Fractional Difference Equations with Real Variable

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 _ℎ^𝑡_𝑠=𝑎𝑥(𝑠)≜[𝑥(𝑎)+𝑥(𝑎+ℎ)+𝑥(𝑎+2ℎ)+⋯+𝑥(𝑡)], (2.1) where 𝑡=𝑎+𝑗ℎ,  𝑗∈𝑁₀={0,1,2,…}.

Definition 2.1. Let 𝑎,  𝑡 be real numbers, and let ℎ be a positive number, we call _𝑎∇_ℎ⁻¹𝑥(𝑡)=_ℎ^𝑡_𝑠=𝑎𝑥(𝑠)ℎ (2.2) one-order backward sum of 𝑥(𝑡), where 𝑡=𝑎+𝑗ℎ,  𝑗∈𝑁₀={0,1,2,…}. We call _𝑎∇_ℎ^−𝑘𝑥(𝑡)=_𝑎∇_ℎ⁻¹_𝑎∇_ℎ^{−(𝑘−1)}𝑥(𝑡) (2.3)𝑘-order backward sum of 𝑥(𝑡), where 𝑘 is a positive integer number.

Definition 2.2. Let 𝑎,  𝑡 be real numbers, and let ℎ be a positive number, we call _𝑎Δ_ℎ⁻¹𝑥(𝑡)=_ℎ^𝑡−ℎ_𝑠=𝑎𝑥(𝑠)ℎ (2.4) one-order forward sum of 𝑥(𝑡), where 𝑡=𝑎+𝑗ℎ,  𝑗∈𝑁₁={1,2,…}. We call _𝑎Δ_ℎ^−𝑘𝑥(𝑡)=_𝑎Δ_ℎ⁻¹_𝑎Δ_ℎ^{−(𝑘−1)}𝑥(𝑡) (2.5)𝑘-order forward difference of 𝑥(𝑡), where 𝑘 is a positive integer number.

Definition 2.3. Let 𝑡 be a real number, and let ℎ be a positive number, we call ∇_ℎ𝑥(𝑡)=𝑥(𝑡)−𝑥(𝑡−ℎ)ℎ (2.6) one-order backward difference of 𝑥(𝑡), where ℎ is step. We call ∇^𝑘_ℎ𝑥(𝑡)=∇_ℎ∇_ℎ^𝑘−1𝑥(𝑡) (2.7)𝑘-order backward difference of 𝑥(𝑡), where 𝑘 is a positive integer number.

Similarly, we can define forward difference as follows.

Definition 2.4. Let 𝑡 be a real number, and let ℎ be a positive number, we call Δ_ℎ𝑥(𝑡)=𝑥(𝑡+ℎ)−𝑥(𝑡)ℎ (2.8) one-order forward difference of 𝑥(𝑡), where ℎ is step. We call Δ^𝑘_ℎ𝑥(𝑡)=Δ_ℎΔ_ℎ^𝑘−1𝑥(𝑡) (2.9)𝑘-order forward difference of 𝑥(𝑡), where 𝑘 is a positive integer number.

Theorem 2.5. The following two equalities hold:(1)∇_ℎ(_𝑎∇_ℎ⁻¹𝑥(𝑡))=𝑥(𝑡), (2)Δ_ℎ(_𝑎Δ_ℎ⁻¹𝑥(𝑡))=𝑥(𝑡).

Definition 2.6. If 𝑘,  𝑡 are real numbers, and let ℎ be a positive number, define 𝑡^𝑘_ℎ=ℎ^𝑘Γ(𝑡/ℎ+𝑘)Γ(),()𝑡/ℎ𝑘∈𝑅 (2.10) rising factorial function, and set 𝑡⁰_ℎ=1. If 𝑘 is a positive integer number, then we have 𝑡^𝑘_ℎ=𝑡(𝑡+ℎ)(𝑡+2ℎ)⋯(𝑡+(𝑘−1)ℎ). (2.11)

Definition 2.7. Let 𝑘,  𝑡 be real numbers, and let ℎ be a positive number, define 𝑡_ℎ^(𝑘)=ℎ^𝑘Γ(𝑡/ℎ+1)Γ(),()𝑡/ℎ+1−𝑘𝑘∈𝑅 (2.12) down factorial function, and set 𝑡_ℎ⁽⁰⁾=1. If 𝑘 is an positive integer number, then 𝑡_ℎ^(𝑘)=𝑡(𝑡−ℎ)(𝑡−2ℎ)⋯(𝑡−(𝑘−1)ℎ). (2.13) In Definitions 2.6 and 2.7, if ℎ=1, we can simply denote 𝑡^𝑘_ℎ,  𝑡_ℎ^(𝑘) as 𝑡^𝑘,  𝑡^(𝑘).

Definition 2.8. For any 𝑘,𝛾∈𝑅,  ℎ>0, we define 𝛾𝑘≜Γ𝑘+𝛾Γ𝛾Γ(),𝛾𝑘+1𝑘_ℎ≜ℎ^𝛾𝛾𝑘ℎ. (2.14) If 𝑘∈𝑁,  ℎ=1, then it is easy to see that 𝛾𝑘=𝛾𝛾+1⋯𝛾+𝑘−1.𝑘! (2.15) If we let 𝑡/ℎ=𝑡̃, or 𝑡=𝑡ℎ̃, then we clearly have the following.

Theorem 2.9. Assume that 𝑘∈𝑅,ℎ>0,  𝑡/ℎ=𝑡̃; then 𝑡^𝑘_ℎ=ℎ^𝑘𝑡^𝑘̃;𝑡_ℎ^(𝑘)=ℎ^𝑘𝑡̃^(𝑘). (2.16)

Theorem 2.10. Let 𝑘∈𝑅,  ℎ>0, then, following equality holds: lim_ℎ→0𝑡^𝑘_ℎ=lim_ℎ→0𝑡_ℎ^(𝑘)=𝑡^𝑘. (2.17)

3. Fractional Sum and Difference with Real Variable

Before giving the definitions of fractional sum _𝑎∇_ℎ^−𝛾𝑥(𝑡),𝛾>0, let us revisit the calculation of the sum of the integer order. By Definition 2.1, we have _𝑎∇_ℎ⁻¹𝑥(𝑡)=_ℎ^𝑡_𝑠=𝑎𝑥(𝑠)ℎ,𝑡=𝑎+𝑗ℎ,𝑗∈𝑁₀, (3.1) then _𝑎∇_ℎ⁻²𝑥(𝑡)=_𝑎∇_ℎ⁻¹_𝑎∇_ℎ⁻¹𝑥(𝑡)=_ℎ^𝑡_{𝑎𝑠=𝑎}∇_ℎ⁻¹𝑥(𝑠)ℎ=ℎ_ℎ^𝑡_ℎ^𝑠_{𝑠=𝑎𝑟=𝑎}𝑥(𝑟)ℎ=ℎ²_ℎ^𝑡_ℎ^𝑡_{𝑟=𝑎𝑠=𝑟}𝑥(𝑟)=ℎ²_ℎ^𝑡_𝑟=𝑎𝑡−𝑟+ℎℎ𝑥(𝑟)=_ℎ^𝑡_𝑟=𝑎(𝑡−𝑟+ℎ)¹_ℎ𝑥(𝑟)ℎ,_𝑎∇_ℎ⁻³𝑥(𝑡)=_𝑎∇_ℎ⁻¹_𝑎∇_ℎ⁻²𝑥(𝑡)=_ℎ^𝑡_{𝑎𝑠=𝑎}∇_ℎ⁻²𝑥(𝑠)ℎ=ℎ²_ℎ^𝑡_ℎ^𝑠_{𝑠=𝑎𝑟=𝑎}𝑡−𝑟+ℎℎ𝑥(𝑟)ℎ=ℎ³2^ℎ𝑡_𝑟=𝑎𝑡−𝑟+ℎℎ𝑡−𝑟+2ℎℎ𝑥(𝑟=1)2!^ℎ𝑡_𝑟=𝑎(𝑡−𝑟+ℎ)²_ℎ𝑥(𝑟)ℎ…. (3.2) By recursive, it is not hard to obtain _𝑎∇_ℎ^−𝑚𝑥(𝑡)=ℎ^𝑚(𝑚−1)!^ℎ𝑡_𝑠=𝑎𝑡−𝑠+ℎℎ𝑡−𝑠+2ℎℎ⋯𝑡−𝑠+(𝑚−1)ℎℎ𝑥(𝑠=1)Γ(𝑚)^ℎ𝑡_𝑠=𝑎(𝑡−𝑠+ℎ)_ℎ^𝑚−1𝑥(𝑠)ℎ=1Γ(𝑚)^ℎ𝑡_𝑠=𝑎𝑡−𝜌_ℎ(𝑠)_ℎ^𝑚−1𝑥(𝑠)ℎ, (3.3) where 𝜌_ℎ(𝑠)=𝑠−ℎ.

