#### Abstract

We independently propose a new kind of the definition of fractional difference, fractional sum, and fractional difference equation, give some basic properties of fractional difference and fractional sum, and give some examples to demonstrate several methods of how to solve certain fractional difference equations.

#### 1. Introduction

Fractional calculus is an emerging field recently drawing attention from both theoretical and applied disciplines. During the last two decades, it has been successfully applied to several fields [1–6], and it is well known that there is a large quantity of research on what is usually called integer-order difference equations [7, 8]. However, discrete fractional calculus and fractional difference equations represent a very new area for scientists. A pioneering work has been done by Atici et al. [9–12], Anastassiou [13, 14], Bastos et al. [15], Abdeljawad et al. [16–20], and Cheng [21–23], and so forth. In this paper, limited to the length of the paper, we will introduce some of our basic works about discrete fractional calculus and fractional difference equations. Some proofs and results of the theorems and examples in Sections 3–5 are well proved by a more concise method. We refer to the monographer [23] for more further results. In [23] we also aim at presenting some basic properties about discrete fractional calculus and, in a systematic manner, results including the existence and uniqueness of solutions for the Cauchy Type and Cauchy problems, involving nonlinear fractional difference equations, explicit solutions of linear difference equations and linear difference system by their deduction to Volterra sum equation and by using operational methods, applications of Z-transform, R-transform, N-transform, Adomian decomposition method, method of undetermined coefficients, Jordan matrix theory method, and by discrete Mittag-Leffler function and discrete Green’ function, and a theory of so-called sequential linear fractional difference equations, as well as some introduction for discrete fractional difference variational problem, and so forth.

#### 2. Integer-Order Difference and Sum with Real Variable

Let us start from sum and difference of the integer order. Define where , .

*Definition 2.1. *Let , be real numbers, and let be a positive number, we call
one-order backward sum of , where , . We call
-order backward sum of , where is a positive integer number.

*Definition 2.2. *Let , be real numbers, and let be a positive number, we call
one-order forward sum of , where , . We call
-order forward difference of , where is a positive integer number.

*Definition 2.3. *Let be a real number, and let be a positive number, we call
one-order backward difference of , where is step. We call
-order backward difference of , where is a positive integer number.

Similarly, we can define forward difference as follows.

*Definition 2.4. *Let be a real number, and let be a positive number, we call
one-order forward difference of , where is step. We call
-order forward difference of , where is a positive integer number.

Theorem 2.5. *The following two equalities hold:*(1)*, *(2)*. *

*Definition 2.6. *If , are real numbers, and let be a positive number, define
rising factorial function, and set . If is a positive integer number, then we have

*Definition 2.7. *Let , be real numbers, and let be a positive number, define
down factorial function, and set . If is an positive integer number, then
In Definitions 2.6 and 2.7, if , we can simply denote , as , .

*Definition 2.8. *For any , , we define
If , , then it is easy to see that
If we let , or , then we clearly have the following.

Theorem 2.9. *Assume that , ; then
*

Theorem 2.10. *Let , , then, following equality holds:
*

#### 3. Fractional Sum and Difference with Real Variable

Before giving the definitions of fractional sum , let us revisit the calculation of the sum of the integer order. By Definition 2.1, we have then By recursive, it is not hard to obtain where .

Obviously, the right side of formula (3.3) is also meaningful for all real , so we define fractional sum as follows.

*Definition 3.1. *Let , , , , , we call
order fractional sum of .

For any positive number order fractional difference, we take the following.

*Definition 3.2. *Let , and assume that , where denotes a positive integer. Define
as order - type backward fractional difference. Meantime, define
as order Caputo type backward fractional difference.

If we start from Definition 2.2, completely in a similar way, we get positive integer -order forward sum where .

The right side of (3.8) is meaningful for all real , so we can define forward fractional sum as follows.

*Definition 3.3. *Let , , , , , define
as order fractional sum of , where .

