Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions

Abstract

A remarkably large number of integral transforms and fractional integral formulas involving various special functions have been investigated by many authors. Very recently, Agarwal gave some integral transforms and fractional integral formulas involving the 𝐹_𝑝^(𝛼,𝛽) (⋅). In this sequel, using the same technique, we establish certain integral transforms and fractional integral formulas for the generalized Gauss hypergeometric functions 𝐹_𝑝^{(𝛼,𝛽,𝑚)} (⋅). Some interesting special cases of our main results are also considered.

1. Introduction and Preliminaries

The theory of special functions has been one of the most rapidly growing research subjects in mathematical analysis. A lot of special functions of mathematical physics and engineering, such as Jacobi and Laguerre polynomials, can be expressed in terms of the generalized hypergeometric functions or confluent hypergeometric functions (see, e.g., [1, pages 66–69]). Therefore, the corresponding extensions of several other familiar special functions are expected to be useful and need to be investigated (see, e.g., [2–7] and, for a very recent work, see also [8]).

Very recently, Parmar [9] introduced and investigated some fundamental properties and characteristics of the more generalized beta type function 𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑥,𝑦) defined by (see [9, page 37, Equation (19)]) 𝐵_𝑝^{(𝛼,𝛽;𝑚)}𝑥,𝑦∶=¹₀𝑡^𝑥−1(1−𝑡)₁^𝑦−1𝐹₁𝛼;𝛽;−𝑝𝑡^𝑚(1−𝑡)^𝑚ℜ𝑝ℜ(𝑥)𝑦(𝛼)𝛽ℜ(𝑚)).𝑑𝑡>0;min,ℜ,ℜ,ℜ>0,>0 (1) It is easy to see that the special case of (1) when 𝑚=1 reduces to the well-known generalized beta type function defined by (see, e.g., [7, page 4602, Equation (4)]; see also [6, page 32, Chapter 4]) 𝐵_𝑝^(𝛼,𝛽)𝑥,𝑦=𝐵_𝑝^{(𝛼,𝛽;1)}𝑥,𝑦=¹₀𝑡^𝑥−1(1−𝑡)₁^𝑦−1𝐹₁𝛼;𝛽;−𝑝𝑡()ℜ𝑝ℜ(𝑥)𝑦(𝛼)𝛽.1−𝑡𝑑𝑡>0;min,ℜ,ℜ,ℜ>0 (2) For 𝛼=𝛽, (2) reduces to the extended beta type function, due to Chaudhry et al. [4, page 591, Equation (1.7)] (see also [3, page 20, Equation (1.7)]), defined by𝐵_𝑝𝑥,𝑦=𝐵_𝑝^(𝛼,𝛼)𝑥,𝑦=¹₀𝑡^𝑥−1(1−𝑡)^𝑦−1exp−𝑝𝑡()ℜ𝑝.1−𝑡𝑑𝑡>0 (3) The classical Euler’s beta function 𝐵(𝑥,𝑦) is defined by (see, e.g., [1, pages 7–11]) 𝐵𝑥,𝑦∶=¹₀𝑡^𝑥−1(1−𝑡)^𝑦−1𝑑𝑡ℜ(𝑥)>0,ℜ𝑦>0. (4) It is easy to see the following relation: 𝐵𝑥,𝑦∶=𝐵₀𝑥,𝑦=𝐵₀^(𝛼,𝛽)𝑥,𝑦=𝐵₀^{(𝛼,𝛽;1)}𝑥,𝑦. (5)

By making use of 𝐵_𝑝(𝑥,𝑦), Chaudhry et al. [4, page 591, Equations (2.1) and (2.2)] extended the Gauss’s hypergeometric function as follows: 𝐹_𝑝(𝑎,𝑏;𝑐;𝑧)∶=^∞_𝑛=0(𝑎)_𝑛𝐵_𝑝(𝑏+𝑛,𝑐−𝑏)𝐵()𝑧𝑏,𝑐−𝑏^𝑛𝑧|),𝑛!(|<1 (6) where ℜ(𝑐)>ℜ(𝑏)>0 and ℜ(𝑝)≥0 and (𝑎)_𝑛 denotes the Pochhammer symbol given in (11).