Obviously, the right side of formula (3.3) is also meaningful for all real 𝑚>0, so we define fractional sum as follows.

Definition 3.1. Let 𝛾>0,  𝑎∈𝑅,  ℎ>0,  𝑡=𝑎+𝑘ℎ,  𝑘∈𝑁₀, we call _𝑎∇_ℎ^−𝛾𝑥(𝑡)=1Γ𝛾^ℎ𝑡_𝑠=𝑎𝑡−𝜌_ℎ(𝑠)_ℎ^𝛾−1𝑥(𝑠)ℎ (3.4)𝛾 order fractional sum of 𝑥(𝑡).

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

Definition 3.2. Let 𝜇>0, and assume that 𝑚−1<𝜇<𝑚, where 𝑚 denotes a positive integer. Define _𝑎∇^𝜇_ℎ𝑥(𝑡)=∇^𝑚_ℎ𝑎∇_ℎ^{−(𝑚−𝜇)}𝑥(𝑡) (3.5) as 𝜇 order 𝑅-𝐿 type backward fractional difference. Meantime, define ^𝐶_𝑎∇^𝜇_ℎ𝑥(𝑡)=_𝑎∇_ℎ^{−(𝑚−𝜇)}∇^𝑚_ℎ𝑥(𝑡) (3.6) as 𝜇 order Caputo type backward fractional difference.

If we start from Definition 2.2, _𝑎Δ_ℎ⁻¹𝑥(𝑡)=_ℎ^𝑡−ℎ_𝑠=𝑎𝑥(𝑠)ℎ,𝑡=𝑎+𝑗ℎ,𝑗∈𝑁₁, (3.7) completely in a similar way, we get positive integer 𝑚-order forward sum _𝑎Δ_ℎ^−𝑚𝑥(𝑡)=1Γ(𝑚)^{ℎ𝑡−𝑚ℎ}_𝑠=𝑎𝑡−𝜎_ℎ(𝑠)_ℎ^(𝑚−1)𝑥(𝑠)ℎ, (3.8) where 𝜎_ℎ(𝑠)=𝑠+ℎ.

The right side of (3.8) is meaningful for all real 𝑚>0, so we can define forward fractional sum as follows.

Definition 3.3. Let 𝛾>0,  𝑎∈𝑅,  ℎ>0,  𝑡=𝑎+𝛾ℎ+𝑘ℎ,  𝑘∈𝑁₀, define _𝑎Δ_ℎ^−𝛾𝑥(𝑡)=1Γ(𝜈)^{ℎ𝑡−𝛾ℎ}_𝑠=𝑎𝑡−𝜎_ℎ(𝑠)_ℎ^(𝛾−1)𝑥(𝑠)ℎ (3.9) as 𝛾 order fractional sum of 𝑥(𝑡), where 𝜎_ℎ(𝑠)=𝑠+ℎ.

Definition 3.4. Let 𝜇>0, and assume that 𝑚−1<𝜇<𝑚, where 𝑚 denotes a positive integer. Define _𝑎Δ^𝜇_ℎ𝑥(𝑡)=Δ^𝑚_ℎ𝑎Δ_ℎ^{−(𝑚−𝜇)}𝑥(𝑡) (3.10) as 𝜇 order 𝑅-𝐿 type forward fractional difference. Meantime, define ^𝐶_𝑎Δ^𝜇_ℎ𝑥(𝑡)=_𝑎Δ_ℎ^{−(𝑚−𝜇)}Δ^𝑚_ℎ𝑥(𝑡) (3.11) as 𝜇 order Caputo type forward fractional difference.

In Definitions 3.1–3.4, if step ℎ=1, it is a kind of important situation. At this time, we simply denote _𝑎∇_ℎ^−𝛾,  _𝑎Δ_ℎ^−𝛾;  ∇^𝜇_ℎ,  Δ^𝜇_ℎ as_𝑎∇^−𝛾,  _𝑎Δ^−𝛾;  ∇^𝜇,  Δ^𝜇. When ℎ=1, backward fractional sum is defined as follows.

Definition 3.5. Let 𝛾>0, and define _𝑎∇^−𝛾𝑥(𝑡)=1Γ𝛾^𝑡_𝑠=𝑎𝑡−𝜌(𝑠)^𝛾−1𝑥(𝑠) (3.12) as 𝛾 order fractional sum of 𝑥(𝑡), where 𝑡=𝑎mod(1),  𝜌(𝑠)=𝑠−1.

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

Definition 3.6. Let 𝜇>0 and assume that 𝑚−1<𝜇<𝑚, where 𝑚 denotes a positive integer. Define _𝑎∇^𝜇𝑥(𝑡)=∇^𝑚_𝑎∇^{−(𝑚−𝜇)}𝑥(𝑡) (3.13) as 𝜇 order 𝑅-𝐿 type backward fractional difference. Meantime, define ^𝐶_𝑎∇^𝜇𝑥(𝑡)=_𝑎∇^{−(𝑚−𝜇)}∇^𝑚𝑥(𝑡) (3.14) as 𝜇 order Caputo type backward fractional difference.

We can define forward fractional sum as follows.

Definition 3.7. Let 𝛾>0, and define _𝑎Δ^−𝛾𝑥(𝑡)=1Γ𝛾^𝑡−𝛾_𝑠=𝑎(𝑡−𝜎(𝑠))^(𝛾−1)𝑥(𝑠) (3.15) as 𝛾 order forward fractional sum of 𝑥(𝑡), where 𝑡−𝛾=𝑎mod(1),  𝜎(𝑠)=𝑠+1.

Definition 3.8. Let 𝜇>0, and assume that 𝑚−1<𝜇<𝑚, where 𝑚 denotes a positive integer. Define _𝑎Δ^𝜇𝑥(𝑡)=Δ^𝑚_𝑎Δ^{−(𝑚−𝜇)}𝑥(𝑡) (3.16) as 𝜇 order 𝑅-𝐿 type forward fractional difference. Meantime, define ^𝐶_𝑎Δ^𝜇𝑥(𝑡)=_𝑎Δ^{−(𝑚−𝜇)}Δ^𝑚𝑥(𝑡) (3.17) as 𝜇 order Caputo type forward fractional difference.

