefined by

𝑡(𝑛)

=

𝑛−1

∏

𝑗=0

(𝑡 − 𝑗) =

Γ (𝑡 + 1)

Γ (𝑡 + 1 − 𝑛)

, (1)

2 Discrete Dynamics in Nature and Society

where Γ denotes the special gamma function and the productis zero when 𝑡 + 1 − 𝑗 = 0 for some 𝑗. More generally, forarbitrary 𝛼, define

𝑡(𝛼)

=

Γ (𝑡 + 1)

Γ (𝑡 + 1 − 𝛼)

, (2)

where the convention of that division at pole yields zero.Given that the forward and backward difference operators aredefined by

Δ𝑓 (𝑡) = 𝑓 (𝑡 + 1) − 𝑓 (𝑡) , ∇𝑓 (𝑡) = 𝑓 (𝑡) − 𝑓 (𝑡 − 1) ,

(3)

respectively, we define iteratively the operatorsΔ𝑚 = Δ(Δ𝑚−1)and ∇𝑚 = ∇(∇𝑚−1), where𝑚 is a natural number.

Here are some properties of the factorial function.

Lemma 1 (see [13]). Assume the following factorial functionsare well defined.

(i) Δ𝑡(𝛼) = 𝛼𝑡(𝛼−1).(ii) (𝑡 − 𝜇)𝑡(𝜇) = 𝑡(𝜇+1), where 𝜇 ∈ R.

(iii) 𝜇(𝜇) = Γ(𝜇 + 1).

(iv) If 𝑡 ≤ 𝑟, then 𝑡(𝛼) ≤ 𝑟(𝛼) for any 𝛼 > 𝑟.

(v) If 0 < 𝛼 < 1, then 𝑡(𝛼]) ≥ (𝑡(]))𝛼.

(vi) 𝑡(𝛼+𝛽) = (𝑡 − 𝛽)(𝛼)𝑡(𝛽).

Also, for our purposes we list down the following twoproperties, the proofs of which are straightforward:

∇𝑠(𝑠 − 𝑡)(𝛼−1)

= (𝛼 − 1) (𝜌 (𝑠) − 𝑡)(𝛼−2)

∇𝑡(𝜌 (𝑠) − 𝑡)(𝛼−1)

= − (𝛼 − 1) (𝜌 (𝑠) − 𝑡)(𝛼−2)

.

(4)

For the sake of the nabla fractional calculus, we have thefollowing definition.

Definition 2 (see [25–28]). (i) For a natural number𝑚, the𝑚rising (ascending) factorial of 𝑡 is defined by

𝑡𝑚=

𝑚−1

∏

𝑘=0

(𝑡 + 𝑘) , 𝑡0= 1. (5)

(ii) For any real number, the 𝛼 rising function is definedby

𝑡𝛼=

Γ (𝑡 + 𝛼)

Γ (𝑡)

, 𝑡 ∈ R − {. . . , −2, −1, 0} , 0𝛼= 0. (6)

Regarding the rising factorial function, we observe thefollowing:

(i)

∇ (𝑡𝛼) = 𝛼𝑡

𝛼−1, (7)

(ii)

(𝑡𝛼) = (𝑡 + 𝛼 − 1)

(𝛼), (8)

(iii)

Δ 𝑡(𝑠 − 𝜌 (𝑡))𝛼= −𝛼(𝑠 − 𝜌 (𝑡))

𝛼−1. (9)

Notation:

(i) For a real 𝛼 > 0, we set 𝑛 = [𝛼] + 1, where [𝛼] is thegreatest integer less than 𝛼.

(ii) For real numbers 𝑎 and 𝑏, we denoteN𝑎 = {𝑎, 𝑎+1, . . .}and 𝑏N = {𝑏, 𝑏 − 1, . . .}.