*Definition 3.4. *Let , and assume that , where denotes a positive integer. Define
as order - type forward fractional difference. Meantime, define
as order Caputo type forward fractional difference.

In Definitions 3.1–3.4, if step , it is a kind of important situation. At this time, we simply denote , ; , as, ; , . When , backward fractional sum is defined as follows.

*Definition 3.5. *Let , and define
as order fractional sum of , where , .

For any positive number order fractional difference, we can take the following way.

*Definition 3.6. *Let and assume that , where denotes a positive integer. Define
as order - type backward fractional difference. Meantime, define
as order Caputo type backward fractional difference.

We can define forward fractional sum as follows.

*Definition 3.7. *Let , and define
as order forward fractional sum of , where , .

*Definition 3.8. *Let , and assume that , where denotes a positive integer. Define
as order - type forward fractional difference. Meantime, define
as order Caputo type forward fractional difference.

By Definition 2.8, it is easy to calculate

By Theorem 2.9 we have

Therefore, if we adopt Definition 2.8, then Definitions 3.1, 3.3, 3.5, and 3.7 can be rewritten as follows.

*Definition 3.9. *Assume that , let , , , , and define
as order backward fractional sum of .

*Definition 3.10. *Assume that , let , , , , and define
as order forward fractional sum of .

*Definition 3.11. *Assume that , , and , and define
as order backward fractional sum of .

*Definition 3.12. *Assume that , , and , and define
as order forward fractional sum of .

Set , , or , and set ; then by Theorem 2.9 and Definitions 3.1–3.4, one obtains the following.

Theorem 3.13. *For any , the following equalities hold:*(1)*; ,*(2)*; ,*(3)*; .*

From Theorem 3.13 we can see, by stretching , the functions and , with common step , can be convert into the functions and with step , respectively. In essence, nothing arises much different, but the latter is more convenient in research.

In view of Definitions 3.1–3.4 and Theorem 2.10, if we let , then we can obtain the following.

Corollary 3.14. *Assume that is integrable, then:*(1)*,
*(2)*,
*(3)*. *

#### 4. Some Basic Properties

We sometimes only list some basic results here, for more detailed results and their proofs can been seen in monographer [23].

Theorem 4.1. *Assume that the following function is well defined; then*(1)*, ,*(2)*, , ,*(3)*If , then , ,*(4)*, ,*(5)*Let , if , then ; If , then .*

Theorem 4.2. *Let , where is defined in , then*(1)*, ,*(2)*, .*

Theorem 4.3. *Let , , is defined in , then*(1)*,
*(2)*. *

Theorem 4.4. *For any real , the following equality holds:*(1)*,
*(2)*. *

Theorem 4.5. *For any real and , the following equality holds:*(1)*,
*(2)*. *

Theorem 4.6. *Let , then*(1)*,
*(2)*. *

In the previous theorems, we only need to consider the simplest case , but actually the methods of proof and conclusions can also be extended for general step . In fact, we only need do a stretching transformation and then make use of Theorem 2.9.

Next, we discusses fractional sum transform such as: transform, transform, transform, and some properties of these transforms.

*Definition 4.7. *Let be defined in , we call
is a transform of , denote it by .

*Definition 4.8. *Let be defined in , and define transform as follows:
If the domain of the function is , then we use the notation .

If we set , define Then, can be regarded as a sequence Under this definition, transform can be simply rewritten as Set , then we have where is transform of sequence .

If , then

Theorem 4.9. *For any , then*(1)*, ,*(2)*, .*

*Proof. *(1) Making use of (4.7), we get
where is transform of sequence ,
Since (see [21–23])
hence

(2) It is only to use
then the proof of (2) follows from the proof of (1).

Theorem 4.10. *Let and be defined in , and define convolution of , as follows:
**
For , then
*

Theorem 4.11. *Let , be defined in , then
*

Theorem 4.12. *For any real , one has
*

Theorem 4.13. *For , one has
**
where is defined in .*

Theorem 4.14. *Let , , then
*

Theorem 4.15. *Let be a real function, , then
*

*Definition 4.16. *Let be defined in , and define transform as follows:
In Definition 4.16, if we set , and define:
then, can be regarded as a sequence
Under this definition, transform can be simply rewritten as
Set , then
where is a transform of sequence .

