Dual identities in fractional difference calculus within Riemann

We Investigate two types of dual identities for Riemann fractional sums and differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. These dual identities insist that in the definition of right fractional differences we have to use both the nabla and delta operators. The solution representation for higher order Riemann fractional difference equation is obtained as well.


Introduction
During the last two decades, due to its widespread applications in different fields of science and engineering, fractional calculus has attracted the attention of many researchers [9,10,11,23,24,25].
Starting from the idea of discretizing the Cauchy integral formula, Miller and Ross [8] and Gray and Zhang [18] obtained discrete versions of left type fractional integrals and derivatives, called fractional sums and differences. Fifteen years later, several authors started to deal with discrete fractional calculus [1,3,4,5,6,14,12,20,21,22], benefiting from the theory of time scales originated by Hilger in 1988 (see [15]).
In this article, we summarize some of the results mentioned in the above references and add more in the Caputo type and right type fractional differences. Throughout the article we almost agree with the previously presented definitions except for the definition of right fractional difference. We shall figure out that these definitions seem to be more convenient than the previously presented ones by proving some dual identities. These identities fall into two kinds. The first relate nabla type fractional differences and sums to delta ones. The second kind represented by the Q-operator relate left and right fractional sums and differences. Also, in the definition of left and right fractional sums we use the ones that agree with the general theory for time scales and not as in [5] or in [18]. This setting enables us to get identities resembling better the ordinary fractional case. Along with the previously mentioned points we are able to fit a reasonable nabla integration by parts formula which remains in accordance with the one obtained in [12] but different from those obtained in [6] and [14].
The article is organized as follows: The remaining part of this section contains summary to some of the basic notations and definitions in delta and nabla calculus. Section 2 contains the definitions in the frame of delta and nabla fractional sums and differences in the Riemann sense. Moreover some essential lemmas about the commutativity of the different fractional sum operators with the usual difference operators are established. These lemmas are vital to proceed in the next sections. The third section contains some dual identities relating nabla and delta fractional sums and differences in the left and right cases. Using these dual identities, power formulae for nabla left and right fractional sums and a commutative law for nabla left and right sums are obtained. In Section 4 an integration by parts formula for nabla fractional sums and differences is obtained with an accordance to the one obtained in [12]. Section 5 is devoted to delta and nabla Caputo fractional differences and their relations with the Riemann ones. Finally, Section 6 contains Caputo type fractional dynamical equations where a nonhomogeneous nabla Caputo fractional difference equation is solved to obtain nabla discrete versions for Mittag-Leffler functions. For the case α = 1 we obtain the discrete nabla exponential function [15]. In addition to this, the Q-operator is used to relate left and right fractional sums in the nabla and delta case. The Q-dual identities obtained in this section expose the validity of the definition of delta and nabla right fractional differences.
For a natural number n, the fractional polynomial is defined by, where Γ denotes the special gamma function and the product is zero when t + 1 − j = 0 for some j. More generally, for arbitrary α, define where the convention that division at pole yields zero. Given that the forward and backward difference operators are defined by respectively, we define iteratively the operators ∆ m = ∆(∆ m−1 ) and ∇ m = ∇(∇ m−1 ), where m is a natural number.
Here are some properties of the factorial function.
) Assume the following factorial functions are well defined.
Also, for our purposes we list down the following two properties, the proofs of which are straightforward.
For the sake of the nabla fractional calculus we have the following definition (ii) For any real number the α rising function is defined by Regarding the rising factorial function we observe the following: Notation: (i) For a real α > 0, we set n = [α] + 1, where [α] is the greatest integer less than α.
(iii) For n ∈ N and real a, we denote a∆ n f (t) (−1) n ∆ n f (t).
(iv) For n ∈ N and real b, we denote ∇ n b f (t) (−1) n ∇ n f (t).