(iii) For 𝑛 ∈ N and real 𝑎, we denote

⊝Δ𝑛𝑓 (𝑡) ≜ (−1)

𝑛Δ𝑛𝑓 (𝑡) . (10)

(iv) For 𝑛 ∈ N and real 𝑏, we denote

∇𝑛

⊝𝑓 (𝑡) ≜ (−1)𝑛∇𝑛𝑓 (𝑡) . (11)

The following definition and the properties followed canbe found in [29] and the references therein.

Definition 3 (see [29]). Let 𝜎(𝑡) = 𝑡+1 and 𝜌(𝑡) = 𝑡−1 be theforward and backward jumping operators, respectively. Then

(i) the (delta) left fractional sum of order 𝛼 > 0 (startingfrom 𝑎) is defined by

Δ−𝛼

𝑎 𝑓 (𝑡) =1

Γ (𝛼)

𝑡−𝛼

∑

𝑠=𝑎

(𝑡 − 𝜎 (𝑠))(𝛼−1)

𝑓 (𝑠) , 𝑡 ∈ N𝑎+𝛼. (12)

(ii) The (delta) right fractional sum of order 𝛼 > 0

(ending at 𝑏) is defined by

𝑏Δ−𝛼𝑓 (𝑡) =

1

Γ (𝛼)

𝑏

∑

𝑠=𝑡+𝛼

(𝑠 − 𝜎 (𝑡))(𝛼−1)

𝑓 (𝑠)

=

1

Γ (𝛼)

𝑏

∑

𝑠=𝑡+𝛼

(𝜌 (𝑠) − 𝑡)(𝛼−1)

𝑓 (𝑠) , 𝑡 ∈ 𝑏−𝛼N .

(13)

(iii) The (nabla) left fractional sumof order𝛼 > 0 (startingfrom 𝑎) is defined by

∇−𝛼

𝑎 𝑓 (𝑡) =1

Γ (𝛼)

𝑡

∑

𝑠=𝑎+1

(𝑡 − 𝜌 (𝑠))𝛼−1

𝑓 (𝑠) , 𝑡 ∈ N𝑎+1. (14)

(iv) The (nabla) right fractional sum of order 𝛼 > 0

(ending at 𝑏) is defined by

𝑏∇−𝛼𝑓 (𝑡) =

1

Γ (𝛼)

𝑏−1

∑

𝑠=𝑡

(𝑠 − 𝜌 (𝑡))𝛼−1

𝑓 (𝑠)

=

1

Γ (𝛼)

𝑏−1

∑

𝑠=𝑡

(𝜎 (𝑠) − 𝑡)𝛼−1

𝑓 (𝑠) , 𝑡 ∈ 𝑏−1N .

(15)

Discrete Dynamics in Nature and Society 3

Regarding the delta left fractional sum, we observe thefollowing:

(i) Δ−𝛼𝑎 maps functions defined on N𝑎 to functionsdefined on N𝑎+𝛼.

(ii) 𝑢(𝑡) = Δ−𝑛𝑎 𝑓(𝑡), 𝑛 ∈ N, satisfies the initial value pro-

blem:Δ𝑛𝑢 (𝑡) = 𝑓 (𝑡) , 𝑡 ∈ 𝑁𝑎,

𝑢 (𝑎 + 𝑗 − 1) = 0, 𝑗 = 1, 2, . . . , 𝑛.

(16)

(iii) The Cauchy function (𝑡 − 𝜎(𝑠))(𝑛−1)/(𝑛 − 1)! vanishesat 𝑠 = 𝑡 − (𝑛 − 1), . . . , 𝑡 − 1.

Regarding the delta right fractional sum, we observe thefollowing:

(i) 𝑏Δ−𝛼 maps functions defined on 𝑏N to functions

defined on 𝑏−𝛼N.(ii) 𝑢(𝑡) = 𝑏Δ

−𝑛𝑓(𝑡), 𝑛 ∈ N, satisfies the initial value

problem:

∇𝑛

⊖𝑢 (𝑡) = 𝑓 (𝑡) , 𝑡 ∈ 𝑏𝑁,

𝑢 (𝑏 − 𝑗 + 1) = 0, 𝑗 = 1, 2, . . . , 𝑛.

