On Solution Representation of Generalized Abel Integral Equation

Abstract

An explicit representation for solution of the generalized Abel integral equation ∑^𝑚_𝑘=0𝜆_𝑘𝐷^−𝛼𝑘_0𝑥𝑢(𝑥)=𝑓(𝑥), where 𝐷^−𝛼𝑘_0𝑥 is the Riemann-Liouville fractional integral, in terms of the Wright function, is constructed.

1. Introduction

Consider the equation ^𝑚_𝑘=0𝜆_𝑘𝐷^−𝛼𝑘_0𝑥𝑢(𝑥)=𝑓(𝑥), (1) where 𝛼_𝑘≥0,  𝜆_𝑘∈ℝ,  𝑥∈(0,𝑙), and  𝐷^𝛽_0𝑥 is the Riemann-Liouville operator of fractional integrodifferentiation of order 𝛽, which is defined as follows: 𝐷^𝛽_0𝑥𝜑(𝑥)=1Γ−𝛽^𝑥₀𝜑(𝑡)(𝑥−𝑡)^−𝛽−1𝐷𝑑𝑡for𝛽<0,_00𝑥𝜑(𝑥)(𝑥),𝐷=𝜑_𝛽0𝑥𝜑(𝑥)=𝑑^𝑛𝑑𝑥^𝑛𝐷^𝛽−𝑛_0𝑥𝜑(𝑥)for𝛽∈(𝑛−1,𝑛],𝑛∈ℕ. (2)

For suitable function 𝑓(𝑥), (1) with 𝑚=0 and 𝛼₀=𝛼>0,  𝜆₀=1; that is, the Abel integral equation of the first kind (see [1]) 𝐷^−𝛼_0𝑥𝑢(𝑥)=𝑓(𝑥) (3) may be inverted by the formula 𝑢(𝑥)=𝐷^𝛼_0𝑥𝑓(𝑥) (4) (see [2]). For 𝑚=1,  𝛼₀=0,  𝛼₁=𝛼>0,  𝜆₀=1, and  𝜆₁=𝜆≠0, (1) has the following form 𝑢(𝑥)+𝜆𝐷^−𝛼_0𝑥𝑢(𝑥)=𝑓(𝑥), (5) which is the Abel equation of the second kind, and its solution may be given by the Hille-Tamarkin formula 𝑢(𝑥)=𝑓(𝑥)−𝜆^𝑥₀(𝑥−𝑡)^𝛼−1𝐸_𝛼,𝛼−𝜆(𝑥−𝑡)^𝛼𝑓(𝑡)𝑑𝑡 (6) (see [3]), which can be rewritten as 𝑢(𝑥)=𝑑𝑑𝑥^𝑥₀𝐸_𝛼,1−𝜆(𝑥−𝑡)^𝛼𝑓(𝑡)𝑑𝑡. (7) Here, 𝐸_𝛼,𝛽(𝑧)=^∞_𝑛=0𝑧^𝑛Γ𝛼𝑛+𝛽 (8) is the Mittag-Leffler function.

Equations of type (1) with rational parameters 𝛼_𝑘 were considered in [4, 5]. Generalized Abel integral equations of the second kind were investigated by operational method in [6]. For a more complete survey on results relating Abel type integral equations and fractional order integral and differential equations and their applications, we refer to [6–8].

In this paper, we obtain a representation for solution of (1). The results cover both cases, the solution of equation of the first kind (𝛼₀>0) and that of the second kind (𝛼₀=0). The solution is constructed in terms of the Wright function. It should be noted that (1) can be reduced to the generalized Abel integral equation of the second kind, and the method developed in [6] can be applied for (1). This provides an alternative approach to the equation under study.

2. Preliminaries

Now, we recall several properties of the operators of fractional integration and differentiation and special functions which are required for what follows.