Similarly, by appealing to 𝐵_𝑝^(𝛼,𝛼)(𝑥,𝑦), Özergin et al. introduced and investigated a further extension of the following potentially useful generalized Gauss hypergeometric functions defined as follows (see, e.g., [7, page 4606, Section3]; see also [6, page 39, Chapter 4]): 𝐹_𝑝^(𝛼,𝛽)(𝑎,𝑏;𝑐;𝑧)=^∞_𝑛=0(𝑎)_𝑛𝐵_𝑝^(𝛼,𝛽)(𝑏+𝑛,𝑐−𝑏)𝐵()𝑧𝑏,𝑐−𝑏^𝑛𝑧|),𝑛!(|<1 (7) where min{ℜ(𝛼),ℜ(𝛽)}>0, ℜ(𝑐)>ℜ(𝑏)>0, and ℜ(𝑝)≥0.

By using the more generalized beta function (1), Parmar [9] introduced and investigated a family of the following potentially useful generalized Gauss hypergeometric functions defined as follows (see [9, page 44]): 𝐹_𝑝^{(𝛼,𝛽;𝑚)}(𝑎,𝑏;𝑐;𝑧)=^∞_𝑛=0(𝑎)_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()𝑧𝑏,𝑐−𝑏^𝑛𝑧|),𝑛!(|<1 (8) where min{ℜ(𝛼),ℜ(𝛽),ℜ(𝑚)}>0;ℜ(𝑐)>ℜ(𝑏)>0 and ℜ(𝑝)≥0.

It is obvious to see that 𝐹_𝑝^{(𝛼,𝛽;1)}(𝑎,𝑏;𝑐;𝑧)=𝐹_𝑝^(𝛼,𝛽)(),𝐹𝑎,𝑏;𝑐;𝑧_{𝑝(𝛼,𝛼;1)}(𝑎,𝑏;𝑐;𝑧)=𝐹_𝑝(),𝐹𝑎,𝑏;𝑐;𝑧_{0(𝛼,𝛼;1)}(𝑎,𝑏;𝑐;𝑧)=₂𝐹₁(𝑎,𝑏;𝑐;𝑧), (9) where the ₂𝐹₁(⋅) is a special case of the well-known generalized hypergeometric series _𝑝𝐹_𝑞(⋅) defined by (see, e.g., [1, Section1.5]; see also [10]) _𝑝𝐹_𝑞𝛼₁,…,𝛼_𝑝;𝑧𝛽₁,…,𝛽_𝑞;=^∞_𝑛=0𝛼_1𝑛⋯𝛼_𝑝𝑛𝛽_1𝑛⋯𝛽_𝑞𝑛𝑧^𝑛=𝑛!_𝑝𝐹_𝑞𝛼₁,…,𝛼_𝑝;𝛽₁,…,𝛽_𝑞;𝑧, (10) where (𝜆)_𝑛 is the Pochhammer symbol defined (for 𝜆∈ℂ) by (see [1, page 2 and pages 4–6]) (𝜆)_𝑛1()𝜆()({=Γ()∶=𝑛=0𝜆+1,…,𝜆+𝑛−1)(𝑛∈ℕ∶=1,2,3,…})𝜆+𝑛Γ(𝜆)𝜆∈ℂ⧵ℤ⁻₀. (11) Here, Γ is the familiar Gamma function and ℂ and ℤ⁻₀ denote sets of complex numbers and nonpositive integers, respectively.

The above-mentioned detailed and systematic investigation was indeed motivated largely by a demonstrated potential for applications of the more generalized Gauss hypergeometric function 𝐹_𝑝^{(𝛼,𝛽;𝑚)} and their special cases to many diverse areas of mathematical, physical, engineering, and statistical sciences (see, for details, [9] and the references cited therein). Very recently, Agarwal [11] gave some interesting integral transform and fractional integral formulas involving (7). In this sequel, using the same technique, we propose to derive some integral transforms and image formulas for the generalized Gauss hypergeometric function (8) by applying certain integral transforms (like beta transform, Laplace transform, and Whittaker transforms) and general pair of fractional integral operators involving Gauss hypergeometric function ₂𝐹₁, which will be introduced in Sections 2 and 3, respectively. We also consider some interesting special cases of our main results.

