Generalized Mittag-Leffler Function Associated with Weyl Fractional Calculus Operators

Abstract

This paper is devoted to study further properties of generalized Mittag-Leffler function 𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝} associated with Weyl fractional integral and differential operators. A new integral operator ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞} depending on Weyl fractional integral operator and containing 𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}(𝑧) in its kernel is defined and studied, namely, its boundedness. Also, composition of Weyl fractional integral and differential operators with the new operator ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞} is established.

1. Introduction

In 1903, the Swedish mathematician Mittag-Leffler [1] introduced the function 𝐸_𝛼(𝑧) as 𝐸_𝛼(𝑧)=^∞_𝑛=0𝑧^𝑛Γ(),𝛼𝑛+1 (1) where 𝑧∈ℂ and Γ(𝑠) is the gamma function; 𝛼≥0.

During the last century and due to its involvement in the problems of physics, engineering, and applied sciences, many authors defined and studied in their research papers different generalizations of Mittag-Leffler type function, namely, 𝐸_𝛼,𝛽(𝑧) introduced by Wiman [2], 𝐸^𝛾_𝛼,𝛽(𝑧) stated by Prabhakar [3], 𝐸^𝛾,𝑞_𝛼,𝛽(𝑧) defined and studied by Shukla and Prajapati [4], and 𝐸^𝛾,𝛿_𝛼,𝛽(𝑧) investigated by Salim and Faraj [5].

Prabhakar studied some properties of generalized Mittag-Leffler type-function 𝐸^𝛾_𝛼,𝛽(𝑧) and the fractional integral operator ℰ^𝛾_{𝛼,𝛽,𝑤,𝑎+}𝜑(𝑥)=^𝑥_𝑎(𝑥−𝑡)^𝛽−1𝐸^𝛾_𝛼,𝛽𝑤(𝑥−𝑡)^𝛼𝜑(𝑡)𝑑𝑡 (2) containing 𝐸^𝛾_𝛼,𝛽(𝑧) in the kernel and applied the result obtained to prove the existence and uniqueness of the solution of corresponding integral equation of the first kind.

Moreover, Kilbas et al. [6] devoted themselves to further investigation of 𝐸^𝛾_𝛼,𝛽(𝑧), and the integral operator defined in (2). They established integral representation, differentiation and integration properties of 𝐸^𝛾_𝛼,𝛽(𝑧) and formulas of its Riemann-Liouville fractional integral and differential operators. For more results and conclusions, one can refer to the work of Srivastava and Tomovski [7].

Recently, Salim and Faraj [5] introduced a new generalization of Mittag-Leffler-type function as 𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}(𝑧)=^∞_𝑛=0𝛾_𝑞𝑛𝑧^𝑛Γ𝛼𝑛+𝛽(𝛿)_𝑝𝑛, (3) where 𝑧,𝛼,𝛽,𝛾,𝛿∈ℂ;minRe(𝛼),Re𝛽,Re𝛾,Re(𝛿)>0,𝑝,𝑞>0. (4) Equation (3) is just a generalized formula of Mittag-Leffler function; its various properties including differentiation, Laplace, Beta, and Mellin transforms, and generalized hypergeometric series form and its relationship with other type of special functions were investigated and established.

On the other hand Salim and Faraj in their research paper defined and studied an integral operator ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,𝑎+} as ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,𝑎+}𝜑(𝑥)=^𝑥_𝑎(𝑥−𝑡)^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤(𝑥−𝑡)^𝛼𝜑(𝑡)𝑑𝑡 (5) containing 𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}(𝑧) in the kernel. Also, composition of Riemann-Liouville fractional integral and differential operators with the integral operator defined in (5) was established.