The following three relations are valid: 𝐷^𝜀_0𝑥𝐷^𝛿_0𝑥𝜑(𝑥)=𝐷^𝜀+𝛿_0𝑥𝜑(𝑥) (9) for 𝜀∈ℝ and 𝛿≤0 or 𝜀∈ℕ and 𝛿∈ℝ; 𝐷^𝜀_0𝑥𝐷^𝛿_0𝑥𝜑(𝑥)=𝐷^𝜀+𝛿_0𝑥𝜑(𝑥)−^𝑛_𝑗=1𝑥^{−𝜀−𝑗}Γ𝐷1−𝜀−𝑗^{𝛿−𝑗0𝑥}𝜑(𝑥)_𝑥=0 (10) for 𝜀∈ℝ and 𝛿∈(𝑛−1,𝑛], 𝑛∈ℕ; 𝐷^𝜀_0𝑥𝑥^𝛿−1Γ(𝛿)=𝑥^{𝛿−𝜀−1}Γ(𝛿−𝜀) (11) for 𝜀∈ℝ and 𝛿>0 or 𝜀∈ℕ and 𝛿∈ℝ (e.g., see [2] and [9, Section 2.1]).

By (𝜑∗𝜓)(𝑥), we denote the Laplace convolution of functions 𝜑(𝑥) and 𝜓(𝑥): 𝜑∗𝜓(𝑥)=^𝑥₀𝜑(𝑥−𝑡)𝜓(𝑡)𝑑𝑡. (12)

Operator 𝐷^𝛽_0𝑥 may be written in the form of convolution 𝐷^𝛽_0𝑥𝜑(𝑥)𝑥=𝜑∗^−𝛽−1Γ𝐷−𝛽for𝛽<0,_𝛽0𝑥𝜑(𝑥)=𝑑^𝑛𝑑𝑥^𝑛𝑥𝜑∗^{𝑛−𝛽−1}Γ(]𝑛−𝛽for𝛽∈𝑛−1,𝑛,𝑛∈ℕ. (13)

Using (11) and (13) and taking into account the associativity and commutativity of convolution, we get 1∏^𝑚_𝑖=1Γ𝜀_𝑖𝑥^𝜀1−1∗⋯∗𝑥^𝜀𝑚−1=𝑥^𝜀−1Γ(𝜀), (14) where 𝜀=∑^𝑚_𝑗=1𝜀_𝑗,  𝜀_𝑗>0, and 𝐷^−𝛿1_0𝑥𝜑₁∗⋯∗𝐷^−𝛿𝑚_0𝑥𝜑_𝑚=𝐷^−𝛿_0𝑥𝜑₁∗⋯∗𝜑_𝑚, (15) with 𝛿=∑^𝑚_𝑗=1𝛿_𝑗 and 𝛿_𝑗≥0. The relation 𝐷^−𝛿_0𝑥𝜑∗𝜓(𝑥)=𝐷^−𝛿_0𝑥𝜑∗𝜓(𝑥)=𝜑∗𝐷^−𝛿_0𝑥𝜓(𝑥) (16) is a particular case of (15).

Recall that the Wright function 𝜙𝛽,𝛿;𝑧=^∞_𝑛=0𝑧^𝑛𝑛!Γ𝛽𝑛+𝛿 (17) satisfies the relations 𝑑𝜙𝑑𝑧𝛽,𝛿;𝑧=𝜙𝛽,𝛿+𝛽;𝑧 (18) (see [10]) and 𝐷^𝜀_0𝑥𝑥^𝛿−1𝜙𝛽,𝛿;𝑧𝑥^𝛽=𝑥^{𝛿−𝜀−1}𝜙𝛽,𝛿−𝜀;𝑧𝑥^𝛽for𝜀∈ℝ,𝛿>0. (19) For 𝛽>0, relation (19) may be proved by termwise applying formula (11) to the representation of the Wright function as a series (17). Combining (18) and (19), we obtain 𝜕𝑥𝜕𝑧^𝛿−1𝜙𝛽,𝛿;𝜆𝑧𝑥^𝛽=𝜆𝐷^−𝛽_0𝑥𝑥^𝛿−1𝜙𝛽,𝛿;𝜆𝑧𝑥^𝛽(𝜆∈ℝ). (20)

3. Auxiliary Results

Here, we investigate properties of functions in terms of which a solution of integral equation (1) will be constructed.