By Definition 2.8, it is easy to calculate 𝛾=1𝑡−𝑠Γ𝛾𝑡−𝜌(𝑠)^𝛾−1,𝛾=1𝑡−𝛾−𝑠Γ𝛾(𝑡−𝜎(𝑠))^𝛾−1. (3.18)

By Theorem 2.9 we have (𝑡−𝑠+ℎ)_ℎ^𝛾−1Γ𝛾=ℎ^𝛾−1((𝑡−𝑠)/ℎ+1)^𝛾−1Γ𝛾=ℎ^𝛾−1𝛾𝑡−𝑠ℎ,(𝑡−𝑠−ℎ)_ℎ^(𝛾−1)Γ𝛾=ℎ^𝛾−1((𝑡−𝑠)/ℎ−1)^(𝛾−1)Γ𝛾=ℎ^𝛾−1𝛾𝑡−𝑠ℎ.−𝛾 (3.19)

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 𝛾>0, let 𝑎∈𝑅,  ℎ>0,  𝑡=𝑎+𝑘ℎ,  𝑘∈𝑁₀, and define _𝑎∇_ℎ^−𝛾𝑥(𝑡)=_ℎ^𝑡_𝑠=𝑎𝛾𝑡−𝑠_ℎ𝑥(𝑠) (3.20) as 𝛾 order backward fractional sum of 𝑥(𝑡).

Definition 3.10. Assume that 𝛾>0, let  𝑎∈𝑅,  ℎ>0,  𝑡=𝑎+𝛾ℎ+𝑘ℎ,  𝑘∈𝑁₀, and define _𝑎Δ_ℎ^−𝛾𝑥(𝑡)=_ℎ^{𝑡−𝛾ℎ}_𝑠=𝑎𝛾𝑡−𝑠−𝛾ℎ_ℎ𝑥(𝑠) (3.21) as 𝛾 order forward fractional sum of 𝑥(𝑡).

Definition 3.11. Assume that 𝛾>0,  𝑡,𝑎∈𝑅, and 𝑡=𝑎mod(1), and define _𝑎∇^−𝛾𝑥(𝑡)=^𝑡_𝑠=𝑎𝛾𝑥(𝑠)𝑡−𝑠 (3.22) as 𝛾 order backward fractional sum of 𝑥(𝑡).

Definition 3.12. Assume that 𝛾>0, 𝑡,𝑎∈𝑅, and 𝑡−𝛾=𝑎mod(1), and define _𝑎Δ^−𝛾𝑥(𝑡)=^𝑡−𝛾_𝑠=𝑎𝛾𝑥(𝑠)𝑡−𝛾−𝑠 (3.23) as 𝛾 order forward fractional sum of 𝑥(𝑡).

Set 𝑎/ℎ=𝑎,  𝑡/ℎ=𝑡̃, or 𝑎=𝑎ℎ,𝑡=𝑡ℎ̃, and set 𝑥(𝑡)=𝑥(𝑡ℎ̃)=𝑦(𝑡)̃; then by Theorem 2.9 and Definitions 3.1–3.4, one obtains the following.

Theorem 3.13. For any 𝛾,𝜇>0, the following equalities hold:(1)_𝑎∇_ℎ^−𝛾𝑥(𝑡)=ℎ^𝛾[_𝑎∇^−𝛾𝑦(𝑡)̃];  _𝑎Δ_ℎ^−𝛾𝑥(𝑡)=ℎ^𝛾[_𝑎Δ^−𝛾𝑦(𝑡)̃],(2)_𝑎∇^𝜇_ℎ𝑥(𝑡)=ℎ^−𝜇[_𝑎∇^𝜇𝑦(𝑡)̃];  _𝑎Δ^𝜇_ℎ𝑥(𝑡)=ℎ^−𝜇[_𝑎Δ^𝜇𝑦(𝑡)̃],(3)^𝐶_𝑎∇^𝜇_ℎ𝑥(𝑡)=ℎ^−𝜇[^𝐶_𝑎∇^𝜇𝑦(𝑡)̃];  ^𝐶_𝑎Δ^𝜇_ℎ𝑥(𝑡)=ℎ^−𝜇[^𝐶_𝑎Δ^𝜇𝑦(𝑡)̃].

From Theorem 3.13 we can see, by stretching 𝑡=𝑡ℎ̃, the functions _𝑎∇_ℎ^−𝛾𝑥(𝑡) and _𝑎∇^𝜇_ℎ𝑥(𝑡), with common step ℎ, can be convert into the functions _𝑎∇^−𝛾𝑦(𝑡)̃ and _𝑎∇^𝜇𝑦(𝑡)̃ with step ℎ=1, 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 ℎ→0, then we can obtain the following.

Corollary 3.14. Assume that 𝑥(𝑡) is integrable, then:(1)lim_ℎ→0(_𝑎∇_ℎ^−𝛾𝑥(𝑡))=lim_ℎ→0(_𝑎∇_ℎ^−𝛾𝑥(𝑡))=(1/Γ(𝛾))∫^𝑡_𝑎(𝑡−𝑠)^𝛾−1𝑥(𝑡)𝑑𝑠≜𝐷_𝑡^−𝛾𝑥(𝑡), (2)lim_ℎ→0(_𝑎∇^𝜇_ℎ𝑥(𝑡))=lim_ℎ→0(_𝑎∇^𝜇_ℎ𝑥(𝑡))=𝐷^𝑚(_𝑎𝐷_𝑡^{−(𝑚−𝜇)}𝑥(𝑡))≜_𝑎𝐷^𝜇_𝑡𝑥(𝑡), (3)lim_ℎ→0(^𝐶_𝑎∇^𝜇_ℎ𝑥(𝑡))=lim_ℎ→0(^𝐶_𝑎∇^𝜇_ℎ𝑥(𝑡))=𝐷^𝑚(_𝑎𝐷_𝑡^{−(𝑚−𝜇)}𝑥(𝑡))≜^𝐶_𝑎𝐷^𝜇_𝑡𝑥(𝑡).

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)∇_ℎ𝑡^𝛾_ℎ=𝛾𝑡_ℎ^𝛾−1,  Δ_ℎ𝑡_ℎ^(𝛾)=𝛾𝑡_ℎ^(𝛾−1),(2)(𝑡+𝛾ℎ)𝑡^𝛾_ℎ=𝑡_ℎ^𝛾+1,  (𝑡−𝛾ℎ)t_ℎ^(𝛾)=𝑡_ℎ^(𝛾+1),  𝛾∈𝑅,(3)If 0<𝛾<1, then 𝑡_ℎ^𝛼𝛾≤(𝑡^𝛼_ℎ)^𝛾,  𝑡_ℎ^(𝛼𝛾)≥(𝑡_ℎ^(𝛼))^𝛾,(4)𝑡_ℎ^𝛼+𝛽=(𝑡+𝛽)^𝛼_ℎ𝑡^𝛽_ℎ,  𝑡_ℎ^(𝛼+𝛽)=(𝑡−𝛽)_ℎ^(𝛼)𝑡_ℎ^(𝛽),(5)Let 0<𝑡≤𝑟,  if 𝛾>0,  then 𝑡^𝛾_ℎ≤𝑟^𝛾_ℎ,𝑡_ℎ^(𝛾)≤𝑟^𝛾_ℎ; If 𝛾<0, then 𝑡^𝛾_ℎ≥𝑟^𝛾_ℎ,𝑡_ℎ^(𝛾)≥𝑟^𝛾_ℎ.

Theorem 4.2. Let 0≤𝑚−1<𝛾≤𝑚,𝑚∈𝑁, where 𝑥(𝑡) is defined in 𝑁_ℎ,𝑎={𝑎,𝑎+ℎ,𝑎+2ℎ,…}, then(1)_𝑎∇_ℎ^−𝛾𝑥(𝑡)=_𝑎Δ_ℎ^−𝛾𝑥(𝑡+𝛾ℎ),  𝑡∈𝑁_ℎ,𝑎,(2)_𝑎∇^𝛾𝑥(𝑡)=_𝑎Δ^𝛾𝑥(𝑡−𝛾ℎ),  𝑡∈𝑁_{ℎ,𝑚+𝑎}.