This paper is devoted for the study of further properties of the generalized Mittag-Leffler function 𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}(𝑧) defined in (3) with another type of fractional calculus operators called Weyl fractional integral and differential operators written as: 𝐼^𝜆₋𝜑(𝑥)=1Γ(𝜆)^∞_𝑥(𝑡−𝑥)^𝜆−1𝜑(𝑡)𝐷𝑑𝑡,_𝜆−𝜑(𝑥)=(−1)^𝑚𝑑𝑑𝑥^𝑚1Γ(𝑚−𝜆)^∞_𝑥(𝑡−𝑥)^{𝑚−𝜆−1}𝜑(𝑡)𝑑𝑡. (6)(7) The last definition can be written in the form 𝐷^𝜆₋𝜑(𝑥)=(−1)^𝑚𝑑𝑑𝑥^𝑚𝐼₋^𝑚−𝜆𝜑(𝑥). (8) Precisely, the authors investigate the basic properties of Weyl fractional integral and differential operator with generalized Mittag-Leffler function 𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}(𝑧); moreover, a new integral operator depending on Weyl fractional integral operator and containing 𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}(𝑧) in its kernel is established as ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑤)=^∞_𝑥(𝑡−𝑥)^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤(𝑡−𝑥)^𝛼𝜑(𝑡)𝑑𝑡. (9) The condition of boundedness of the integral operator (9) is discussed and stated in the space 𝐿(𝑎,∞) of Lebesgue-measurable functions on (𝑎,∞)𝐿(𝑎,∞)=𝑔(𝑥)∶𝑔₁=^∞_𝑎𝑔(𝑥)𝑑𝑥<∞. (10) Also, composition of Weyl fractional integration and differentiation with the operator defined in (9) is established.

Throughout this paper, we need the following well-known facts and rules.(i) Fubini's theorem (Dirichlet formula) [8] ^𝑏_𝑎𝑑𝑥^𝑥_𝑎𝑓(𝑥,𝑡)𝑑𝑡=^𝑏_𝑎𝑑𝑡^𝑏_𝑡𝑓()𝑑𝑥,𝑡𝑑𝑥,𝑑𝑥^𝑥_𝑎ℎ(𝑥,𝑡)𝑑𝑡=^𝑥_𝑎𝜕ℎ()()𝜕𝑥𝑥,𝑡𝑑𝑡+ℎ𝑥,𝑥. (11)(12)(ii) The Riemann-Liouville fractional integral [8] 𝐼^𝜆_𝑎+𝜑(𝑥)=1Γ(𝜆)^𝑥_𝑎(𝑥−𝑡)^𝜆−1𝜑(𝑡)((𝛼)).𝑑𝑡𝛼∈ℂ,Re>0 (13)(iii) The Riemann-Liouville fractional derivative [8] 𝐷^𝜆_𝑎+𝜑(𝑥)=𝑑𝑑𝑥^𝑛𝐼_𝑎^𝑛−𝜆₊𝜑(𝑥)𝑛=[Re(𝛼)]+1. (14)(iv) Beta transform (Sneddon [9]) 𝐵𝑓(𝑧);𝑎,𝑏=¹₀𝑧^𝑎−1(1−𝑧)^𝑏𝑓(𝑧)𝑑𝑧, (15) where Re(𝑎)>0, Re(𝑏)>0.(v) The Beta function is written as: 𝛽=Γ(𝛼)Γ𝛽𝛼,𝛽Γ𝛼+𝛽. (16)(vi) The difference property of the Gamma function is Γ(𝛼+1)=𝛼Γ(𝛼). (17)

2. Further Properties of Weyl Fractional Integral Related to Mittag-Leffler Function

In this section, we consider composition of Weyl fractional integral and derivative (6) and (7) with generalized Mittag-Leffler function 𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}(𝑧) defined in (3).

Theorem 1. Let 𝛼,𝛽,𝛾,𝛿,𝜆,𝑤∈ℂ; min{Re(𝛼),Re(𝛽),Re(𝛾),Re(𝛿),Re(𝜆)}>0 and 𝑝,𝑞>0, then 𝐼^𝜆₋𝑡^{−𝜆−𝛽}𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝑡^−𝛼(𝑥)=𝑥^−𝛽𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽+𝜆,𝑝}𝑤𝑥^−𝛼. (18)

