More Properties of Fractional Proportional Differences

Developing new differential and integral operators that generates the classical operators is an important branch of mathematical analysis [1, 2, 3, 4, 5, 6, 7, 8, 9]. Recently, the discrete fractional operators are investigated thoroughly to develop operators that can better describe some real world problems. In [10, 11], the authors introduced conformable derivatives and integrals which are local-type derivatives and integrals with arbitrary order. The authors in [12, 13] presented a type of proportional derivatives that yields the original function and its derivatives directly when the parameter tends to 0 and 1. In [14, 15, 16, 17, 18, 19], the nonsingular case is studied, authors defined new types of fractional operators with nonsingular exponential and Mittag-Leffler kernels. Fractional h-differences with discrete exponential kernels are discussed along with their monotonicity properties in [20, 21, 22]. The authors in [23], studied nabla fractional

Recently, the discrete fractional operators are investigated thoroughly to develop operators that can better describe some real world problems. In [10,11], the authors introduced conformable derivatives and integrals which are local-type derivatives and integrals with arbitrary order. The authors in [12,13] presented a type of proportional derivatives that yields the original function and its derivatives directly when the parameter tends to 0 and 1. In [14,15,16,17,18,19], the nonsingular case is studied, authors defined new types of fractional operators with nonsingular exponential and Mittag-Leffler kernels. Fractional h-differences with discrete exponential kernels are discussed along with their monotonicity properties in [20,21,22]. The authors in [23], studied nabla fractional sums and differences using the discrete Laplace transform on the time scale hZ. They employed a local nabla proportional difference to generate left and right generalized types of fractional differences with memory.
Local type derivatives and integrals are beneficial when generating new types of fractional derivatives and integrals with memory using different types of kernels [24,25,26,27]. In this paper, we reintroduce the discrete Laplace transform on the time scale hZ and extend the theoretical framework of the nabla discrete version of proportional differences to generate new types of generalized fractional differences and sums presented in [23]. The kernel of the generalized fractional sum and difference operators is then discussed. The convolution theorem for the discrete h-Laplace transform is utilized to introduce the discrete fractional operators and propose a solution for the Cauchy linear fractional difference type problems with step 0 < h 1.
The paper is organized as follows: In Section 2, we review the nabla fractional sums and differences on the time scale hZ. The generation of the fractional differences and sums with memory is discussed in Section 3. Section 4 dedicated to study Riemann fractional proportional sums and differences using h-discrete Laplace transforms. In Section 5, we present Caputo fractional proportional difference. Section 6 concludes the paper.

The nabla fractional sums and differences and Laplace transforms on hZ
This section is devoted to setting some essential definitions and assertions that will be used throughout the remarkaining part of the paper. Definition 2.1. [5] The following identities are valid.
(i) Let m be a natural number, then the m rising factorial of t is written as (2.1) (ii) For any real number, the α rising function becomes In addition, we have Hence t α is increasing on N 0 .
The backward difference operator on hZ is given by . For h = 1, we get the backward and forward difference operators ∇f(t) = f(t) − f(t − 1) and ∆f(t) = f(t + 1) − f(t), respectively. The forward jumping operator on the time scale hZ is σ h (t) = t + h and the backward jumping operator is ρ h (t) = t − h. For a, b ∈ R and h > 0 we use the notation N a,h = {a, a + h, a + 2h, .

Definition 2.2.
For arbitrary t, α ∈ R and h > 0, the nabla h−factorial function is defined by .
The proof of the above statement follows by using the definitions and direct calcula-

Lemma 2.4. For the time scale T = N a,h , one has the nabla Taylor polynomial
was used in [7,8] to connect left and right fractional sums and differences. In our manuscript, we also use the discrete Q−operator to relate left and right h− fractional sums and differences and hence confirm our definitions. Definition 2.5. (Nabla Discrete Mittag-Leffler)( [7,8,9]) For λ ∈ R, |λ| < 1 and α, β, z ∈ C with Re(α) > 0, the nabla discrete Mittag-Leffler functions is The following definition generalizes Definition 2.5.
Definition 2.6. (Nabla h−discrete Mittag-Leffler) For λ ∈ R, such that |λh α | < 1 and α, β, z ∈ C with Re(α) > 0, the nabla discrete Mittag-Leffler functions is The nabla right h−fractional sum of order α > 0 (ending and the nabla right h−fractional difference of order α > 0 (ending at b) is defined as The left h−Caputo fractional difference of order α starting at a h (α) is defined by The right h−Caputo fractional difference of order α ending at b h (α) is defined by For h = 1, we obtain the definitions given in [7,8].
The proof is a modification to the case h = 1 in [7,8] by making use of relation (2.4).
Following the time scale calculus, we have the following definition for the discrete Laplace transform on N a,h .
In case a = 0 and we write Definition 2.12. [9] Let s ∈ R, 0 < α < 1 and f, g : N a,h → R be a functions. The nabla h−discrete convolution of f with g is defined by The following Lemma is a generalization of Lemma 2 in [9] to hZ. However, the reader should note that the nabla discrete Laplace which is used here is slightly different. Lemma 2.14. [23] Let f be defined on N a,h . Then, Proof. Using Definition 2.11 and identities for hypergeometric functions in [30], we have The time scale algebraic operations on hZ [29] can be used to generate some of the present results in this paper. Her we mention some of the useful operations as follows: where, w = −w 1−wh . Using the above operations, we can assure that z z = 0 as follows: Proof. By Definition 2.11 and using time scale algebraic operations presented in [29], we get .24) is straightforward using the same steps when replacing h e λ (t, a) in (2.23) by h e λ (t, a).

