On delta and nabla Caputo fractional differences and dual identities

We Investigate two types of dual identities for Caputo fractional 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. Two types of Caputo fractional differences are introduced, one of them (dual one) is defined so that it obeys the investigated dual identities. The relation between Rieamnn and Caputo fractional differences is investigated and the delta and nabla discrete Mittag-Leffler functions are confirmed by solving Caputo type linear fractional difference equations. A nabla integration by parts formula is obtained for Caputo fractional differences 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].
In [1], the concept of Caputo fractional difference was introduced and investigated. In this article we proceed deeply to investigate Caputo fractional differences under two kinds of dual identities. The first kind relates nabla and delta type Caputo fractional differences and the second one, represented by the Q-operator, relates left and right ones. Arbitrary order Riemann and Caputo fractional differences are related as well. By the help of the previously obtained results in [27] and [28] an integration by parts formula for Caputo fractional differences is originated.
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. The third section contains some dual identities relating nabla and delta type fractional sums and differences in Riemann sense as previously investigated in [28]. In Section 4 Caputo fractional differences are given and related to the Riemann ones. In section 5, slightly different modified (dual) Caputo fracctional differences are introduced and investigated under some dual identities. Section 6 is devoted to the integration by parts for delta and nabla Caputo fractional differences. Finally, Section 7 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 Caputo fractional differences 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 Caputo 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 t (α) = Γ(t + 1) Γ(t + 1 − α) , (2) 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.
(ii) For real numbers a and b, we denote Na = {a, a + 1, .
(iv) For n ∈ N and real b, we denote 2 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 (ρ(s)−t) (n−1) (n−1)! vanishes at s = t + 1, t + 2, ..., t + (n − 1).
Regarding the nabla right fractional sum we observe the following: satisfies the n-th order discrete initial value problem 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 s k=s+1 = 0. (iii) The Cauchy function satisfies ⊖ ∆ n y(t) = 0.
Definition 2.2. (i) [8] The (delta) left fractional difference of order α > 0 (starting from a ) is defined by: (ii) [12] The (delta) right fractional difference of order α > 0 (ending at b ) is defined by: 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+n (on Na if we think f defined at some points before a).
(iv) The nabla right fractional difference b ∇ α maps functions defined on b N to functions defined on b−n N (on b N if we think f defined at some points after b).
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. Actually, the nabla fractional sums defined in this article and those in [7] are related. For more details we refer to [27]. [27] and [28]) For any α > 0, the following equality holds: Remark 2.1. (see [27] and [28]) Let α > 0 and n = [α] + 1. Then, by the help of Lemma 2 Then, using the identity we infer that (26) is valid for any real α.
Theorem 2.5. (see [27] and [28]) For any real number α and any positive integer p, the following equality holds: where f is defined on Na and some points before a .
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. (see [27] and [28]) For any real number α and any positive integer p, the following equality holds: where f is defined on b N and some points after b.

Dual identities for fractional sums and Riemann fractional 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 the author in [28] obtained: Lemma 3.3. Let y(t) be defined on b+1 N. Then the following statements are valid. ( Lemma 3.4. [28] Let 0 ≤ n − 1 < α ≤ n and let y(t) be defined on n−α N. Then the following statements are valid.
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].
[28] 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. (see [27] and [28]) Let α > 0, µ > −1. Then, for t ∈ Na , we have Proposition 3.10. (see [27] and [28]) 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 (48), interchanging the order of summation and using the power formula (47). 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.

Caputo fractional differences
In analogous to the usual fractional calculus we can formulate the following definition Definition 4.1. 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 and some points before a, is defined by (iv) the nabla α−order Caputo right fractional difference of a function f defined on b N and some points after b, 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 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 4.3. 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 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 (64) follows by the definition and applying Proposition 3.10 and (89) of Proposition 6.2. The proof of (65) follows by the definition and applying Proposition 3.7 and (92) of Proposition 6.3.
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 However, In the above formulae (67) and (68), 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.