Theorem 4.3. Let 0≤𝑚−1<𝛾≤𝑚,  𝑚∈𝑁,  𝑥(𝑡) is defined in 𝑁_ℎ,𝑎={𝑎,𝑎+ℎ,𝑎+2ℎ,…}, then(1)_𝑎Δ_ℎ^−𝛾𝑥(𝑡)=_𝑎∇_ℎ^−𝛾𝑥(𝑡−𝛾ℎ),𝑡∈𝑁_{ℎ,𝑎+𝛾}, (2)_𝑎Δ^𝛾_ℎ𝑥(𝑡)=_𝑎∇^𝛾_ℎ𝑥(𝑡+𝛾ℎ),𝑡∈𝑁_{ℎ,𝑎−𝛾+𝑚}.

Theorem 4.4. For any real 𝛾, the following equality holds:(1)_𝑎∇_ℎ^−𝛾∇_ℎ𝑥(𝑡)=∇_ℎ(_𝑎∇_ℎ^−𝛾)𝑥(𝑡)−((𝑡−𝑎−1)_ℎ^𝛾−1/Γ(𝛾))𝑥(𝑎−ℎ), (2)_𝑎Δ_ℎ^−𝛾Δ_ℎ𝑥(𝑡)=Δ_ℎ(_𝑎Δ_ℎ^−𝛾)𝑥(𝑡)−((𝑡−𝑎)_ℎ^(𝛾−1)/Γ(𝛾))𝑥(𝑎).

Theorem 4.5. For any real 𝛾 and 𝑝>0, the following equality holds:(1)_𝑎∇_ℎ^−𝛾∇^𝑝_ℎ𝑥(𝑡)=∇^𝑝_ℎ(_𝑎∇_ℎ^−𝛾𝑥(𝑡))−∑^𝑝−1_𝑘=0((𝑡−𝑎+1)_ℎ^{𝛾−𝑝+𝑘}/Γ(𝛾+𝑘−𝑝+1))∇^𝑘_ℎ𝑥(𝑎−ℎ), (2)_𝑎Δ_ℎ^−𝛾Δ^𝑝_ℎ𝑥(𝑡)=Δ^𝑝_ℎ(_𝑎Δ_ℎ^−𝛾𝑥(𝑡))−∑^𝑝−1_𝑘=0((𝑡−𝑎)_ℎ^{(𝛾−𝑝+𝑘)}/Γ(𝛾+𝑘−𝑝+1))Δ^𝑘_ℎ𝑥(𝑎).

Theorem 4.6. Let 𝑝,𝛾>0, then(1)∇^𝑝_ℎ(_𝑎∇_ℎ^−𝛾𝑥(𝑡))=_𝑎∇_ℎ^{−(𝛾−𝑝)}𝑥(𝑡), (2)Δ^𝑝_ℎ(_𝑎Δ_ℎ^−𝛾𝑥(𝑡))=_𝑎Δ_ℎ^{−(𝛾−𝑝)}𝑥(𝑡).

In the previous theorems, we only need to consider the simplest case ℎ=1, but actually the methods of proof and conclusions can also be extended for general step ℎ>0. 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: 𝑍 transform, 𝑁 transform, 𝑅 transform, and some properties of these transforms.

Definition 4.7. Let 𝑓(𝑡) be defined in 𝑁₀={0,1,2,…}, we call 𝑓(𝑡)=^∞_𝑡=0𝑓(𝑡)𝑧^−𝑡 (4.1) is a 𝑍 transform of 𝑓(𝑡), denote it by 𝑍[𝑓(𝑡)].

Definition 4.8. Let 𝑓(𝑡) be defined in 𝑁_𝑡0={𝑡₀,𝑡₀+1,𝑡₀+2,…},𝑡₀∈𝑅, and define 𝑁 transform as follows: 𝑁_𝑡0𝑓(𝑡)(𝑠)=^∞_𝑡=𝑡0(1−𝑠)^𝑡−1𝑓(𝑡). (4.2) If the domain of the function 𝑓(𝑡) is 𝑁₁, then we use the notation 𝑁(𝑓(𝑡)).

If we set 𝑡−𝑡₀=𝑛∈𝑁₀, define 𝑓^{𝑡0}_𝑛=𝑓𝑛+𝑡₀=𝑓(𝑡),𝑓^{𝑡0}_𝑛−1=𝑓𝑛−1+𝑡₀()𝑓=𝑓𝑡−1,…,_{𝑡0}0=𝑓0+𝑡₀=𝑓𝑡₀. (4.3) Then, 𝑓(𝑡₀),𝑓(𝑡₀+1),…,𝑓(𝑡),… can be regarded as a sequence 𝑓^{𝑡0}₀,𝑓^{𝑡0}₁,…𝑓^{𝑡0}_𝑛,…. (4.4) Under this definition, 𝑁 transform can be simply rewritten as 𝑁₀𝑓(𝑡)(𝑠)=^∞_𝑡=𝑡0(1−𝑠)^𝑡−1𝑓(𝑠)=^∞_𝑛=0(1−𝑠)^{𝑛+𝑡0−1}𝑓𝑛+𝑡₀=(1−𝑠)^𝑡0∞−1_𝑛=0(1−𝑠)^𝑛𝑓^{𝑡0}_𝑛. (4.5) Set 𝑧=1/(1−𝑠), then we have 𝑁₀𝑓(𝑡)(𝑠)=𝑧^1−𝑡0∞_𝑛=0𝑓^{𝑡0}_𝑛𝑧^−𝑛=𝑧^1−𝑡0𝐹(𝑧), (4.6) where 𝐹(𝑧) is 𝑍 transform of sequence 𝑓^{𝑡0}_𝑛.

If 𝑡₀=1, then 𝑁𝑓(𝑡(𝑧),1)=𝐹𝑧=.1−𝑠 (4.7)

Theorem 4.9. For any 𝛾∈𝑅⧵{…,−2,−1,0}, then(1)𝑁(𝑡^𝛾−1)(𝑠)=Γ(𝛾)/𝑠^𝛾,  |1−𝑠|<1,(2)𝑁(t^𝛾−1𝛼^−𝑡)(𝑠)=𝛼^𝛾−1Γ(𝛾)/(𝑠+𝛼−1)^𝛾,  |1−𝑠|<𝛼.

Proof. (1) Making use of (4.7), we get 𝑁𝑡^𝛾−1Γ𝛾=𝐹(𝑧), (4.8) where 𝐹(𝑧) is 𝑍 transform of sequence 𝑓_𝑛^{1}=𝑓(𝑛+1), 𝑓_𝑛^{1}()=()=𝑓𝑛+1𝑛+1^𝛾−1Γ𝛾=𝛾𝑛. (4.9) Since (see [21–23]) 𝐹𝛾𝑛=𝑧−1𝑧^−𝛾=1𝑠^𝛾,(|𝑧|>1,|1−𝑠|<1), (4.10) hence 𝑁𝑡^𝛾−1Γ𝛾=1𝑠^𝛾,(|1−𝑠|<1). (4.11)
(2) It is only to use ^∞_𝑡=1(1−𝑠)^𝑡−1𝑡^𝛾−1𝛼^−𝑡=1𝛼^∞_𝑡=11−𝑠+𝛼−1𝛼^𝑡−1𝑡^𝛾−1, (4.12) then the proof of (2) follows from the proof of (1).