The proportional differences and sums with memory
In [12], Anderson et al. introduced the modified conformable derivative by Then, the modified conformable differential operator of order ρ is defined by The derivative given in Definition 3.1 is called a proportional derivative. For more details about the control theory meanings of the proportional derivatives and its component functions κ 0 and κ 1 we refer to [12,13].
Of special interest, we shall consider in this article the case when κ 1 (ρ, t) = 1 − ρ and The reader should note that lim ρ→0 + D ρ f(t) = f(t) and lim ρ→1 − D ρ f(t) = f (t) which is an advantage over the conformable derivative since the conformable derivative does not tend to the original function as ρ tends to 0.
In view of(3.1), the h−discrete proportional derivative (proportional difference) of order 0 < ρ 1 for a function f defined on N a,h = {a, a + h, a + 2h, ...} is given by [23] ( where (∇ h f)(t) = f(t) − f(t − h) and the regressivity condition insists that 1 − h ρ−1 ρ = 0 or ρ = h 1−h . The proportional sum associated to ∇ ρ by The authors in [23] iterated the local fractional proportional sum to generate the following nonlocal fractional proportional sums and differences.
As in the classical fractional calculus, the right fractional sum ending at b can be then defined by where p = ρ−1 ρ .
Remark 3.4. [23] Using the discrete convolutions, we can express the left proportional fractional sum as (3.5) For the special case h = 1, we have Remark 3.5. [23] To deal with the right fractional proportional case we shall use the notation We shall also write ( ∆ n,ρ h g)(t) Definition 3.6. [23] For ρ > 0 and α ∈ C, Re(α) > 0, we define the left (proportional) fractional difference of f by The right (proportional) fractional difference ending at b is defined by where n = [Re(α)] + 1.

Proof. From Definition 3.3 and by the help of Theorem 3.7, we have
Lemma 3.11. For 0 < ρ 1, α > 0, and ξ defined on N a,h . Assume ξ is of discrete exponential order e c (t, 0) . Then, we have where N a,h {ξ(t)}(s) = ξ a (s) and p = ρ−1 ρ . Proof. First, by Remark 3.4, we observe that . (3.12) Apply the discrete Laplace transform, and make use of Theorem 2.13 and Lemma 3.11, to see that Then, the result follows by using that If in (3.11), we set ρ = 1 then we recover the identity  17) and (3.19) Then, applying the inverse discrete Laplace and by making use of (2.23) in Lemma 2.17, we reach (3.16), and hence the proof is completed.
. In particular, if m = 1, then
Remark 3.14. We have the following important observation for the discrete proportional operators. For a function f defined on N a−(n−1)h , we have for α > 0 and if α = n ∈ N, Alternatively, one can say that when f is only defined on N a we start our fractional proportional operators from a h (α).

(4.2)
Proof. By using Lemma 2.14, we have The statement of the theorem follows by applying (4.3) inductively.
Proof. By applying Theorem 4.2 and Lemma 3.11, we have Basing on Remark 3.14, we shall generate the following initial value problem in the sense of Riemann: a−h ∇ α,ρ h y(t) = f(t, y(t)) for t = a + h, a + 2h, · · · , (4.5) where 0 < α < 1 and a is any real number. Applying the operator a ∇ −α,ρ h to each side of the equation (4.5) we obtain Then using the definition of the fractional difference and sum operators, we obtain It follows from Lemma 3.13 (3.21) that Using that we obtain Finally, substituting back in (4.7) and making use of the fact that ∇ ρ h c 1 e p (t, a − h) = 0, we obtain the solution representation Indeed, we have proved the following theorem.

The Caputo fractional proportional difference
Definition 5.1. [23] For ρ ∈ (0, 1] and α ∈ C with Re(α) > 0. Assume f is defined on N a,h = {a, a + h, a + 2h, . . .} and on b,h N = {b, b − h, b − 2h, . . .}. We usually have b = a + kh for some k ∈ N. Then, we define the left Caputo fractional proportional difference starting at a by The right Caputo fractional proportional difference ending at b In the right case, we get Proof. By the help of (3.23) in Remark 3.14, we have a h (α)).
In the proof, we have used that η(h, ρ, n) = e p (h(n − 1), 0). For the proof of (5.4), one may either follow similar arguments or use the action of the Q−operator.
Proof. By using Lemma 3.11 and Theorem 4.2, we obtain From which (5.5) follows and the proof is completed.

6)
Similarly, by applying the Q−operator, we have In fact, if we apply N a,h to (5.8) and make use of Theorem 5.5 with n = 1, then we have Hence, Applying the inverse of N a,h and using Theorem 2.13 and Lemma 2.16, we reach the representation (5.9). Conversely, if y(t) has the representation (5.9), then by the help of Example 5.3 it satisfies (5.8).

Conclusions
Proposing new fractional operators has become one of the most important tackles in the field of mathematical analysis. This is due to the need of different types of fractional operators that serve to study some real world phenomena. The proportional fractional operators, both in continuous and discrete versions were recently proposed. In this work, we presented the discrete fractional operators defined on hZ. These operators covers some known discrete fractional operators such as the Riemann-Liouville fractional sum, Riemann-Liouville fractional difference and the Caputo fractional difference as ρ tends to 1. In addition, we found the discrete h-Laplace transforms of the fractional operators discussed and used them to solve some linear type problems. On the top of this, we corrected a mistake done previously in the literature concerning the h-Laplace transforms of these operators.