Definitions and essential lemmas
Definition 2.1. Let σ(t) = t + 1 and ρ(t) = t − 1 be the forward and backward jumping operators, respectively. Then (i) The (delta) left fractional sum of order α > 0 (starting from a) is defined by: (ii) The (delta) right fractional sum of order α > 0 (ending at b) is defined by: (iii) The (nabla) left fractional sum of order α > 0 (starting from a) is defined by: (iv)The (nabla) right fractional sum of order α > 0 (ending at b) is defined by: Regarding the delta left fractional sum we observe the following: (iii) The Cauchy function Regarding the delta right fractional sum we observe the following: (iii) the Cauchy function satisfies ∇ n y(t) = 0.
Regarding the nabla right fractional sum we observe the following: The proof can be done inductively. Namely, assuming it is true for n, we have By the help of (10), it follows that The other part is clear by using the convention that t k=s = 0, s > t.
Regarding the domains of the fractional type differences we observe: (i) The delta left fractional difference ∆ α a maps functions defined on Na to functions defined on N a+(n−α) .
(ii) The delta right fractional difference b ∆ α maps functions defined on b N to functions defined on b−(n−α) N.
(iii) The nabla left fractional difference ∇ α a maps functions defined on Na to functions defined on N a+1−n .
(iv) The nabla right fractional difference b ∇ α maps functions defined on b N to functions defined on b−1+n N.
For any α > 0, the following equality holds: Lemma 2.2. [12] For any α > 0, the following equality holds: For any α > 0, the following equality holds: The result of Lemma 2.3 was obtained in [7] by applying the nabla left fractional sum starting from a not from a + 1. Next will provide the version of Lemma 2.3 by applying the definition in this article.
Lemma 2.4. For any α > 0, the following equality holds: Proof. By the help of the following by parts identity On the other hand Then, using the identity we infer that (26) is valid for any real α.
By the help of Lemma 2.4, Remark 2.1 and the identity we arrive inductively at the following generalization.
Theorem 2.5. For any real number α and any positive integer p, the following equality holds: where f is defined on Na .
Lemma 2.6. For any α > 0, the following equality holds: Proof. By the help of the following discrete by parts formula: On the other hand, where the identity and the convention that (0) α−1 = 0 are used.
Then, using the identity we infer that (35) is valid for any real α.
By the help of Lemma 2.6, Remark 2.2 and the identity ∆ if we follow inductively we arrive at the following generalization Theorem 2.7. For any real number α and any positive integer p, the following equality holds:

Dual identities for right fractional sums and differences
The dual relations for left fractional sums and differences were investigated in [5]. Indeed, the following two lemmas are dual relations between the delta left fractional sums (differences) and the nabla left fractional sums (differences).
[5] Let 0 ≤ n−1 < α ≤ n and let y(t) be defined on Na. Then the following statements are valid.
[5] Let 0 ≤ n − 1 < α ≤ n and let y(t) be defined on N α−n . Then the following statements are valid.
We remind that the above two dual lemmas for left fractional sums and differences were obtained when the nabla left fractional sum was defined by Now, in analogous to Lemma 3.1 and Lemma 3.2, for the right fractional summations and differences we obtain Lemma 3.3. Let y(t) be defined on b+1 N. Then the following statements are valid.
Proof. We prove only (i). The proof of (ii) is similar and easier.
We prove (i), the proof of (ii) is similar. By the definition of right nabla difference we have Note that the above two dual lemmas for right fractional differences can not be obtained if we apply the definition of the delta right fractional difference introduced in [14] and [6].
The following commutative property for delta right fractional sums is Theorem 9 in [12].
Proposition 3.7. Let f be a real valued function defined on b N, and let α, β > 0. Then Proof. The proof follows by applying Lemma 3.3(ii) and Theorem 3.6 above. Indeed, The following power rule for nabla right fractional differences plays an important rule.

Proof. By the dual formula (ii) of Lemma 3.3, we have
Then by the identity t α = (t + α − 1) (α−1) and using the change of variable r = s − µ + 1, it follows that Which by Lemma 3.5 leads to Similarly, for the nabla left fractional sum we can have the following power formula and exponent law Proposition 3.9. Let α, µ > 0. Then, for t ∈ Na , we have Proposition 3.10. Let f be a real valued function defined on Na, and let α, β > 0. Then Proof. The proof can be achieved as in Theorem 2.1 [5], by expressing the left hand side of (58), interchanging the order of summation and using the power formula (57). Alternatively, the proof can be done by following as in the proof of Proposition 3.7 with the help of the dual formula for left fractional sum in Lemma 3.1 after its arrangement according to our definitions.

Integration by parts for fractional sums and differences
We first state the integration by parts for delta fractional sums and differences.
[12] Let α > 0, a, b ∈ R such that a < b and b ≡ a + α (mod 1). If f is defined on Na and g is defined on b N , then we have Proposition 4.2. [12] Let α > 0 be non-integer and assume that b ≡ a + (n − α) (mod 1). If f is defined on b N and g is defined on Na, then b−n+1 Proof. By the definition of the nabla left fractional sum we have If we interchange the order of summation we reach at ( 61).
By the help of Theorem 2.5, Proposition 3.10, (17) and that ∇ −(n−α) a f (a) = 0, we can, for the nabla left sums and differences, obtain Proposition 4.4. For α > 0, and f defined in a suitable domain Na, we have and We recall that (64) is valid in the usual fractional case for sufficiently good functions such as continuous functions. As a result of this it was possible to obtain an integration by parts in the Riemann fractional derivative case for certain class of functions (see [11] page 76, and for more details see [10]). Since discrete functions are continuous we see that the term ∇ −(1−α) a f (t)|t=a, for 0 < α < 1 disappears in (64), with the application of the convention that a s=a+1 f (s) = 0. However, in the Caputo case the initial type conditions starting from a appear as we shall see in the next sections. By this connection, we would like to ask the reader to compare with [7], where the term ∇ −(1−α) a f (t)|t=a = f (a) appears and hence it was possible to employ the Riemann type initial value fractional difference equation to obtain a discrete fractional version of Gronwall's inequality. However, we remind that the authors there obey [18] in defining the nabla left fractional sum. In our case the Caputo fractional difference will be the more suitable tool to obtain such fractional version of Gronwall's inequality.
By the help of Theorem 2.7, Proposition 3.7, (18) and that b ∇ −(n−α) f (b) = 0, we can, for the nabla right sums and differences, obtain Proposition 4.6. Let α > 0 be non-integer. If f is defined on b N and g is defined on Na, then Proof. By the help of equation (67) of Proposition 4.5 we can write and by Proposition 4.1 we have Then the result follows by equation (64) of Proposition 4.4.