Theorem 4.10. Let 𝑓(𝑡) and 𝑔(𝑡) be defined in 𝑁_𝑎, and define convolution of 𝑓(𝑡),  𝑔(𝑡) as follows: ℎ∗𝑔_𝑎(𝑡)=^𝑡_𝑠=𝑎ℎ𝑡−𝜌(𝑠)𝑔(𝑠). (4.13) For ℎ(𝑡)=𝑡^𝛾−1/Γ(𝛾), then ℎ∗𝑔_𝑎(𝑡)=1Γ𝛾^𝑡_𝑠=𝑎𝑡−𝜌(𝑠)^𝛾−1𝑔(𝑠)=_𝑎∇^−𝛾𝑔(𝑡). (4.14)

Theorem 4.11. Let 𝑓,  𝑔 be defined in 𝑁_𝑎, then 𝑁_𝑎𝑓∗𝑔=𝑁₁𝑓𝑁_𝑎𝑔. (4.15)

Theorem 4.12. For any real 𝛾, one has 𝑁_𝑎𝑎∇^−𝛾𝑓(𝑡)=𝑠^−𝛾𝑁_𝑎𝑓(𝑡). (4.16)

Theorem 4.13. For 0<𝛾≤1, one has 𝑁_𝑎+1𝑎∇^−𝛾𝑓(𝑡)=𝑠^𝛾𝑁_𝑎𝑓(𝑡)(𝑠)−(1−𝑠)^𝛼−1𝑓(𝑎), (4.17) where 𝑓 is defined in 𝑁_𝑎.

Theorem 4.14. Let 𝜇∈𝑅⧵{…,−2,−1,0},  𝛾>0, then ₁∇^−𝛾𝑡^𝜇Γ=𝑡𝜇+1Γ^𝜇+𝛾.𝜇+𝛾+1 (4.18)

Theorem 4.15. Let 𝑓 be a real function, 𝜇,𝛾>0, then _𝑎∇^−𝛾_𝑎∇^−𝜇𝑓(𝑡)=_𝑎∇^{−(𝜇+𝛾)}𝑓(𝑡)=_𝑎∇^−𝜇_𝑎∇^−𝛾𝑓(𝑡). (4.19)

Definition 4.16. Let 𝑓(𝑡) be defined in 𝑁_𝑡0, and define 𝑅 transform as follows: 𝑅_𝑡0𝑓(𝑡)=^∞_𝑡=𝑡01𝑠+1^𝑡+1𝑓(𝑡). (4.20) In Definition 4.16, if we set 𝑡−𝑡₀=𝑛∈𝑁₀, and define: 𝑓^{𝑡0}_𝑛=𝑓𝑛+𝑡₀=𝑓(𝑡),𝑓^{𝑡0}_𝑛−1=𝑓𝑛−1+𝑡₀()𝑓=𝑓𝑡−1,…,_{𝑡0}0=𝑓0+𝑡₀=𝑓𝑡₀, (4.21) then, 𝑓(𝑡₀),𝑓(𝑡₀+1),…,𝑓(𝑡),… can be regarded as a sequence 𝑓^{𝑡0}₀,𝑓^{𝑡0}₁,…𝑓^{𝑡0}_𝑛,…. (4.22) Under this definition, 𝑅 transform can be simply rewritten as 𝑅_𝑡0𝑓(𝑡)(𝑠)=^∞_𝑡=𝑡01𝑠+1^𝑡+1𝑓(𝑡)=^∞_𝑛=01𝑠+1^𝑛+𝑡0+1𝑓𝑛+𝑡₀=1𝑠+1^𝑡0∞+1_𝑛=011+𝑠^𝑛𝑓^{𝑡0}_𝑛. (4.23) Set 𝑧=1+𝑠, then 𝑅_𝑡0𝑓(𝑡)(𝑠)=𝑧^{−1−𝑡0∞}_𝑛=0𝑓^{𝑡0}_𝑛𝑧^−𝑛=𝑧^−1−𝑡0𝐹(𝑧), (4.24) where 𝐹(𝑧) is a 𝑍 transform of sequence 𝑓^{𝑡0}_𝑛.

Theorem 4.17. For any 𝛾∈𝑅⧵{…,−2,−1,0}, then(1)𝑅_𝛾−1(𝑡^(𝛾−1))(𝑠)=Γ(𝛾)/𝑠^𝛾,(2)𝑅_𝛾−1(𝑡^(𝛾−1)𝛼^𝑡)(𝑠)=𝛼^𝛾−1Γ(𝛾)/(𝑠+1−𝛼)^𝛾.

Proof. (1) let 𝑡₀=𝛾−1, then 𝑅_𝛾−1𝑡^(𝛾−1)Γ𝛾=𝑧^−𝛾𝐹(𝑧), (4.25) where 𝐹(𝑧) is a 𝑍 transform of sequence 𝑓_𝑛^{𝛾−1}. Since 𝑓_𝑛^{𝛾−1}=𝑓𝑛+𝛾−1=𝑛+𝛾−1^(𝛾−1)Γ𝛾=𝛾𝑛, (4.26) and (see [22, 23]) 𝐹𝛾𝑛=𝑧−1𝑧^−𝛾, (4.27) hence 𝑅_𝛾−1𝑡^(𝛾−1)Γ𝛾=(𝑧−1)^−𝛾=𝑠^−𝛾,(|1+𝑠|<1), (4.28) or 𝑅_𝛾−1𝑡^(𝛾−1)Γ𝛾=1𝑠^𝛾,(|1+𝑠|<1). (4.29)
(2) The proof of (2) follows from the proof of (1).

Definition 4.18. Define convolution of ℎ(𝑡) and 𝑔(𝑡) as follows: ℎ∗𝑔(𝑡)=^𝑡−𝛾_𝑠=𝑎ℎ(𝑡−𝜎(𝑠))𝑔(𝑠). (4.30) If ℎ(𝑡)=𝑡^(𝛾−1)/Γ(𝛾), then ℎ∗𝑔_𝑎(𝑡)=1Γ𝛾^𝑡−𝛾_𝑠=𝑎𝑡−𝜌(𝑠)^(𝛾−1)𝑔(𝑠)=_𝑎Δ^−𝛾𝑔(𝑡). (4.31)

Theorem 4.19. For any 𝛾∈𝑅⧵{…,−2,−1,0}, then 𝑅_𝛾+𝑎ℎ∗𝑔=𝑅_𝛾−1(ℎ)𝑅_𝑎𝑔. (4.32)

Theorem 4.20. Let 𝜇>0,  𝑚−1<𝜇≤𝑚∈𝑁₁, and let 𝑓(𝑡) be defined in 𝑁_𝜇−𝑚={𝜇−𝑚,𝜇−𝑚+1,…}, then 𝑅₀Δ^𝜇𝑓(𝑡)(𝑠)=𝑠^𝜇𝑅_𝜇−𝑚𝑓(𝑡)(𝑠)−^𝑚−1_𝑘=0𝑠^{𝑚−𝑘−1}Δ^{𝑘−𝑚+𝜇}𝑓(𝑡)_𝑡=0. (4.33)

Theorem 4.21. Let 𝜇∈𝑅⧵{…,−2,−1,0}, 𝛾>0, then Δ^−𝛾𝑡^(𝜇)Γ=𝑡𝜇+1Γ^(𝜇+𝛾).𝜇+𝛾+1 (4.34)

Theorem 4.22. Let 𝑓 be a real function, 𝜇,𝛾>0, then for all 𝑡=𝜇+𝛾mod(1), one has Δ^−𝛾Δ^−𝜇𝑓(𝑡)=Δ^{−(𝜇+𝛾)}𝑓(𝑡)=Δ^−𝜇Δ^−𝛾𝑓(𝑡). (4.35)

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 𝜇∈𝑅, 𝛾∈𝑅, then(1)∇^𝛾𝑡^𝜇=𝜇^(𝛾)𝑡^𝜇−𝛾,  Δ^𝛾𝑡^(𝜇)=𝜇^(𝛾)𝑡^{(𝜇−𝛾)},(2)Δ^𝛾𝑡^𝜇=𝜇^(𝛾)(𝑡+𝛾)^𝜇−𝛾,  ∇^𝛾𝑡^(𝜇)=𝜇^(𝛾)(𝑡−𝛾)^{(𝜇−𝛾)}.