Consider the function 𝑆^𝜇_𝑚𝑥;𝑧₁,…,𝑧_𝑚;𝛽₁,…,𝛽_𝑚=ℎ₁∗ℎ₂∗⋯∗ℎ_𝑚(𝑥) (21) (see [11]), where ℎ_𝑘=ℎ_𝑘(𝑥)=𝑥^𝜇𝑘−1𝜙𝛽_𝑘,𝜇_𝑘;𝑧_𝑘𝑥^𝛽𝑘. (22) In what follows, the parameters and arguments of function (21) satisfy 𝑥>0,𝑧_𝑘∈ℝ,𝛽_𝑘𝜇>0,_𝑘>0,𝜇=^𝑚_𝑘=1𝜇_𝑘. (23)

Remark 1. The function 𝑆^𝜇_𝑚(𝑥;𝑧₁,…,𝑧_𝑚;𝛽₁,…,𝛽_𝑚) is independent of the distribution of parameters 𝜇_𝑘 but depends only on their sum 𝜇. Indeed, let 𝑆 and 𝑆 be the values of functions 𝑆^𝜇_𝑚 corresponding to the sets {𝜇₁,…,𝜇_𝑚} and {𝜇₁,…,𝜇_𝑚}, with same sets of 𝑥, 𝑧_𝑘, and 𝛽_𝑘, and 𝜇=^𝑚_𝑘=1𝜇_𝑘=^𝑚_𝑘=1𝜇_𝑘. (24) Let 𝜈=min𝜇₁,…,𝜇_𝑚,𝜇₁,…,𝜇_𝑚,𝑔_𝑘=𝑥^𝜈−1𝜙𝛽_𝑘,𝜈;𝑧_𝑘𝑥^𝛽𝑘. (25) Then, using (15) and (19), we obtain 𝑆=𝐷^𝜈−𝜇1_0𝑥𝑔₁∗⋯∗𝐷^{𝜈−𝜇𝑚}_0𝑥𝑔_𝑚=𝐷^{𝑚𝜈−𝜇}_0𝑥𝑔₁∗⋯∗𝑔_𝑚=𝐷^𝜈−₁₀𝜇_𝑥𝑔₁∗⋯∗𝐷^𝜈−_𝑚0𝜇_𝑥𝑔_𝑚=𝑆. (26)

Lemma 2. If 𝑍=max𝑧₁,…,𝑧₂,𝜌=min𝛽₁,…,𝛽_𝑚,𝜏=max𝛽₁,…,𝛽_𝑚, (27) then 𝑆^𝜇_𝑚𝑥;𝑧₁,…,𝑧_𝑚;𝛽₁,…,𝛽_𝑚≤𝐶𝑥^𝜇−1exp𝜎𝑍^1/(1+𝜌)𝑥^{𝜏/(1+𝜏)} (28) with positive constants 𝐶 and 𝜎 independent of 𝑥 and 𝑧_𝑘.

Proof. Using the asymptotic formula of the Wright function (see [10]), we have 𝜙𝛽_𝑘,𝜇_𝑘;𝑧_𝑘≤𝐶_𝑘𝑥^𝜇−1exp𝜎_𝑘𝑧_𝑘^{1/(1+𝛽𝑘)}𝑥^{𝛽𝑘/(1+𝛽𝑘)}_𝑘, (29) where 𝐶_𝑘 and 𝜎_𝑘 are positive constants independent of 𝑥 and 𝑧_𝑘. If we recall the definition (21), we get 𝑆^𝜇_𝑚𝑥;𝑧₁,…,𝑧_𝑚;𝛽₁,…,𝛽_𝑘≤𝐶𝑥^𝜇1−1∗⋯∗𝑥^𝜇𝑚−1exp𝜎𝑍^1/(1+𝜌)𝑥^{𝜏/(1+𝜏)}. (30) Using (14), we obtain (28).

Lemma 3. If 𝜇>𝜈, then 𝐷^𝜈_0𝑥𝑆^𝜇_𝑚𝑥;𝑧₁,…,𝑧_𝑚;𝛽₁,…,𝛽_𝑚=𝑆_𝑚^𝜇−𝜈𝑥;𝑧₁,…,𝑧_𝑚;𝛽₁,…,𝛽_𝑚. (31)