Proof. 𝐼^𝜆₋𝑡^{−𝜆−𝛽}𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝑡^−𝛼𝑥)=1(Γ(𝜆)^∞_𝑥(𝑡−𝑥)^𝜆−1𝑡^{−𝜆−𝛽}×^∞_𝑛=0𝛾_𝑞𝑛𝑤^𝑛𝑡^−𝛼𝑛(𝛿)_𝑝𝑛Γ=1𝛼𝑛+𝛽𝑑𝑡Γ(𝜆)^∞_𝑛=0𝛾_𝑞𝑛𝑤^𝑛(𝛿)_𝑝𝑛Γ×𝛼𝑛+𝛽_∞𝑥(𝑡−𝑥)^𝜆−1𝑡^{−𝜆−𝛽}𝑡^−𝛼𝑑𝑡. (19) Let 𝑢=(𝑡−𝑥)/𝑡, then 𝐼^𝜆₋𝑡^{−𝜆−𝛽}𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝑡^−𝛼𝑥)=1(Γ(𝜆)^∞_𝑛=0𝛾_𝑞𝑛𝑤^𝑛𝑥^{−𝛽−𝛼𝑛}(𝛿)_𝑝𝑛Γ×𝛼𝑛+𝛽₁₀𝑢^𝜆−1(1−𝑢)^{𝛼𝑛+𝛽−1}=1𝑑𝑢Γ(𝜆)^∞_𝑛=0𝛾_𝑞𝑛𝑤^𝑛𝑥^{−𝛽−𝛼𝑛}(𝛿)_𝑝𝑛Γ𝐵𝛼𝑛+𝛽𝜆,𝛼𝑛+𝛽=𝑥^−𝛽𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽+𝜆,𝑝}𝑤𝑥^−𝛼. (20)

Theorem 2. Let 𝛼,𝛽,𝛾, 𝛿,𝜆, 𝑤∈ℂ; min{Re(𝛼), Re(𝛽), Re(𝛾), Re(𝛿), Re(𝜆)}>0, Re(𝛽)>[Re(𝜆)]+1 and 𝑝,𝑞>0, then 𝐷^𝜆₋𝑡^𝜆−𝛽𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝑡^−𝛼(𝑥)=𝑥^−𝛽𝐸^𝛾,𝛿_{𝛼,𝛽−𝜆,𝑝}𝑤𝑥^−𝛼. (21)

Proof. Making use of (7), we get 𝐷^𝜆₋𝑡^𝜆−𝛽𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝑡^−𝛼𝑥)=()(−1^𝑚𝑑𝑑𝑥^𝑚1Γ()×𝑚−𝜆_∞𝑥(𝑡−𝑥)^{𝑚−𝜆−1}𝑡^{∞𝜆−𝛽}_𝑛=0𝛾_𝑞𝑛𝑤𝑡^−𝛼𝑛(𝛿)_𝑝𝑛Γ=()𝛼𝑛+𝛽𝑑𝑡−1^𝑚𝑑𝑑𝑥^𝑚1Γ()×𝑚−𝜆_∞𝑛=0𝑤^𝑛𝛾_𝑞𝑛(𝛿)_𝑝𝑛Γ×𝛼𝑛+𝛽_∞𝑥(𝑡−𝑥)^{𝑚−𝜆−1}𝑡^𝜆−𝛽𝑡^−𝛼𝑛𝑑𝑡. (22) Let 𝑢=(𝑡−𝑥)/𝑡, then 𝐷^𝜆₋𝑡^𝜆−𝛽𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝑡^−𝛼𝑥)=()(−1^𝑚𝑑𝑑𝑥^𝑚1Γ()×𝑚−𝜆_∞𝑛=0𝛾_𝑞𝑛𝑤^𝑛𝑥^{𝑚−𝛼𝑛−𝛽}(𝛿)_𝑝𝑛Γ𝛼𝑛+𝛽¹₀𝑢^{𝑚−𝜆−1}(1−𝑢)^{𝛼𝑛+𝛽−𝑚−1}=()𝑑𝑢−1^𝑚𝑑𝑑𝑥^𝑚1Γ()×𝑚−𝜆_∞𝑛=0𝛾_𝑞𝑛𝑤^𝑛𝑥^{𝑚−𝛼𝑛−𝛽}(𝛿)_𝑝𝑛Γ𝐵=()𝛼𝑛+𝛽𝑚−𝜆,𝛼𝑛+𝛽−𝑚−1^𝑚∞_𝑛=0𝛾_𝑞𝑛𝑤^𝑛Γ𝛼𝑛+𝛽−𝑚(𝛿)_𝑝𝑛ΓΓ⋅⋯𝑥𝛼𝑛+𝛽𝛼𝑛+𝛽−𝜆𝑚−𝜆𝑛−𝛽𝑚−𝛼𝑛−𝛽−𝑚+1^{𝑚−𝛼𝑛−𝛽−𝑚}=(−1)^𝑚∞_𝑛=0𝛾_𝑞𝑛𝑤^𝑛Γ𝛼𝑛+𝛽−𝑚(𝛿)_𝑝𝑛ΓΓ⋅()𝛼𝑛+𝛽𝛼𝑛+𝛽−𝜆−1^𝑚𝛼𝑛+𝛽−𝑚_𝑚𝑥^{−𝛼𝑛−𝛽}=𝑥^∞−𝛽_𝑛=0𝛾_𝑞𝑛𝑤𝑥^−𝛼𝑛(𝛿)_𝑝𝑛Γ𝛼𝑛+𝛽−𝜆=𝑥^−𝛽𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽−𝜆,𝑝}𝑤𝑥^−𝛼. (23)