2. Integral Transform and the Generalized Gauss Hypergeometric Functions

In this section, we will prove three theorems, which exhibit the connections between the Euler, Laplace, and Whittaker integral transforms and the generalized Gauss hypergeometric type functions 𝐹_𝑝^{(𝛼,𝛽;𝑚)}(𝑎,𝑏;𝑐;𝑧) defined by (8).

Theorem 1. Suppose that ℜ(𝑝)≥0, min{ℜ(𝛼),ℜ(𝛽),ℜ(𝑚)}>0, ℜ(𝑐)>ℜ(𝑏)>0, and 𝑙∈ℂare parameters. Then, the following beta transform formula holds: 𝐵𝐹_𝑝^{(𝛼,𝛽;𝑚)}()𝑙+𝑚,𝑏;𝑐;𝑦𝑧∶𝑙,𝑚=𝐵𝑙,𝑚^∞_𝑛=0(𝑙)_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()𝑦𝑏,𝑐−𝑏^𝑛()𝐹𝑛!=𝐵𝑙,𝑚_𝑝^{(𝛼,𝛽;𝑚)}𝑙,𝑏;𝑐;𝑦𝑦<1, (12) where the beta transform of 𝑓(𝑧) is defined as (see [12]) 𝐵𝑓(𝑧)∶𝑎,𝑏=¹₀𝑧^𝑎−1(1−𝑧)^𝑏−1𝑓(𝑧)𝑑𝑧. (13) Further, it is assumed that the involved Euler (beta) transforms 𝐹_𝑝^{(𝛼,𝛽;𝑚)} exist.

Proof. Using (13) and applying (8) to the Euler (beta) transform of (12), we get ¹₀𝑧^𝑙−1(1−𝑧)^𝑚−1𝐹_𝑝^{(𝛼,𝛽;𝑚)}=𝑙+𝑚,𝑏;𝑐;𝑦𝑧𝑑𝑧₁₀𝑧^𝑙−1(1−𝑧)^∞𝑚−1_𝑛=0(𝑙+𝑚)_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵(𝑏,𝑐−𝑏)𝑦𝑧^𝑛𝑛!𝑑𝑧. (14) By changing the order of integration and summation and using beta integral, we obtain ¹₀𝑧^𝑙−1(1−𝑧)^𝑚−1𝐹_𝑝^{(𝛼,𝛽;𝑚)}=𝑙+𝑚,𝑏;𝑐;𝑦𝑧𝑑𝑧_∞𝑛=0(𝑙+𝑚)_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()Γ()Γ(𝑚)𝑏,𝑐−𝑏𝑙+𝑛Γ()𝑦𝑙+𝑚+𝑛^𝑛=Γ(𝑙)Γ(𝑚)𝑛!Γ(𝑙+𝑚)^∞_𝑛=0(𝑙)_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()𝑦𝑏,𝑐−𝑏^𝑛,𝑛! (15) which, upon using (8), yields our desired result (12).

Theorem 2. Suppose that 𝑦≥0, ℜ(𝑠)>0, min{ℜ(𝛼),ℜ(𝛽),ℜ(𝑚)}>0, ℜ(𝑐)>ℜ(𝑏)>0, ℜ(𝑝)≥0, and |𝑦/𝑠|<1. Then, the following Laplace transform formula holds: 𝐿𝑧^𝑙−1𝐹_𝑝^{(𝛼,𝛽;𝑚)}=Γ(𝑙)𝑎,𝑏;𝑐;𝑦𝑧𝑠^𝑙1𝐹_𝑝^{(𝛼,𝛽;𝑚)}𝑦𝑎,𝑙,𝑏;𝑐;𝑠, (16) where the Laplace transform of 𝑓(𝑧) is defined as (see [12]) 𝐿𝑓(𝑧)=^∞₀𝑒^−𝑠𝑧𝑓(𝑧)𝑑𝑧, (17) and assume that both sides of (16) exist.