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 Δ^𝛾𝑡^𝜇=∇^𝛾𝑡+𝛾^𝜇=𝜇^(𝛾)𝑡+𝛾^𝜇−𝛾,∇^𝛾𝑡^(𝜇)=Δ^𝛾𝑡−𝛾^(𝜇)=𝜇^(𝛾)𝑡−𝛾^{(𝜇−𝛾)}. (5.1)

Example 5.2. Consider Euler type fractional difference equations 𝑡^2𝛼Δ^2𝛼𝑥(𝑡)+𝑎𝑡^𝛼Δ^𝛼𝑥(𝑡)+𝑏𝑥(𝑡)=0,(0<𝛼<1). (5.2) Set 𝑥(𝑡)=𝑡^𝛾, and take it into previous equation, we get 𝑡^2𝛼𝛾^(2𝛼)(𝑡+2𝛼)^𝛾−2𝛼+𝑎𝑡^𝛼𝛾^(𝛼)(𝑡+𝛼)^𝛾−𝛼+𝑏𝑡^𝛾=0. (5.3) By Theorem 4.1 (4), we obtain 𝛾^(2𝛼)𝑡^𝛾+𝑎𝛾^(𝛼)𝑡^𝛾+𝑏𝑡^𝛾=0, (5.4) and get indicator equation 𝛾^(2𝛼)+𝑎𝛾^(𝛼)+𝑏=0. (5.5) Therefore, we can transform Euler type fractional difference equations into its indicator equation.

Example 5.3. Consider initial value problem of homogeneous linear 𝛾 order (0<𝛾≤1) fractional difference equation with constant coefficient ∇^𝛾𝑦(𝑡)+𝑎∇⁰𝑦(𝑡)=0,𝑡∈𝑁₀,∇^𝛾−1(𝑡)_𝑡=−1=𝑎₀. (5.6) Note that ∇^𝛾−1𝑦(𝑡) is defined in 𝑁₋₁={−1,0,1,2,…}, since ₋₁∇^𝛾−1𝑓(𝑡)_𝑡=−1=1Γ1−𝛾^𝑡_𝑠=−1𝑡−𝜌(𝑠)^−𝛾𝑦(𝑠)=1^−𝛾Γ𝑦()().1−𝛾−1=𝑦−1 (5.7) Therefore, initial problem of (5.6) is equivalent to initial problem ∇^𝛾𝑦(𝑡)+𝑎∇⁰𝑦(𝑡)𝑦()=0,𝑡∈𝑁,−1=𝑎₀. (5.8) The solution of initial problem of (5.6) is equivalent to the solution of sum equations 𝑦(𝑡)=(𝑡+1)^𝛾−1Γ𝛾𝑎₀+𝑎^𝑡_𝑠=0𝑡−𝜌(𝑠)^𝛾−1𝑦(𝑠). (5.9)
We use approximation method to solve these sum equations. Set 𝑦₀(𝑡)=(𝑡+1)^𝛾−1Γ𝛾𝑎₀,𝑦_𝑚(𝑡)=𝑦₀(𝑡)+𝑎Γ𝛾^𝑡_𝑠=0𝑡−𝜌(𝑠)^𝛾−1𝑦_𝑚−1(𝑠)=𝑦₀(𝑡)+𝑎∇^−𝛾𝑦_𝑚−1(𝑡),𝑚=1,2,…. (5.10) Applying power law (Theorem 4.22), we get 𝑦₁(𝑡)=𝑦₀(𝑡)+𝑎∇^−𝛾𝑦₀(𝑡)=𝑎₀(𝑡+1)^𝛾−1Γ𝛾()+𝑎𝑡+1^2𝛾−1Γ.2𝛾 (5.11) Applying power law repeatedly, and by recursion, we obtain 𝑦_𝑚(𝑡)=𝑎₀^𝑚_𝑖=0𝑎^𝑖𝑡^{𝑖𝛾+𝛾−1}Γ(𝑖+1)𝛾,𝑚=0,1,2,…. (5.12) Let 𝑚→∞, then 𝑦(𝑡)=𝑎₀^∞_𝑖=0𝑎^𝑖(𝑡+1)^{𝑖𝛾+𝛾−1}Γ(𝑖+1)𝛾=𝑎₀^∞_𝑖=0𝑎^𝑖𝑡.𝑖𝛾+𝛾 (5.13)

Example 5.4. Let 𝛾=1/𝑞,  𝑞∈𝑁, we call ∇^𝛾𝑦(𝑡)−𝑎∇⁰𝑦(𝑡)=0,𝑡∈𝑁₀, (5.14) the fractional difference equation of order (1,𝑞).

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

Definition 5.5. Define function Λ𝑡,𝛾,𝜆=_𝑎∇^−𝛾𝜆^𝑡,𝛾∈𝑅, (5.15) where 𝑡=𝑎mod(1). Sometimes denote it Λ(𝛾,𝜆) or Λ(𝑡,𝛾,𝜆;𝑎).

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)Λ(𝑡,𝛾,𝜆)=(1−1/𝜆)Λ(𝑡,𝛾+1,𝜆)+(𝑡−𝑎+1)^𝛾/Γ(𝛾+1), (2)∇Λ(𝑡,𝛾+1,𝜆)=Λ(𝑡,𝛾,𝜆), (3)∇^𝑝Λ(𝑡,𝛾+𝑡,𝜆)=Λ(𝑡,𝛾,𝜆), where 𝑝=0,1,2,…,(4)∇^𝜇Λ(𝑡,𝛾,𝜆)=Λ(𝑡,𝛾−𝜇,𝜆), where 𝑝−1<𝜇≤𝑝,(5)∇^−𝜇Λ(𝑡,𝛾,𝜆)=Λ(𝑡,𝛾+𝜇,𝜆).

Now we will use the method of undetermined coefficients to solve Example 5.4. By Theorem 5.6, we notice that ∇^𝛾Λ(),∇𝑡,0,𝜆=Λ𝑡,−𝛾,𝜆_𝛾Λ,⋮∇𝑡,−𝛾,𝜆=Λ𝑡,−2𝛾,𝜆_𝛾Λ,∇𝑡,−𝑞−2𝛾,𝜆=Λ𝑡,−𝑞−1𝛾,𝜆_𝛾Λ()=1𝑡,−𝑞−1𝛾,𝜆=Λ𝑡,−1,𝜆1−𝜆Λ(𝑡,0,𝜆). (5.16)