(17)

(iii) The Cauchy function (𝜌(𝑠) − 𝑡)(𝑛−1)/(𝑛 − 1)! vanishesat 𝑠 = 𝑡 + 1, 𝑡 + 2, . . . , 𝑡 + (𝑛 − 1).

Regarding the nabla left fractional sum, we observe thefollowing:

(i) ∇−𝛼𝑎 maps functions defined on N𝑎 to functionsdefined on N𝑎.

(ii) ∇−𝑛𝑎 𝑓(𝑡) satisfies the 𝑛th-order discrete initial valueproblem:

∇𝑛𝑦 (𝑡) = 𝑓 (𝑡) , ∇

𝑖𝑦 (𝑎) = 0, 𝑖 = 0, 1, . . . , 𝑛 − 1.

(18)

(iii) The Cauchy function (𝑡 − 𝜌(𝑠))𝑛−1

/Γ(𝑛) satisfies∇𝑛𝑦(𝑡) = 0.

Regarding the nabla right fractional sum we observe thefollowing:

(i) 𝑏∇−𝛼 maps functions defined on 𝑏N to functions

defined on 𝑏N.(ii) 𝑏∇

−𝑛𝑓(𝑡) satisfies the 𝑛th-order discrete initial value

problem:

⊖Δ𝑛𝑦 (𝑡) = 𝑓 (𝑡) , ⊖Δ

𝑖𝑦 (𝑏) = 0, 𝑖 = 0, 1, . . . , 𝑛 − 1.

(19)

The proof can be done inductively. Namely, assumingit is true for 𝑛, we have

⊖Δ𝑛+1

𝑏∇−(𝑛+1)

𝑓 (𝑡) = ⊖Δ𝑛[−Δ 𝑏∇

−(𝑛+1)𝑓 (𝑡)] . (20)

By the help of (9), it follows that

⊖Δ𝑛+1

𝑏∇−(𝑛+1)

𝑓 (𝑡) = ⊖Δ𝑛

𝑏∇−𝑛𝑓 (𝑡) = 𝑓 (𝑡) . (21)

The other part is clear by using the convention that∑𝑠

𝑘=𝑠+1 = 0.

(iii) The Cauchy function (𝑠 − 𝜌(𝑡))𝑛−1

/Γ(𝑛) satisfies⊖Δ𝑛𝑦(𝑡) = 0.

Definition 4. (i) [12] The (delta) left fractional difference oforder 𝛼 > 0 (starting from 𝑎) is defined by

Δ𝛼

𝑎𝑓 (𝑡) = Δ𝑛Δ−(𝑛−𝛼)

𝑎 𝑓 (𝑡)

=

Δ𝑛

Γ (𝑛 − 𝛼)

𝑡−(𝑛−𝛼)

∑

𝑠=𝑎

(𝑡 − 𝜎 (𝑠))(𝑛−𝛼−1)

𝑓 (𝑠) ,

𝑡 ∈ N𝑎+(𝑛−𝛼).

(22)

(ii) [19]The (delta) right fractional difference of order 𝛼 >0 (ending at 𝑏) is defined by

𝑏Δ𝛼𝑓 (𝑡) = ∇

𝑛

⊝ 𝑏Δ−(𝑛−𝛼)

𝑓 (𝑡)

=

(−1)𝑛∇𝑛

Γ (𝑛 − 𝛼)

𝑏

∑

𝑠=𝑡+(𝑛−𝛼)

(𝑠 − 𝜎 (𝑡))(𝑛−𝛼−1)

𝑓 (𝑠) ,

𝑡 ∈ 𝑏−(𝑛−𝛼)N.