3. Weyl Integral Operator with Generalized Mittag-Leffler Function in the Kernel

Consider the Weyl integral operator defined in (9) containing 𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}(𝑧) in the kernel. First of all, we prove that the operator ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞} is bounded on 𝐿(𝑎,∞).

Theorem 3. Let 𝛼,𝛽,𝛾,𝛿,𝜆,𝑤∈ℂ; with min{Re(𝛼), Re(𝛽), Re(𝛾), Re(𝛿), Re(𝜆)}>0 and 𝑝,𝑞>0, then the operator ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞} is bounded on 𝐿(𝑎,∞) and ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑₁≤𝛽𝜑₁, (24) where 𝛽=(𝑏−𝑎)^∞Re(𝛽)_𝑛=0𝛾_𝑞𝑛𝑤(𝑏−𝑎)^𝛼𝑛(𝛿)_𝑝𝑛Γ(𝛼)𝛽.𝛼𝑛+𝛽Re𝑛+Re (25)

Proof. Let 𝐶_𝑛 denote the 𝑛th term of (25), then 𝐶_𝑛+1𝐶_𝑛=𝛾_{𝑞𝑛+𝑞}𝛾_𝑞𝑛Γ𝛼𝑛+𝛽Γ(𝛿)𝛼𝑛+𝛽+𝛼(𝛿)^𝑝𝑛_{𝑝𝑛+𝑝}×Re(𝛼)𝑛+Re𝛽(𝛼)(𝛼)𝛽𝑤()Re𝑛+Re+Re𝑏−𝑎^Re(𝛼)≈𝑤(𝑏−𝑎)^Re(𝛼)𝑞𝑛^𝑞(|𝛼|𝑛)^Re(𝛼)𝑝𝑛^𝑝as𝑛⟶∞. (26) Hence, |𝐶_𝑛+1/𝐶_𝑛|→0 as 𝑛→∞ and 𝑞<𝑝+Re(𝛼) which means that the right-hand side of (25) is convergent and finite under the given condition.
Now, according to (9), (10), and (11), we get ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑₁=^∞_𝑎^∞_𝑥(𝑡−𝑥)^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤(𝑡−𝑥)^𝛼𝜑(𝑡)≤𝑑𝑡𝑑𝑥_{∞𝑎𝑡𝑎}(𝑡−𝑥)^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤(𝑡−𝑥)^𝛼𝜑(𝑡=𝑑𝑥)𝑑𝑡_{∞𝑎0𝑡−𝑎}𝑢^Re(𝛽)−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝑢^𝛼𝜑(𝑡≤𝑑𝑢)𝑑𝑡byletting𝑢=𝑡−𝑥_{∞𝑎0𝑏−𝑎}𝑢^Re(𝛽)−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝑢^𝛼𝑑𝑢𝜑(𝑡)𝑑𝑡. (27) Let ₀^𝑏−𝑎𝑢^Re(𝛽)−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝑢^𝛼𝑑𝑢=𝛽, (28) then 𝛽=^∞_𝑛=0𝛾_𝑞𝑛𝑤^𝑛(𝛿)_𝑝𝑛Γ𝛼𝑛+𝛽^0𝑏−𝑎𝑢^{Re(𝛼)𝑛+Re(𝛽)−1}=()𝑑𝑢𝑏−𝑎^∞Re(𝛽)_𝑛=0𝛾_𝑞𝑛𝑤(𝑏−𝑎)^Re(𝛼)𝑛(𝛿)_𝑝𝑛Γ(𝛼)𝛽.𝛼𝑛+𝛽Re𝑛+Re (29) Hence, ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑₁≤^∞_𝑎𝛽𝜑(𝑡)𝑑𝑡≤𝛽𝜑₁. (30)