Proof. If 𝜈<0, then combining (15), (19), and (21), we get 𝐷^𝜈_0𝑥𝑆^𝜇_𝑚𝑥;𝑧₁,…,𝑧_𝑚;𝛽₁,…,𝛽_𝑚=𝐷^𝜈_0𝑥ℎ₁∗ℎ₂∗⋯∗ℎ_𝑚(𝑥)=𝑆_𝑚^𝜇−𝜈𝑥;𝑧₁,…,𝑧_𝑚;𝛽₁,…,𝛽_𝑚. (32)
Now, let 𝜈∈(𝑞−1,𝑞],  𝑞∈ℕ. As follows from Remark 1, in (21), we can take a parameter 𝜇₁ such that 𝜇₁>𝜈. Therefore, in view of the equality 𝑑(𝑥)=𝜑𝑑𝑥𝜙∗𝜓^′∗𝜓−𝜑(0)𝜓(𝑥), (33) formula (19), and the relation lim_𝑥→0𝐷^𝜈−𝑗_0𝑥ℎ₁(𝑥)=0,1≤𝑗≤𝑞, (34) we see that 𝐷^𝜈_0𝑥𝑆^𝜇_𝑚𝑥;𝑧₁,…,𝑧_𝑚;𝛽₁,…,𝛽_𝑚=𝑑^𝑞𝑑𝑥^𝑞𝐷^𝜈−𝑞_0𝑥ℎ₁∗ℎ₂∗⋯∗ℎ_𝑚𝑥)=𝐷_𝜈0𝑥(ℎ₁∗ℎ₂∗⋯∗ℎ_𝑚(𝑥)=𝑆_𝑚^𝜇−𝜈𝑥;𝑧₁,…,𝑧_𝑚;𝛽₁,…,𝛽_𝑚. (35) This completes the proof.

Now, we set 𝐺^𝜇_𝑚𝑥;𝛾₁,…,𝛾_𝑚;𝛽₁,…,𝛽_𝑚=^∞₀𝑒^−𝑡𝑆^𝜇_𝑚𝑥;𝛾₁𝑡,…,𝛾_𝑚𝑡;𝛽₁,…,𝛽_𝑚𝑑𝑡. (36) The convergence of the integral in (36) follows from (28).

Lemma 4. The function 𝐺^𝜇_𝑚=𝐺^𝜇_𝑚(𝑥;𝛾₁,…,𝛾_𝑚;𝛽₁,…,𝛽_𝑚) satisfies the relations 𝐺^𝜇_𝑚=𝑂𝑥^𝜇−1𝐷𝑎𝑠𝑥→0,_𝜈0𝑥𝐺^𝜇_𝑚=𝐺_𝑚^𝜇−𝜈𝑓𝑜𝑟𝜇>𝜈. (37)(38)

Proof. From (28) and (36), we obtain 𝐺^𝜇_𝑚≤𝐶𝑥^𝜇−1∞₀exp−𝑡+𝜎𝛾𝑡^1/(1+𝜌)𝑥^{𝜏/(1+𝜏)}𝑑𝑡≤𝐶𝑥^𝜇−1expsup_𝑡>0−𝑡2+𝜎𝛾𝑡^1/(1+𝜌)𝑥^{𝜏/(1+𝜏)}×^∞₀𝑒^−𝑡/2𝑑𝑡, (39) where 𝜌=min𝛽₁,…,𝛽_𝑚,𝜏=max𝛽₁,…,𝛽_𝑚,𝛾=max𝛾₁,…,𝛾_𝑚, (40) and 𝐶 and 𝜎 are positive constants independent of 𝑥, 𝛾, and 𝑡. By the relation sup_𝑡>0−𝑡+𝑎𝑡^𝜀=𝑎^1/(1−𝜀)𝜀^{𝜀/(1−𝜀)}(1−𝜀), (41) which holds for 𝑎>0 and 𝜀∈(0,1), (39) yields (37).
Relation (38) follows from (28), (31), and definition (36).

Lemma 5. Let 𝜆₀≠0, 0≤𝛼₀<𝛼_𝑘 for 𝑘=1,2,…,𝑚, and 𝜇>𝛼₀. Denote that 𝑤_𝜇(𝑥)=1𝜆₀𝐺^𝜇−𝛼0_𝑚𝜆𝑥;−¹𝜆₀𝜆,…,−^𝑚𝜆₀;𝛼₁−𝛼₀,…,𝛼_𝑚−𝛼₀. (42) Then, the function 𝑤_𝜇(𝑥) is a solution of the integral equation ^𝑚_𝑘=0𝜆_𝑘𝐷^−𝛼𝑘_0𝑥𝑤_𝜇(𝑥)=𝑥^𝜇−1Γ𝜇. (43)