(23)

(iii) [20]The (nabla) left fractional difference of order 𝛼 >0 (starting from 𝑎) is defined by

∇𝛼

𝑎𝑓 (𝑡) = ∇𝑛∇−(𝑛−𝛼)

𝑎 𝑓 (𝑡)

=

∇𝑛

Γ (𝑛 − 𝛼)

𝑡

∑

𝑠=𝑎+1

(𝑡 − 𝜌 (𝑠))𝑛−𝛼−1

𝑓 (𝑠) ,

𝑡 ∈ N𝑎+1.

(24)

(iv) [29, 30] The (nabla) right fractional difference oforder 𝛼 > 0 (ending at 𝑏) is defined by

𝑏∇𝛼𝑓 (𝑡) = ⊝Δ

𝑛

𝑏∇−(𝑛−𝛼)

𝑓 (𝑡)

=

(−1)𝑛Δ𝑛

Γ (𝑛 − 𝛼)

𝑏−1

∑

𝑠=𝑡

(𝑠 − 𝜌 (𝑡))𝑛−𝛼−1

𝑓 (𝑠) ,

𝑡 ∈ 𝑏−1N .

(25)

Regarding the domains of the fractional type differenceswe observe the following.

(i) The delta left fractional difference Δ𝛼𝑎 maps functionsdefined on N𝑎 to functions defined on N𝑎+(𝑛−𝛼).

(ii) The delta right fractional difference 𝑏Δ𝛼 maps func-

tions defined on 𝑏N to functions defined on 𝑏−(𝑛−𝛼)N.

(iii) The nabla left fractional difference ∇𝛼𝑎 maps functionsdefined on N𝑎 to functions defined on N𝑎+𝑛.

(iv) The nabla right fractional difference 𝑏∇𝛼 maps func-

tions defined on 𝑏N to functions defined on 𝑏−𝑛N.


Lemma 5 (see [15]). Let 0 ≤ 𝑛 − 1 < 𝛼 ≤ 𝑛, and let 𝑦(𝑡) bedefined on N𝑎. Then the following statements are valid:

(i) (Δ𝛼𝑎)𝑦(𝑡 − 𝛼) = ∇𝛼𝑎−1𝑦(𝑡) for 𝑡 ∈ N𝑛+𝑎,

(ii) (Δ−𝛼𝑎 )𝑦(𝑡 + 𝛼) = ∇−𝛼𝑎−1𝑦(𝑡) for 𝑡 ∈ N𝑎.

Lemma 6 (see [29]). Let 𝑦(𝑡) be defined on 𝑏+1N. Then thefollowing statements are valid:

(i) (𝑏Δ𝛼)𝑦(𝑡 + 𝛼) = 𝑏+1∇

𝛼𝑦(𝑡) for 𝑡 ∈ 𝑏−𝑛N,

(ii) (𝑏Δ−𝛼)𝑦(𝑡 − 𝛼) = 𝑏+1∇

−𝛼𝑦(𝑡) for 𝑡 ∈ 𝑏N.

If 𝑓(𝑠) is defined on 𝑁𝑎 ∩ 𝑏𝑁 and 𝑎 ≡ 𝑏 (mod1) then(𝑄𝑓)(𝑠) = 𝑓(𝑎 + 𝑏 − 𝑠). The Q-operator generates a dualidentity by which the left type and the right type fractionalsums and differences are related. Using the change of variable𝑢 = 𝑎 + 𝑏 − 𝑠, in [18] it was shown that

Δ−𝛼

𝑎 𝑄𝑓 (𝑡) = 𝑄 𝑏Δ−𝛼𝑓 (𝑡) , (26)

and, hence,

Δ𝛼

𝑎𝑄𝑓 (𝑡) = (𝑄 𝑏Δ𝛼𝑓) (𝑡) . (27)

The proof of (27) follows by (26) and by noting that

−𝑄∇𝑓 (𝑡) = Δ𝑄𝑓 (𝑡) . (28)

Similarly, in the nabla case we have

∇−𝛼

𝑎 𝑄𝑓 (𝑡) = 𝑄 𝑏∇−𝛼𝑓 (𝑡) , (29)

and, hence,

∇𝛼

𝑎𝑄𝑓 (𝑡) = (𝑄 𝑏∇𝛼𝑓) (𝑡) . (30)

The proof of (30) follows by (29) and that

−𝑄Δ𝑓 (𝑡) = ∇𝑄𝑓 (𝑡) . (31)

For more details about the discrete version of the Q-operatorwe refer to [29].