We consider now composition of Weyl fractional integration and differentiation 𝐼^𝜆₋,𝐷^𝜆₋ with the operator ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞} defined in (9) contained in the next two theorems.

Remark 4. One can use the result of the next lemma for the proof of the stated theorems (see [5]).

Lemma 5. Let 𝑎∈ℝ₊,𝛼,𝛽,𝛾,𝛿,𝜆∈ℂ; min{Re(𝛼), Re(𝛽), Re(𝛾), Re(𝛿), Re(𝜆)}>0 and 𝑝,𝑞>0, then for 𝑥>𝑎, one has 𝐼^𝜆_𝑎+(𝑡−𝑎)^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤(𝑡−𝑎)^𝛼𝑥)=()(𝑥−𝑎^{𝛽+𝜆−1}𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽+𝜆,𝑝}𝑤(𝑥−𝑎)^𝛼. (31)

Theorem 6. Let 𝛼,𝛽,𝛾,𝛿,𝜆,𝑤∈ℂ; with min{Re(𝛼), Re(𝛽), Re(𝛾), Re(𝛿)}>0 and 𝑝,𝑞>0, then 𝐼^𝜆₋ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑥)=ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽+𝜆,𝑝,𝑤,∞}𝜑(𝑥)=ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝐼^𝜆₋𝜑(𝑥). (32)

Proof. Applying (8) and (9), and by using Dirichlet formula (11) yields 𝐼^𝜆₋ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑥)=1Γ(𝜆)×^∞_𝑥(𝑢−𝑥)^𝜆−1×^∞_𝑢(𝑡−𝑢)^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤(𝑡−𝑢)^𝛼𝜑(𝑡)=𝑑𝑡𝑑𝑢_∞𝑥1Γ(𝜆)×^𝑡_𝑥(𝑢−𝑥)^𝜆−1(𝑡−𝑢)^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤(𝑡−𝑢)^𝛼(𝑡)𝑑𝑢×𝜑𝑑𝑡. (33) Let 𝜏=(𝑡−𝑢), (34) then 𝐼^𝜆₋ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑥)=^∞_𝑥1Γ(𝜆)₀^𝑡−𝑥(𝑡−𝑥−𝜏)^𝜆−1×𝜏^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝜏^𝛼𝑑𝑡𝜑(𝑡)𝑑𝑡. (35) Applying (13) and the result of Lemma 5, we get 𝐼^𝜆₋ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑥)=^∞_𝑥𝐼^𝜆₀𝜏^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝜏^𝛼)𝜑(𝑡)=(𝑡−𝑥𝑑𝑡_∞𝑥(𝑡−𝑥)^{𝛽+𝜆−1}𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽+𝜆,𝑝}𝑤(𝑡−𝑥)^𝛼𝜑(𝑡)=ℰ𝑑𝑡_{𝛾,𝛿,𝑞𝛼,𝛽+𝜆,𝑝,𝑤,∞}𝜑(𝑥). (36) On the other hand, ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝐼^𝜆₋𝜑(𝑥)=^∞_𝑥(𝑡−𝑥)^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤(𝑡−𝑥)^𝛼1Γ(𝜆)×^∞_𝑡(𝑢−𝑡)^𝜆−1𝜑(𝑢)=𝑑𝑢𝑑𝑡_∞𝑥1Γ(𝜆)^𝑢_𝑥(𝑡−𝑥)^𝛽−1(𝑢−𝑡)^𝜆−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤(𝑡−𝑥)^𝛼(𝑢)𝑑𝑡×𝜑𝑑𝑢. (37) Let 𝜏=𝑡−𝑥, we get ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝐼^𝜆₋𝜑(𝑥)=^∞_𝑥1Γ(𝜆)₀^𝑢−𝑥(𝑢−𝑥−𝜏)^𝜆−1×𝜏^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝜏^𝛼𝑑𝜏𝜑(𝑢)𝑑𝑢. (38) Returning to (13) and Lemma 5, we have ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝐼^𝜆₋𝜑(𝑥)=^∞_𝑥𝐼^𝜆₀𝜏^𝛽−1𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝}𝑤𝜏^𝛼)𝜑(𝑢)=(𝑢−𝑥𝑑𝑢_∞𝑥(𝑢−𝑥)^{𝛽+𝜆−1}𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽+𝜆,𝑝}𝑤(𝑢−𝑥)^𝛼𝜑(𝑢)=ℰ𝑑𝑢_{𝛾,𝛿,𝑞𝛼,𝛽+𝜆,𝑝,𝑤,∞}𝜑(𝑥) (39) which ends the proof.