Caputo fractional differences
In analogous to the usual fractional calculus we can formulate the following definition Definition 5.1. Let α > 0, α / ∈ N. Then, (i) [1] the delta α−order Caputo left fractional difference of a function f defined on Na is defined by (iii) the nabla α−order Caputo left fractional difference of a function f defined on Na is defined by It is clear that C ∆ α a maps functions defined on Na to functions defined on N a+(n−α) , and that C b ∆ α maps functions defined on b N to functions defined on b−(n−α) N. Also, it is clear that the nabla left fractional difference ∇ α a maps functions defined on Na to functions defined on N a+1−n and the nabla right fractional difference b ∇ α maps functions defined on b N to functions defined on b−1+n N.
Riemann and Caputo delta fractional differences are related by the following theorem Theorem 5.1.
[1] For any α > 0, we have In particular, when 0 < α < 1, we have One can note that the Riemann and Caputo fractional differences, for 0 < α < 1, coincide when f vanishes at the end points.
The following identity is useful to transform delta type Caputo fractional difference equations into fractional summations.
In particular, if 0 < α ≤ 1 then Similar to what we have above, for the nabla fractional differences we obtain Theorem 5.3. For any α > 0, we have In particular, when 0 < α < 1, we have Proof. The proof follows by replacing α by n − α and p by n in Theorem 2.5 and Theorem 2.7, respectively.
One can see that the nabla Riemann and Caputo fractional differences, for 0 < α < 1, coincide when f vanishes at the end points.
In particular, if 0 < α ≤ 1 then Proof. The proof of (87) follows by the definition and applying Proposition 3.10 and (65) of Proposition 4.4. The proof of (88) follows by the definition and applying Proposition 3.7 and (68) of Proposition 4.5.
Using the definition and Proposition 3.9 and Proposition 3.8, we can find the nabla type Caputo fractional differences for certain power functions. For example, for 1 = β > 0 and α ≥ 0 we have and However, In the above formulae (90) and (91), we apply the convention that dividing over a pole leads to zero. Therefore the fractional difference when β − 1 = α − j, j = 1, 2, ..., n is zero.
Remark 5.1. The results obtained in Theorem 5.1and afterward agree with those in the usual continuous case (See [11] pages 91,96).

The Q-operator and fractional difference equations
If f (s) is defined on Na ∩ b N and a ≡ b (mod 1) then (Qf )(s) = f (a + b − s). The Q-operator generates a dual identity by which the left type and the right type fractional sums and differences are related. Using the change of variable u = a + b − s, in [1] it was shown that and (95) The proof of (95) follows by the definition, (94) and by noting that Similarly, in the nabla case we have and hence (97) The proof of (97) follows by the definition, (96) and that −Q∆f (t) = ∇Qf (t). The Q-dual identities (95) and( 97) are still valid for the delta and nabla (Riemann) fractional differences, respectively. The proof is similar to the Caputo case above.
It is worthwhile to mention that the Q-dual identity (95) can not be obtained if the definition of the delta right fractional difference introduced by Nuno R.O. Bastos et al. in [14] or by Atıcı F. et al. in [6]. Thus, the definition introduced in [1] and [12] is more convenience . Analogously, the Q-dual identity (97) indicates that the nabla right Riemann and Caputo fractional differences presented in this article are also more convenient.
It is clear from the above argument that, the Q-operator agrees with its continuous counterpart when applied to left and right fractional Riemann Integrals and the Caputo and Riemann derivatives. More generally, this discrete version of the Q-operator can be used to transform the discrete delay-type fractional functional difference dynamic equations to advanced ones. For details in the continuous counterparts see [2]. Example 6.1.
(104) If we apply ∇ −α a on equation (104) then by (89) we see that To obtain an explicit solution, we apply the method of successive approximation. Set y 0 (t) = a 0 and ym(t) = a 0 + ∇ −α a [λy m−1 (t) + f (t)], m = 1, 2, 3, .... For m = 1, we have by the power formula (57) For m = 2, we also see by the help of Proposition 3.10 that Then, (t − ρ(s)) kα+α−1 f (s). (105) Interchanging the order of sums in (105), we conclude that That is Remark 6.1. If we solve the nabla discrete fractional system (104) with α = 1 and a 0 = 1 we obtain the solution The first part of the solution is the nabla discrete exponential function e λ (t, a). For the sake of more comparisons see ( [15], chapter 3).