From the difference calculus or time scale calculus, for anatural 𝑛 and a sequence 𝑓, we recall

Δ𝑛𝑓 (𝑡) =

𝑛

∑

𝑘=0

(−1)𝑘(

𝑛

𝑘)𝑓 (𝑡 + 𝑛 − 𝑘) ,

∇𝑛𝑓 (𝑡) =

𝑛

∑

𝑘=0

(−1)𝑘(

𝑛

𝑘)𝑓 (𝑡 − 𝑛 + 𝑘) .

(32)

2. The Fractional Differences and Sums withBinomial Coefficients

We first give the definition of fractional order of (32) in theleft and right sense.

Definition 7. The (binomial) delta left fractional differenceand sum of order 𝛼 > 0 for a function 𝑓 defined on N𝑎 aredefined by

(a)

𝐵Δ𝛼

𝑎 =

𝛼+𝑡−𝑎

∑

𝑘=0

(−1)𝑘(

𝛼

𝑘)𝑓 (𝑡 + 𝛼 − 𝑘) , 𝑡 ∈ N𝑎+𝑛−𝛼, (33)

(b)

𝐵Δ−𝛼

𝑎 =

𝛼+𝑡−𝑎

∑

𝑘=0

(−1)𝑘(

−𝛼

𝑘)𝑓 (𝑡 − 𝛼 − 𝑘) , 𝑡 ∈ N𝑎+𝛼, (34)

where (−1)𝑘 ( −𝛼𝑘 ) = ( 𝛼+𝑘−1𝑘 ).

Definition 8. The (binomial) nabla left fractional differenceand sum of order 𝛼 > 0 for a function 𝑓 defined on N𝑎, aredefined by

(a)

𝐵∇𝛼

𝑎 =

𝑡−𝑎−1

∑

𝑘=0

(−1)𝑘(

𝛼

𝑘)𝑓 (𝑡 − 𝑘) , 𝑡 ∈ N𝑎+𝑛, (35)

(b)

𝐵∇−𝛼

𝑎 =

𝑡−𝑎−1

∑

𝑘=0

(−1)𝑘(

−𝛼

𝑘)𝑓 (𝑡 − 𝑘) , 𝑡 ∈ N𝑎. (36)

Analogously, in the right case we can define the following.

Definition 9. The (binomial) delta right fractional differenceand sum of order 𝛼 > 0 for a function 𝑓 defined on 𝑏N aredefined by

(a)

𝑏Δ𝐵𝛼=

𝛼+𝑏−𝑡

∑

𝑘=0

(−1)𝑘(

𝛼

𝑘)𝑓 (𝑡 − 𝛼 + 𝑘) , 𝑡 ∈ 𝑏 − 𝑛 + 𝛼N,

(37)

(b)

𝑏Δ𝐵−𝛼

=

−𝛼+𝑏−𝑡

∑

𝑘=0

(−1)𝑘(

−𝛼

𝑘)𝑓 (𝑡 + 𝛼 + 𝑘) , 𝑡 ∈ 𝑏 − 𝛼N.

(38)

Definition 10. The(binomial) nabla right fractional differenceand sum of order 𝛼 > 0 for a function 𝑓 defined on 𝑏N aredefined by

(a)

𝑏∇𝐵𝛼=

𝑏−𝑡−1

∑

𝑘=0

(−1)𝑘(

𝛼

𝑘)𝑓 (𝑡 − 𝑘) , 𝑡 ∈ 𝑏−𝑛N, (39)

(b)

𝑏∇𝐵−𝛼

=

𝑏−𝑡−1

∑

𝑘=0

(−1)𝑘(

−𝛼

𝑘)𝑓 (𝑡 − 𝑘) , 𝑡 ∈ 𝑏N . (40)