The significance of these applications is that if we apply the operator ∇^𝛾 to Λ(𝑡,0,𝜆),Λ𝑡,−𝛾,𝜆,…,Λ𝑡,−𝑞−1𝛾,𝜆, (5.17) 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 𝑦(𝑡)=𝑏₀Λ(𝑡,0,𝜆)+𝑏₁Λ𝑡,−𝛾,𝜆+…+𝑏_𝑞−2Λ𝑡,−𝑞−2𝛾,𝜆+𝑏_𝑞−1Λ𝑡,−𝑞−1𝛾,𝜆. (5.18) Then ∇^𝛾𝑦(𝑡)=𝑏₀Λ𝑡,−𝛾,𝜆+𝑏₁Λ𝑡,−2𝛾,𝜆+…+𝑏_𝑞−2Λ𝑡,−𝑞−1𝛾,𝜆+𝑏_𝑞−111−𝜆Λ(𝑡,0,𝜆). (5.19) Taking 𝑦(𝑡),  ∇^𝛾𝑦(𝑡) into the left side of (5.14), we obtain ∇^𝛾𝑦(𝑡)−𝑎𝑦(𝑡)=𝑏_𝑞−111−𝜆−𝑎𝑏₀Λ()+𝑏𝑡,0,𝜆₀−𝑎𝑏₁Λ𝑡,−𝛾,𝜆+⋯+𝑏_𝑞−2−𝑎𝑏_𝑞−1Λ𝑡,−𝑞−1𝛾,𝜆. (5.20) In order to make the right side equate zero, set 𝑏_𝑘=𝑐𝛼^−𝑘,𝑘=1,2,…,𝑞−1. (5.21) Then 𝑏_𝑘−1−𝑎𝑏_𝑘=𝑐𝛼^−𝑘+1−𝑎𝛼^−𝑘=𝑐𝛼^−𝑘(𝛼−𝑎). (5.22) If we let 𝛼 be a root of the indicial equation 𝑃(𝑥)=𝑥−𝑎=0, (5.23) or 𝛼=𝑎, then we have 𝑏_𝑘−1−𝑎𝑏_𝑘=𝑐𝑎^−𝑘𝑃(𝑎)=0𝑘=1,2,…,𝑞−1. (5.24) Since we also need 0=𝑏_𝑞−111−𝜆−𝑎𝑏₀1=𝑎𝑐1−𝜆𝑎^−𝑞−1, (5.25) so let us set 11−𝜆=𝑎^𝑞1,𝜆=1−𝛼^𝑞. (5.26) Since 𝑐 is an arbitrary number, set 𝑐=𝑎^𝑞−1, then 𝑏_𝑘=𝑎^{𝑞−1−𝑘}. (5.27) Therefore, we obtain a solution of fractional difference of order (1,𝑞) as 𝑦(𝑡)=^𝑞−1_𝑘=0𝑏_𝑘Λ=𝑡,−𝑘𝛾,𝜆_{𝑞−1𝑘=0}𝑎^{𝑞−1−𝑘}Λ1𝑡,−𝑘𝛾,1−𝑎^𝑞≜𝜆_𝑎(𝑡). (5.28)

The fractional difference equation of order (1,𝑞) in Example 5.4 can be solved by the method of 𝑁₀ transform. Make 𝑁₁ transform to the following equation: ∇^𝛾𝑦(𝑡)−𝑎∇⁰𝑦(𝑡)=0. (5.29) We have 𝑠^𝛾𝑁₀𝑓(𝑡)−(1−𝑠)⁻¹𝑓(0)+𝑎𝑁₁𝑓(𝑡𝑁)=0,₁𝑓(𝑡)=^∞_𝑡=1(1−𝑠)^𝑡−1𝑓(𝑡)=^∞_𝑡=0(1−𝑠)^𝑡−1𝑓(𝑡)−(1−𝑠)⁻¹𝑓(0). (5.30) Taking them into previos equation, we get 𝑠^𝛾𝑁₀𝑓(𝑡)−(1−𝑎)(1−𝑠)⁻¹𝑓(0)−𝑎𝑁₀𝑓(𝑡)=0, (5.31) and we have 𝑁₀𝑓(𝑡=()𝑦(0)1)1−𝑎(1−𝑠)(𝑠^𝛾)=()𝑦(0)∑−𝑎1−𝑎^{𝑞−1𝑘=0}𝑎^{𝑞−1−𝑘}𝑠^𝑘𝛾(1−𝑠)(𝑠^𝛾−𝑎)∑^𝑞−1_𝑘=0𝑎^{𝑞−1−𝑘}𝑠^𝑘𝛾=()𝑦(0)∑1−𝑎^{𝑞−1𝑘=0}𝑎^{𝑞−1−𝑘}𝑠^𝑘𝛾(1−𝑠)(𝑠−𝑎^𝑞). (5.32) In [23], we have the following

Theorem 5.7. The following equality holds:(1)𝑁₀(Λ(𝑡,0,𝜆))=𝑁₀(𝜆^𝑡)=1/(1−𝑠)⋅1/(1−(1−𝑠)𝜆), (2)𝑁₀(Λ(𝑡,−𝑘𝛾,𝜆))=𝑁₀(∇^𝑘𝛾𝜆^𝑡)=1/(1−𝑠)⋅𝑠^𝑘𝛾/(1−(1−𝑠)𝜆)⋅(𝑘=1,2,…,𝑞−1).
Set 𝜆=1/(1−𝑎^𝑞), then 𝑁₀Λ1𝑡,0,1−𝑎^𝑞=1⋅1−𝑠1−𝑎^𝑞𝑠−𝑎^𝑞,𝑁₀Λ1𝑡,−𝑘𝛾,1−𝑎^𝑞=𝑠^𝑘𝛾⋅1−𝑠1−𝑎^𝑞𝑠−𝑎^𝑞.𝑘=1,2,…,𝑞−1. (5.33) By Theorem 5.7 and (5.33), we know that 𝑦(𝑡)=(1−𝑎)𝑦(0)^𝑞−1_𝑘=0𝑎^{𝑞−1−𝑘}Λ1𝑡,−𝑘𝛾,1−𝑎^𝑞 (5.34) is a solution of (5.14). Except a constant, the solution 𝑦(𝑡) is the same as the solution (5.28), where 𝑦(𝑡)=𝜆_𝑎(𝑡), (5.35) 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 ℎ=1_𝑎∇_𝑡^−𝛾𝑓(𝑡)=1Γ𝛾^𝑡_𝑠=𝑎𝑡−𝜌(𝑠)^𝛾−1𝑓(𝑠), (6.1) where 𝑡∈𝑁_𝑎={𝑎,𝑎+1,𝑎+2,…}. If we set 𝑡−𝑎=𝑛∈𝑁₀,𝑠−𝑎=𝑟∈𝑁₀,𝑓_𝑟^{𝑎}=𝑓(𝑟+𝑎)=𝑓(𝑠),𝑓_𝑛^{𝑎}=𝑓(𝑛+𝑎)=𝑓(𝑡), (6.2) then 1Γ𝛾^𝑡_𝑠=𝑎𝑡−𝜌(𝑠)^𝛾−1𝑓(𝑠)=1Γ𝛾^𝑛+𝑎_𝑠=𝑎𝑛+𝑎−𝜌(𝑠)^𝛾−1𝑓(𝑠)=1Γ𝛾^𝑛_𝑟=0(𝑛+𝑎−(𝑟+𝑎+1))^𝛾−1𝑓()=1𝑟+𝑎Γ𝛾^𝑛_𝑟=0(𝑛−𝑟+1)^𝛾−1𝑓_𝑟^{𝑎}=₀∇_𝑛^−𝛾𝑓_𝑛^{𝑎}. (6.3) And it is easy to prove that _𝑎∇^𝜇_𝑡𝑓(𝑡)=₀∇^𝜇_𝑛𝑓_𝑛^{𝑎},𝜇>0. (6.4) Therefore, we have the following.