Proof. Let 𝑔_𝑘=𝑥^𝜇𝑘−1𝜙𝛼_𝑘−𝛼₀,𝜇_𝑘𝜆;−^𝑘𝑡𝜆₀𝑥^{𝛼𝑘−𝛼0},𝜇=^𝑚_𝑘=1𝜇_𝑘. (44) Using (20), (21), and (31) and bearing in mind Remark 1, we get 𝜕𝑆𝜕𝑡^𝜇𝑚𝜆𝑥;−¹𝑡𝜆₀𝜆,…,−^𝑚𝑡𝜆₀;𝛼₁−𝛼₀,…,𝛼_𝑚−𝛼₀=^𝑚_𝑘=1𝑔₁∗⋯∗𝑔_𝑘−1∗𝜕𝑔_𝑘𝜕𝑡∗𝑔_𝑘+1∗⋯∗𝑔_𝑚1=−𝜆₀^𝑚_𝑘=1𝜆_𝑘𝑔₁∗⋯∗𝑔_𝑘−1∗𝐷^{𝛼0−𝛼𝑘}_0𝑥𝑔_𝑘∗𝑔_𝑘+1∗⋯∗𝑔_𝑚1=−𝜆₀^𝑚_𝑘=1𝜆_𝑘𝐷^−𝛼𝑘_0𝑥𝑆^𝜇−𝛼0_𝑚𝜆𝑥;−¹𝑡𝜆₀𝜆,…,−^𝑚𝑡𝜆₀;𝛼₁−𝛼₀,…,𝛼_𝑚−𝛼₀. (45) Hence, in view of (36) and (42), ^𝑚_𝑘=1𝜆_𝑘𝐷^−𝛼𝑘_0𝑥𝑤_𝜇(𝑥)=−^∞₀𝑒^−𝑡𝜕𝑆𝜕𝑡^𝜇𝑚𝜆𝑥;−¹𝑡𝜆₀𝜆,…,−^𝑚𝑡𝜆₀;𝛼₁−𝛼₀,…,𝛼_𝑚−𝛼₀𝑑𝑡=𝑆^𝜇_𝑚𝜆𝑥;−¹𝑡𝜆₀𝜆,…,−^𝑚𝑡𝜆₀;𝛼₁−𝛼₀,…,𝛼_𝑚−𝛼_0𝑡=0−𝜆₀𝐷^−𝛼0_0𝑥𝑤_𝜇(𝑥). (46) Taking account of the equality 𝑆^𝜇_𝑚𝜆𝑥;−¹𝑡𝜆₀𝜆,…,−^𝑚𝑡𝜆₀;𝛼₁−𝛼₀,…,𝛼_𝑚−𝛼_0𝑡=0=𝑥^𝜇1−1Γ𝜇₁𝑥∗⋯∗^𝜇𝑚−1Γ𝜇_𝑚 (47) and (14), the relation (46) yields (43).

4. Main Result

Theorem 6. Let 𝜆₀≠0, 0≤𝛼₀<𝛼_𝑘 for 𝑘=1,2,…,𝑚, 𝑓(𝑥)∈𝐿(0,𝑙), and there exists a function 𝑔(𝑥)∈𝐿(0,𝑙) such that 𝑓(𝑥)=𝐷^−𝛼0_0𝑥𝑔(𝑥). Then, (1) has a unique integrable solution 𝑢(𝑥), which for each 𝜇>𝛼₀ can be represented as follows: 𝑢(𝑥)=𝐷^𝜇_0𝑥𝑓∗𝑤_𝜇(𝑥), (48) where 𝑤_𝜇=𝑤_𝜇(𝑥) is defined by (42).