Theorem 4.17. *For any , then*(1)*,*(2)*. *

*Proof. * (1) let , then
where is a transform of sequence . Since
and (see [22, 23])
hence
or

(2) The proof of (2) follows from the proof of (1).

*Definition 4.18. *Define convolution of and as follows:
If , then

Theorem 4.19. *For any , then
*

Theorem 4.20. *Let , , and let be defined in , then
*

Theorem 4.21. *Let , , then
*

Theorem 4.22. *Let be a real function, , then for all , one has
*

#### 5. The Solution of the Fractional Difference Equations with Real Variable

In this section, we give examples to demonstrate the solving method of fractional difference equations and reveal the inner relationship between fractional differential equations and fractional differential equations.

Theorem 5.1. *Let , , then*(1)*, ,*(2)*, .*

*Proof. *(1) The proof of (1) directly follows from Theorem 4.1 and Theorem 4.2.

(2) By Theorem 4.2 and (1), we have

*Example 5.2. * Consider Euler type fractional difference equations
Set , and take it into previous equation, we get
By Theorem 4.1 (4), we obtain
and get indicator equation
Therefore, we can transform Euler type fractional difference equations into its indicator equation.

*Example 5.3. *Consider initial value problem of homogeneous linear order () fractional difference equation with constant coefficient
Note that is defined in , since
Therefore, initial problem of (5.6) is equivalent to initial problem
The solution of initial problem of (5.6) is equivalent to the solution of sum equations

We use approximation method to solve these sum equations. Set
Applying power law (Theorem 4.22), we get
Applying power law repeatedly, and by recursion, we obtain
Let , then

*Example 5.4. *Let , , we call
the fractional difference equation of order .

In order to solve this equation, we need to introduce some special functions.

*Definition 5.5. *Define function
where . Sometimes denote it or .

In view of Theorems 4.2 and 4.3, we can establish the following theorem.

Theorem 5.6. *Assume the following function is well defined; then*(1)*,
*(2)*,
*(3)*, where ,*(4)*, where ,*(5)*. *

Now we will use the method of undetermined coefficients to solve Example 5.4. By Theorem 5.6, we notice that

The significance of these applications is that if we apply the operator to then we get a cyclic permutation of the same functions. That is, no new functions are introduced. Therefore, we will choose a linear combination of these functions as a candidate for a solution of (5.14). Say Then Taking , into the left side of (5.14), we obtain In order to make the right side equate zero, set Then If we let be a root of the indicial equation or , then we have Since we also need so let us set Since is an arbitrary number, set , then Therefore, we obtain a solution of fractional difference of order as

The fractional difference equation of order in Example 5.4 can be solved by the method of transform. Make transform to the following equation: We have Taking them into previos equation, we get and we have In [23], we have the following

Theorem 5.7. *The following equality holds:*(1)*,
*(2)*. **Set , then
**
By Theorem 5.7 and (5.33), we know that
**
is a solution of (5.14). Except a constant, the solution is the same as the solution (5.28), where
**
which is solved by the method of undetermined coefficients before.*

#### 6. Relationship between the Fractional Difference Equations and the Fractional Differential Equations

In this section, we only give an example to demonstrate the relationship between integers order difference equations and integral order differential equation.

Let us recall the definition of fractional sum when step where . If we set then And it is easy to prove that Therefore, we have the following.

Theorem 6.1. *Let , and set , then
*

*Example 6.2. *(1) Set , , and solve the fractional difference equation of order ,

(2) Let , and solve the equation

(3) Let , and solve the equation

(4) If we let , we ask whether the limit solution of (6.8) is equivalent to that of the following fractional differential equation? Consider

*Solution 1. *(1) By a result in Chapter 7 of book [23], the solution of (6.6) is