Discrete Dynamics in Nature and Society 5

We next proceed to show that the Riemann fractionaldifferences and sums coincide with the binomial ones definedabove. We will use the dual identities in Lemma 5 andLemma 6, and the action of the discrete version of the Q-operator to follow easy proofs and verifications. In [20], theauthor used a delta Leibniz’s rule to obtain the following alter-native definition for Riemann delta left fractional differences:

Δ𝛼

𝑎𝑓 (𝑡) =1

Γ (−𝛼)

𝑡+𝛼

∑

𝑠=𝑎

(𝑡 − 𝜎 (𝑠))(−𝛼−1)

𝑓 (𝑠) ,

𝛼 ∉ N, 𝑡 ∈ N𝑎+𝑛−𝛼,

(41)

then proceeded with long calculations and showed, actually,that

Δ𝛼

𝑎𝑓 (𝑡) = 𝐵Δ𝛼

𝑎𝑓 (𝑡) , Δ−𝛼

𝑎 𝑓 (𝑡) = 𝐵Δ−𝛼

𝑎 𝑓 (𝑡) . (42)

Theorem 11. Let 𝑓 be defined on suitable domains and 𝛼 > 0.Then,

(1)

Δ𝛼

𝑎𝑓 (𝑡) = 𝐵Δ𝛼

𝑎𝑓 (𝑡) , Δ−𝛼

𝑎 𝑓 (𝑡) = 𝐵Δ−𝛼

𝑎 𝑓 (𝑡) , (43)

(2)

𝑏Δ𝛼𝑓 (𝑡) = 𝑏Δ𝐵

𝛼𝑓 (𝑡) , 𝑏Δ

−𝛼𝑓 (𝑡) = Δ𝐵

−𝛼

𝑎 𝑓 (𝑡) , (44)

(3)

∇𝛼

𝑎𝑓 (𝑡) = 𝐵∇𝛼

𝑎𝑓 (𝑡) , ∇−𝛼

𝑎 𝑓 (𝑡) = 𝐵∇−𝛼

𝑎 𝑓 (𝑡) , (45)

(4)

𝑏∇𝛼𝑓 (𝑡) = 𝑏∇𝐵

𝛼𝑓 (𝑡) , 𝑏∇

−𝛼𝑓 (𝑡) = 𝑏∇𝐵

−𝛼𝑓 (𝑡) .

(46)

Proof. (1) follows by (42).(2) By the discrete Q-operator action we have

𝑏Δ𝛼𝑓 (𝑡) = 𝑄Δ 𝑎 (𝑄𝑓) (𝑡)

= 𝑄

𝛼+𝑡−𝑎

∑

𝑘=0

(−1)𝑘(

𝛼

𝑘) (𝑄𝑓) (𝑡 + 𝛼 − 𝑘)

= 𝑏Δ𝐵𝛼𝑓 (𝑡) .

(47)

The fractional sum part is also done in a similar way by usingthe Q-operator.

(3) By the dual identity in Lemma 5 (i) and (42), we have

∇𝛼

𝑎𝑓 (𝑡) = Δ𝛼

𝑎+1𝑓 (𝑡 + 𝛼) = 𝐵Δ𝛼

𝑎+1𝑓 (𝑡 + 𝛼) = 𝐵∇𝛼

𝑎𝑓 (𝑡) .

(48)

The fractional sum part can be proved similarly by usingLemma 5 (ii) and (42).

(4) The proof can be achieved by either (2) and Lemma 6or, alternatively, by (3) and the discrete Q-operator.