Proof. Let 𝑢(𝑥) be a solution of (1). Taking into account (16) and (43), we obtain 𝑓∗𝑤_𝜇=^𝑚_𝑘=0𝜆_𝑘𝐷^−𝛼𝑘_0𝑥𝑢∗𝑤_𝜇=𝑢∗^𝑚_𝑘=0𝜆_𝑘𝐷^−𝛼𝑘_0𝑥𝑤_𝜇𝑥=𝑢∗^𝜇−1Γ𝜇, (49) or, in view of (13), 𝑓∗𝑤_𝜇(𝑥)=𝐷^−𝜇_0𝑥𝑢(𝑥). (50) Applying the operator 𝐷^𝜇_0𝑥 to both sides of this equality, bearing in mind (9), we obtain (48).
Now, we prove that a function 𝑢(𝑥) that is defined by (48) is a solution of (1). Let 𝜇∈(𝑞−1,𝑞],  𝑞∈ℕ. Consider the function 𝑣(𝑥)=𝐷^𝜇−𝑞_0𝑥𝑓∗𝑤_𝜇(𝑥). (51) Bearing in mind (16), (38), and (42), we get 𝑣(𝑥)=𝐷^𝜇−𝑞_0𝑥𝐷^−𝛼0_0𝑥𝑔∗𝑤_𝜇(𝑥)=𝑔∗𝑤_𝑞+𝛼0(𝑥). (52) Using (33), (37), and (38), we obtain 𝐷^𝜇−𝑗_0𝑥𝑓∗𝑤_𝜇(𝑥)=𝑑^𝑞−𝑗𝑑𝑥^𝑞−𝑗𝑣(𝑥)=𝑔∗𝑤_𝑗+𝛼0(𝑥)for𝑗=1,2,…,𝑞. (53) Taking this and relation (37) into account, we may conclude that the function 𝐷^𝜇−𝑗_0𝑥(𝑓∗𝑤_𝜇)(𝑥) is absolutely continuous and lim_𝑥→0𝐷^𝜇−𝑗_0𝑥𝑓∗𝑤_𝜇(𝑥)=𝑑^𝑞−𝑗𝑑𝑥^𝑞−𝑗𝑣(𝑥)_𝑥=0=0 (54) for 𝑗=1,2,…,𝑞, and 𝐷^𝜇_0𝑥𝑓∗𝑤_𝜇(𝑥)=𝑑^𝑞𝑑𝑥^𝑞𝑣(𝑥)∈𝐿(0,𝑙). (55) Using (10) and (54), we obtain 𝐷^−𝛼𝑘_0𝑥𝐷^𝜇_0𝑥𝑓∗𝑤_𝜇(𝑥)=𝐷^𝜇_0𝑥𝐷^−𝛼𝑘_0𝑥𝑓∗𝑤_𝜇(𝑥)=𝐷^𝜇_0𝑥𝑓∗𝐷^−𝛼𝑘_0𝑥𝑤_𝜇(𝑥)for𝑘=0,1,…,𝑚. (56) Then, it follows that ^𝑚_𝑘=0𝜆_𝑘𝐷^−𝛼𝑘_0𝑥𝐷^𝜇_0𝑥𝑓∗𝑤_𝜇(𝑥)=𝐷^𝜇_0𝑥𝑓∗^𝑚_𝑘=0𝜆_𝑘𝐷^−𝛼𝑘_0𝑥𝑤_𝜇(𝑥)=𝐷^𝜇_0𝑥𝑥𝑓∗^𝜇−1Γ𝜇=𝐷^𝜇_0𝑥𝐷^−𝜇_0𝑥𝑓(𝑥)=𝑓(𝑥). (57) This completes the proof.

Remark 7. Let 𝛼₀=𝛼, 𝜆₀=1, and 𝜆_𝑘=0 for 𝑘≥1. In this case, (1) will appear like (3), and we get 𝑤_𝜇(𝑥)=𝐺_𝑚^𝜇−𝛼𝑥;0,…,0;𝛼₁−𝛼,…,𝛼_𝑚=𝑥−𝛼_{𝜇−𝛼−1}Γ.𝜇−𝛼 (58) In addition, it follows from the equalities 𝐺^𝜇₁𝑥;𝜆;𝛽=^∞₀𝑒^−𝑡𝑥^𝜇−1𝜙𝛽,𝜇;𝜆𝑡𝑥^𝛽=𝑑𝑡_∞𝑘=0𝜆^𝑘𝑥^{𝛽𝑘+𝜇−1}𝑘!Γ𝛽𝑘+𝜇^∞₀𝑒^−𝑡𝑡^𝑘𝑑𝑡=𝑥^𝜇−1𝐸_𝛽,𝜇𝜆𝑥^𝛽 (59) that if 𝑚=1,  𝛼₀=0,  𝛼₁=𝛼>0,  𝜆₀=1, and 𝜆₁=𝜆≠0, then 𝑤_𝜇(𝑥)=𝐺^𝜇₁(𝑥;−𝜆;𝛼)=𝑥^𝜇−1𝐸_𝛼,𝜇−𝜆𝑥^𝛼. (60) Thus, formula (48) coincides with (4) and (7) for the cases of (3) and (5) (i.e., for 𝑚=0 and 𝑚=1), respectively.