Theorem 6.1. Let 𝑡∈𝑁_𝑎, and set 𝑡−𝑎=𝑛∈𝑁₀,𝑓_𝑛^{𝑎}=𝑓(𝑛+𝑎)=𝑓(𝑡), then _𝑎∇_𝑡^−𝛾𝑓(𝑡)=₀∇_𝑛^−𝛾𝑓_𝑛^{𝑎};_𝑎∇^𝜇_𝑡𝑓(𝑡)=₀∇^𝜇_𝑛𝑓_𝑛^{𝑎},𝜇,𝛾>0. (6.5)

Example 6.2. (1) Set 𝛾=1/𝑞,  𝑞∈𝑁,𝑛∈𝑁, and solve the fractional difference equation of order (1,𝑞), ∇^𝛾𝑥(𝑛)−𝛼𝑥(𝑛)=0. (6.6)
(2) Let 𝑡∈𝑅, and solve the equation ∇^𝛾𝑥(𝑡)−𝛼𝑥(𝑡)=0. (6.7)
(3) Let ℎ∈𝑅⁺,𝑡∈𝑅, and solve the equation ∇^𝛾_ℎ𝑥(𝑡)−𝛼𝑥(𝑡)=0. (6.8)
(4) If we let ℎ→0, we ask whether the limit solution of (6.8) is equivalent to that of the following fractional differential equation? Consider 𝐷^𝛾𝑥(𝑡)−𝛼𝑥(𝑡)=0,(𝑡∈𝑅). (6.9)

Solution 1. (1) By a result in Chapter 7 of book [23], the solution of (6.6) is 𝑥(𝑛)=𝜆_𝛼(𝑛)=^𝑞−1_𝑘=0𝛼^{𝑞−𝑘−1}Λ_𝑛1−𝑘𝛾,1−𝛼^𝑞𝑛. (6.10)
(2) Set 𝑡−𝑡₀=𝑛∈𝑁₀, and define 𝑥^{𝑡0}_𝑛=𝑥𝑛+𝑡₀=𝑥(𝑡),𝑥^{𝑡0}_𝑛−1=𝑥𝑛−1+𝑡₀()𝑥=𝑥𝑡−1,…,_{𝑡0}0=𝑥0+𝑡₀=𝑥𝑡₀. (6.11) Hence, we can regard the following 𝑥(𝑡₀),𝑥(𝑡₀+1),…,𝑥(𝑡),… as a sequence 𝑥^{𝑡0}₀,𝑥^{𝑡0}₁,…𝑥^{𝑡0}_𝑛,…. (6.12) Under this definition, (6.7) is actually equivalent to the following integer variable difference equation: ∇^𝛾𝑥^{𝑡0}_𝑛−𝛼𝑥^{𝑡0}_𝑛=0. (6.13) By (1), we know that its solution is 𝑥^{𝑡0}_𝑛=^𝑞−1_𝑘=0𝛼^{𝑞−𝑘−1}Λ_𝑛1−𝑘𝛾,1−𝛼^𝑞𝑛=^𝑞−1_𝑘=0𝛼^{𝑞−𝑘−1}Λ1𝑡,−𝑘𝛾,1−𝛼^𝑞𝑡. (6.14) That is 𝑥(𝑡)=^𝑞−1_𝑘=0𝛼^{𝑞−𝑘−1}Λ1𝑡,−𝑘𝛾,1−𝛼^𝑞𝑡≜𝜆_𝛼(𝑡). (6.15)
(3) Set 𝑡=𝑠ℎ,𝑥(𝑡)=𝑥(𝑠ℎ)=𝑦(𝑠), then (6.8) is equivalent to ℎ^−𝛾∇^𝛾𝑦(𝑠)−𝛼𝑦(𝑠)=0. (6.16) By (2), we obtain that the solution of (6.16) is 𝑥(𝑡)=𝑦(𝑠)=𝜆_𝛼ℎ^𝛾(𝑠)=^𝑞−1_𝑘=0𝛼ℎ^{𝛾𝑞−𝑘−1}Λ1𝑠,−𝑘𝛾,1−𝛼ℎ^𝛾𝑞𝑠. (6.17) Since Λ1𝑠,−𝑘𝛾,1−𝛼ℎ^𝛾𝑞𝑠=ℎ^𝑘𝛾Λ_ℎ1𝑡,−𝑘𝛾,1−𝛼ℎ^{𝛾𝑞𝑡/ℎ}, (6.18) hence we have 𝑥(𝑡)=^𝑞−1_𝑘=0𝛼ℎ^{𝛾𝑞−𝑘−1}ℎ^𝑘𝛾Λ_ℎ1𝑡,−𝑘𝛾,1−𝛼ℎ^{𝛾𝑞𝑡/ℎ}. (6.19)
(4) Let ℎ→0, and since 11−𝛼^𝑞ℎ^𝑡/ℎ→𝑒^𝛼𝑞,ℎ^𝑘𝛾Λ_ℎ1𝑡,−𝑘𝛾,1−𝛼^𝑞ℎ^𝑡/ℎ=ℎ^𝑘𝛾∇_ℎ^𝑘𝛾11−𝛼^𝑞ℎ^𝑡/ℎ→𝐷^𝑘𝛾𝑒^𝛼𝑞=𝐸−𝑘𝛾,𝛼^𝑞. (6.20) We then obtain 𝑥(𝑡)=𝑒_𝛼(𝑡)=^𝑞−1_𝑘=0𝛼^{𝑞−𝑘−1}𝐸−𝑘𝛾,𝛼^𝑞, (6.21) and this is exactly the solution of (6.9). (See Chapter 5 in monographer [2]).

Remark 6.3. If we take 𝛾=1/2,  𝑞=2, then the followong occurs.
(1) The solution of (6.19) reduces to 𝑥(𝑡)1=𝛼1−𝛼^2𝑡+∇^1/211−𝛼^2𝑡1=𝛼𝐹𝑡,0,1−𝛼²1+𝐹𝑡,−2,11−𝛼², (6.22) and this result is consistent with the solution (5.28) or (5.34) in Example 5.4 in Section 5.
(2) The solution (6.21) reduces to 𝑥(𝑡)=𝛼ℎ^1/211−𝛼^𝑞ℎ^𝑡/ℎ+ℎ^1/2∇_ℎ^1/211−𝛼^𝑞ℎ^𝑡/ℎ=ℎ^1/2𝛼11−𝛼^𝑞ℎ^𝑡/ℎ+∇_ℎ^1/211−𝛼^𝑞ℎ^𝑡/ℎ. (6.23)
Let ℎ→0, then 𝛼11−𝛼^𝑞ℎ^𝑡/ℎ+∇_ℎ^1/211−𝛼^𝑞ℎ^𝑡/ℎ (6.24) tend to 𝛼𝑒^𝛼𝑞𝑡+𝐷^1/2𝑒^𝛼𝑞𝑡=𝑒_𝛼(𝑡). (6.25) The results perfectly coincide with the monographer [2].

From Theorem 6.1, we see that if we take 𝑡 as 𝑎,𝑎+1,𝑎+2,…, it is only a sequence with step 1, but the initial time is not zero but 𝑎. If we make a translation variable transformation, set 𝑡=𝑛+𝑎,  𝑛∈𝑁₀, then we can change the definition of fractional sum and fractional difference with real variable into the definition of fractional sum and difference with integer variable. But, no doubt, it will be more convenient for us to study fractional sum and difference with integer variable.

7. Conclusion

This work reveals some results in discrete fractional calculus and fractional ℎ-difference equations. This study also provides a reference for researchers in this area. First, this paper gives the definition of the fractional ℎ-difference from the difference of integer order. Then some integral transforms are proposed, that is, 𝑍 transform, 𝑁 transform, and 𝑅 transform. These integral transforms are applied to linear fractional ℎ-difference equations, and approximate solutions are obtained. At last, the study explains the relationship between the fractional difference equations and the fractional differential equations.

Acknowledgment

The authors wish to thank the referee for the very careful reading of the manuscript and fruitful comments and suggestions.