Remark 12. In analogous to (41), the authors in [31] used anabla Leibniz’s rule to prove that

∇𝛼

𝑎𝑓 (𝑡) =1

Γ (−𝛼)

𝑡

∑

𝑠=𝑎+1

(𝑡 − 𝜌 (𝑠))−𝛼−1

𝑓 (𝑠) . (49)

In [30], the authors used a delta Leibniz’s Rule to provethe following formula for nabla right fractional differences:

𝑏∇𝛼𝑓 (𝑡) =

1

Γ (−𝛼)

𝑏−1

∑

𝑠=𝑡

(𝑠 − 𝜌 (𝑡))−𝛼−1

𝑓 (𝑠) . (50)

Similarly, we can use a nabla Leibniz’s rule to prove the fol-lowing formula for the delta right fractional differences:

𝑏Δ𝛼𝑓 (𝑡) =

1

Γ (−𝛼)

𝑏

∑

𝑠=𝑡−𝛼

(𝑠 − 𝜎 (𝑡))(−𝛼−1)

𝑓 (𝑠) . (51)

We here remark that the proofs of the last three parts ofTheorem 11 can be done alternatively by proceeding as in [20]starting from (49), (50), and (51). Also, it is worthmentioningthat mixing both delta and nabla operators in defining deltaand nabla right Riemann fractional differences was essentialin proceeding, through the dual identities and the discrete Q-operator or delta and nabla type Leibniz’s rules, to obtain themain results in this paper [29].

3. Conclusion

The impact of fractional calculus in both pure and appliedbranches of science and engineering started to increasesubstantially.Themain idea of iterating an operator and thengeneralizing to any order (real or complex) started to be usedin the last decade to obtain appropriate discretization for thefractional operators. We mention, from the theory of timescales view point, that how to obtain the fractional operatorswas a natural question and it was not correlated to the well-known Grunwald-Letnikov approach. We believe that thediscretizations obtained recently in the literature for the frac-tional operators are different from the one reported withinGrunwald-Letnikov method. Bearing all of these thinks inmind we proved that the discrete operators via binomialtheorem will lead to the same results as the ones by usingthe discretization of theRiemann-Liouville operators via timescales techniques. The discrete version of the impressive dualtool Q-operator has been used to prove the equivalency.

References

[1] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Marichev, Frac-tional Integrals and Derivatives: Theory and Applications, Gor-don and Breach, Yverdon, Switzerland, 1993.

[2] I. Podlubny, Fractional Differential Equations, vol. 198, Aca-demic Press., San Diego, Calif, USA, 1999.

[3] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,Theory and Ap-plication of Fractional Differential Equations, vol. 204 of North-HollandMathematics Studies, Elsevier Science, Amsterdam,TheNetherland, 2006.


[4] B. J. West, M. Bologna, and P. Grigolini, Physics of FractalOperators, Springer, New York, NY, USA, 2003.

[5] R. L. Magin, Fractional Calculus in Bioengineering, BegellHouse, West Redding, Conn, USA, 2006.

[6] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, FractionalCalculus Models and Numerical Methods, vol. 3 of Series onComplexity, Nonlinearity and Chaos, World Scientific, Hacken-sack, NJ, USA, 2012.

[7] M. D. Ortigueira, “Fractional central differences and deriva-tives,” Journal of Vibration and Control, vol. 14, no. 9-10, pp.1255–1266, 2008.

[8] P. Lino and G. Maione, “Tuning PI(]) fractional order con-trollers for position control of DC-servomotors,” in Proceedingsof the IEEE International Symposium on Industrial Electronics(ISIE ’10), pp. 359–363, July 2010.

[9] J. A. T. Machado, C. M. Pinto, and A. M. Lopes, “Power lawand entropy analysis of catastrophic phenomena,”MathematicalProblems in Engineering, vol. 2013, Article ID 562320, 10 pages,2013.

[10] C. F. Lorenzo, T. T. Hartley, and R. Malti, “Application of theprincipal fractional metatrigonometric functions for the solu-tion of linear commensurate-order time-invariant fractionaldifferential equations,” Philosophical Transactions of the RoyalSociety A, vol. 371, no. 1990, Article ID 20120151, 2013.

[11] H. L. Gray and N. F. Zhang, “On a new definition of the frac-tional difference,”Mathematics of Computation, vol. 50, no. 182,pp. 513–529, 1988.