A similar result concerning the Weyl fractional differentiation is stated in the following theorem.

Theorem 7. If the condition of Theorem 6 is satisfied, then 𝐷^𝜆₋ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑥)=ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽−𝜆,𝑝,𝑤,∞}𝜑(𝑥). (40)

Proof. Making use of (8), we get 𝐷^𝜆₋ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑥)=(−1)^𝑛𝑑𝑑𝑥^𝑛𝐼₋^𝑛−𝜆ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑥). (41) and applying Theorem 6 yields 𝐷^𝜆₋ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑥)=(−1)^𝑛𝑑𝑑𝑥^𝑛×^∞_𝑥(𝑡−𝑥)^{𝛽+𝑛−𝜆−1}𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽−𝑛−𝜆,𝑝}𝑤(𝑡−𝑥)^𝛼𝜑(𝑡)𝑑𝑡. (42) By using Dirichlet formula (12), we get 𝐷^𝜆₋ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑥)=(−1)^𝑛𝑑𝑑𝑥^𝑛−1×^∞_𝑥𝜕()𝜕𝑥𝑡−𝑥^{𝛽+𝑛−𝜆−1}𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽+𝑛−𝜆,𝑝}𝑤(𝑡−𝑥)^𝛼𝜑(𝑡)𝑑𝑡+lim_𝑡→𝑥+(𝑡−𝑥)^{𝛽+𝑛−𝜆−1}𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽+𝑛−𝜆,𝑝}𝑤(𝑡−𝑥)^𝛼𝜑(𝑡)=(−1)^𝑛𝑑𝑑𝑥^𝑛−1×^∞_𝑥^∞_𝑛=0𝛾_𝑞𝑛𝑤^𝑛𝛼𝑛+𝛽+𝑛−𝜆−1(𝛿)_𝑝𝑛Γ×()𝛼𝑛+𝛽−𝑛−𝜆𝑡−𝑥^{𝛼𝑛+𝛽+𝑛−𝜆−2}(𝑡)=()⋅𝜑𝑑𝑡−1^𝑛𝑑𝑑𝑥^𝑛−1∞_𝑥(−1)(𝑡−𝑥)^{𝛽+𝑛−𝜆−2}×𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽+𝑛−𝜆−1,𝑝}𝑤(𝑡−𝑥)^𝛼⋅𝜑(𝑡)𝑑𝑡. (43) Repeating this process 𝑛−1 times, we get 𝐷^𝜆₋ℰ^{𝛾,𝛿,𝑞}_{𝛼,𝛽,𝑝,𝑤,∞}𝜑(𝑥)=(−1)^𝑛(−1)^𝑛∞_𝑥(𝑡−𝑥)^{𝛽−𝜆−1}𝐸^{𝛾,𝛿,𝑞}_{𝛼,𝛽−𝜆,𝑝}𝑤(𝑡−𝑥)^𝛼𝜑(𝑡)=ℰ𝑑𝑡_{𝛾,𝛿,𝑞𝛼,𝛽−𝜆,𝑝,𝑤,∞}𝜑(𝑥). (44)