A dual nabla Caputo fractional difference
In the previous section the nabla Caputo fractional difference is defined under the assumption that f is known before a in the left case and under the assumption that f is known after b in the right case. In this section we define other nabla Caputo fractional differences for which not necessary to request any information about f before a or after b. Since we shall show that these Caputo fractional differences are the dual ones for the delta Caputo fractional differences, we call them dual nabla Caputo fractional differences.
Notice that the Caputo and the dual Caputo differences coincide when 0 < α ≤ 1 and differ for higher order. That is for 0 < α ≤ 1 The following proposition states a dual relation between left delta Caputo fractional differences and left nabla (dual) Caputo fractional differences.
Proof. For t ∈ N a+n , we have Analogously, the following proposition relates right delta Caputo fractional differences and right nabla (dual) Caputo fractional differences.
The following theorem modifies Theorem when f is only defined at Na.
Theorem 5.3. For any real number α and any positive integer p, the following equality holds: where f is defined on only Na .
The proof follows by applying Remark 2.1 inductively.
Similarly, in the right case we have Theorem 5.4. For any real number α and any positive integer p, the following equality holds: where f is defined on b N only.

Now by using the modified Theorem 5.3 and Theorem 5.4 we have
Theorem 5.5. For any α > 0, we have In particular, when 0 < α < 1, then a(α) = a and b(α) = b and hence we have Also, by using the modified Theorem 5.3 and Theorem 5.4 we have Proposition 5.6. Assume α > 0 and f is defined on suitable domains Na and b N. Then In particular, if 0 < α ≤ 1 then a(α) = a and b(α) = b and hence

Integration by parts for Caputo fractional differences
In this section we state the integration by parts formulas for nabla fractional sums and differences obtained in [27], then use the dual identities to obtain delta integration by part formulas.
Proposition 6.1. [27] For α > 0, a, b ∈ R, f defined on Na and g defined on b N, we have Proof. By the definition of the nabla left fractional sum we have If we interchange the order of summation we reach at ( 85).
By the help of Theorem 5, Proposition 3.10, (17) and that ∇ −(n−α) a f (a) = 0, the authors in [27] obtained the following left important tools which lead to a nabla integration by parts formula for fractional differences. and By the help of Theorem 2.7, Proposition 3.7, (18) and that b ∇ −(n−α) f (b) = 0, the authors also in [27] also obtained the following right important tool: Proposition 6.4. [27] Let α > 0 be non-integer and a, b ∈ R such that a < b and b ≡ a (mod 1).If f is defined on b N and g is defined on Na, then The proof was achieved by making use of Proposition 6.1 and the tools Proposition 6.2 and Proposition 6.3.
By the above nabla integration by parts formulas and the dual identities in Lemma 3.1 adjusted to our definitions and Lemma 3.3, in [28] the following delta integration by parts formulas were obtained.
Proposition 6.5. [28] Let α > 0, a, b ∈ R such that a < b and b ≡ a (mod 1). If f is defined on Na and g is defined on b N , then we have Proposition 6.6. [28] Let α > 0 be non-integer and assume that b ≡ a (mod 1). If f is defined on b N and g is defined on Na, then Now, we proceed in this section to obtain nabla and delta integration by parts formulas for Caputo fractional differences. Theorem 6.7. Let 0 < α < 1 and f, g be functions defined on Na where Proof. From the definition of Caputo fractional difference and Proposition 6.1 we have By integration by parts from difference calculus, ∇f (s) = ∆f (s − 1) and the definition of nabla right fractional difference, we reach at From which ( 96) follows.

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 The proof of (101) follows by the definition, (100) and by noting that Similarly, in the nabla case we have and hence The proof of (103) follows by the definition, (102) and that −Q∆f (t) = ∇Qf (t).
The Q-dual identities (101) and( 103) are still valid for the delta and nabla (Riemann) fractional differences, respectively. The proof is similar to the Caputo case above.
It is remarkable to mention that the Q-dual identity (101) 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] is used. Thus, the definition introduced in [1] and [12] is more convenience . Analogously, the Q-dual identity (103) 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].
If we apply ∇ −α a on equation (110) then by (66) we see that To obtain an explicit solution, we apply the method of successive approximation. Set y 0 (t) =  Then, Interchanging the order of sums in (111), we conclude that That is Remark 7.1. If we solve the nabla discrete fractional system (110) 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).
[1] Let 0 < α ≤ 1, a = α − 1 and b such that a ≡ b (mod 1). Let y(t) be defined on Na ∩ b N. Consider the following Caputo right fractional difference equation b ∆ α Qy(t) = λy(2a + b − t), +f (a + b − t) (Qy)(b) = a 0 . (114) If we apply the Q operator on the Caputo fractinal difference equation (114), then we obtain the left Caputo fractional difference equation (104). For more about the Q-operator and its use in functional fractional differential equations we refer to [26].
Similar to Eaxample 7.5 above, we can use the Q-operator to transform, as well, nabla left type fractional difference equations to right ones and vise versa.