[12] K. S. Miller and B. Ross, “Fractional difference calculus,” in Pro-ceedings of the International Symposium on Univalent Functions,Fractional Calculus and Their Applications, pp. 139–152, NihonUniversity, Koriyama, Japan, 1989.

[13] F. M. Atıcı and P.W. Eloe, “A transformmethod in discrete frac-tional calculus,” International Journal of Difference Equations,vol. 2, no. 2, pp. 165–176, 2007.

[14] F. M. Atıcı and P. W. Eloe, “Initial value problems in discretefractional calculus,” Proceedings of the American MathematicalSociety, vol. 137, no. 3, pp. 981–989, 2009.

[15] F. M. Atıcı and P. W. Eloe, “Discrete fractional calculus withthe nabla operator,” Electronic Journal of Qualitative Theory ofDifferential Equations, no. 3, pp. 1–12, 2009.

[16] F. M. Atıcı and S. Sengul, “Modeling with fractional differenceequations,” Journal of Mathematical Analysis and Applications,vol. 369, no. 1, pp. 1–9, 2010.

[17] F. M. Atıcı and P. W. Eloe, “Gronwall’s inequality on discretefractional calculus,” Computerand Mathematics with Applica-tions, vol. 64, no. 10, pp. 3193–3200, 2012.

[18] T. Abdeljawad, “On Riemann and Caputo fractional differ-ences,” Computers and Mathematics with Applications, vol. 62,no. 3, pp. 1602–1611, 2011.

[19] T. Abdeljawad and D. Baleanu, “Fractional differences andintegration by parts,” Journal of Computational Analysis andApplications, vol. 13, no. 3, pp. 574–582, 2011.

[20] M. Holm,The theory of discrete fractional calculus developmentand application [dissertation], University of Nebraska, Lincoln,Neb, USA, 2011.

[21] G. A. Anastassiou, “Principles of delta fractional calculus ontime scales and inequalities,” Mathematical and ComputerModelling, vol. 52, no. 3-4, pp. 556–566, 2010.

[22] G. A. Anastassiou, “Nabla discrete fractional calculus and nablainequalities,”Mathematical and ComputerModelling, vol. 51, no.5-6, pp. 562–571, 2010.

[23] G. A. Anastassiou, “Foundations of nabla fractional calculus ontime scales and inequalities,” Computers and Mathematics withApplications, vol. 59, no. 12, pp. 3750–3762, 2010.

[24] N. R. O. Bastos, R. A. C. Ferreira, andD. F.M. Torres, “Discrete-time fractional variational problems,” Signal Processing, vol. 91,no. 3, pp. 513–524, 2011.

[25] M. Bohner and A. Peterson,Advances in Dynamic Equations onTime Scales, Birkhauser, Boston, Mass, USA, 2003.

[26] G. Boros and V. Moll, Iresistible Integrals, Symbols, Analysis andExpreiments in the Evaluation of Integrals, Cambridge Univer-sity Press, Cambridge, UK, 2004.

[27] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Math-ematics: A Foundation for Copmuter Science, Addison-Wesley,Reading, Mass, USA, 2nd edition, 1994.

[28] J. Spanier and K. B. Oldham, “The pochhammer polynomials(x)𝑛,” in An Atlas of Functions, pp. 149–156, Hemisphere,Washington, DC, USA, 1987.

[29] T. Abdeljawad, “Dual identities in fractional difference calculuswithin Riemann,” Advances in Difference Equations, vol. 2013,article 36, 2013.

[30] T. Abdeljawad and F. M. Atıcı, “On the definitions of nablafractional operators,” Abstract and Applied Analysis, vol. 2012,Article ID 406757, 13 pages, 2012.

[31] K. Ahrendt, L. Castle, M. Holm, and K. Yochman, “Laplacetransforms for the nabla-difference operator and a fractionalvariation of parameters formula,” Communications in AppliedAnalysis. In press.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

research article fractional sums and differences with...

Documents