Proof. Using (17) and applying (8), we get ^∞₀𝑧^𝑙−1𝑒^−𝑠𝑧𝐹_𝑝^{(𝛼,𝛽;𝑚)}=𝑎,𝑏;𝑐;𝑦𝑧𝑑𝑧_∞0𝑧^𝑙−1𝑒^{∞−𝑠𝑧}_𝑛=0(𝑎)_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵(𝑏,𝑐−𝑏)𝑦𝑧^𝑛𝑛!𝑑𝑧. (18) By changing the order of integration and summation and using Laplace transform, we get ^∞₀𝑧^𝑙−1𝑒^−𝑠𝑧𝐹_𝑝^{(𝛼,𝛽;𝑚)}=𝑎,𝑏;𝑐;𝑦𝑧𝑑𝑧_∞𝑛=0(𝑎)_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()Γ()𝑏,𝑐−𝑏𝑙+𝑛𝑠^𝑙+𝑛𝑦^𝑛,𝑛! (19) which, upon using (8), yields our desired result (16).

Theorem 3. Suppose that 𝑤≥0, ℜ(𝑝)≥0, 𝜌 and 𝛿∈ℂ. Then, the following Whittaker transform formula holds: ^∞₀𝑡^𝜌−1𝑒^{−𝛿𝑡/2}𝑊_𝜆,𝜇(𝛿𝑡)𝐹_𝑝^{(𝛼,𝛽;𝑚)}(𝑎,𝑏;𝑐;𝑤𝑡)𝑑𝑡=𝛿^−𝜌Γ(1/2)+𝜇+𝜌Γ(1/2)−𝜇+𝜌Γ×1−𝜆+𝜌₂𝐹^{(𝛼,𝛽;𝑚)}_𝑝,11𝑎,21+𝜇+𝜌,2𝑤−𝜇+𝜌,𝑏;𝑐,1−𝜆+𝜌;𝛿, (20) provided that the integral involving Whittaker transform 𝐹_𝑝^{(𝛼,𝛽;𝑚)} converges.

Proof. Applying (8) and setting 𝛿𝑡=𝜈 in the left-hand side of (20), we get ^∞₀𝜈𝛿^𝜌−1𝑒^−𝜈/2𝑊_𝜆,𝜇(𝜈)×^∞_𝑛=0(𝑎)_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()()𝑏,𝑐−𝑏𝑤𝜈^𝑛𝛿^𝑛𝛿𝑛!𝑑𝜈. (21) By changing the order of integration and summation, we obtain 𝛿^∞−𝜌_𝑛=0(𝑎)_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()𝑤𝑏,𝑐−𝑏𝛿^𝑛𝑛×𝑛!_∞0𝜈^{𝜌+𝑛−1}𝑒^−𝜈/2𝑊_𝜆,𝜇(𝜈)𝑑𝜈. (22) Next, we can use the following integral formula involving the Whittaker function: ^∞₀𝑡^𝜈−1𝑒^−𝑡/2𝑊_𝜆,𝜇(𝑡)=Γ()Γ()𝑑𝑡1/2+𝜇+𝜈1/2−𝜇+𝜈Γ))ℜ1((1/2−𝜆+𝜈𝜈±𝜇>−2. (23) Then, after a little simplification, (22) becomes the following form: ^∞₀𝑡^𝜌−1𝑒^{−𝛿𝑡/2}𝑊_𝜆,𝜇(𝛿𝑡)𝐹_𝑝^{(𝛼,𝛽;𝑚)}(𝑎,𝑏;𝑐;𝑤𝑡)𝑑𝑡=𝛿^∞−𝜌_𝑛=0(𝑎)_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()𝑤𝑏,𝑐−𝑏𝛿^𝑛𝑛×Γ)Γ)𝑛!(1/2+𝜇+𝜌+𝑛(1/2−𝜇+𝜌+𝑛Γ,1−𝜆+𝜌+𝑛 (24) which, upon using (8), yields our desired result (20).

It may be noted in passing that the special cases of Theorems 1 to 3 when 𝑚=1 immediately reduce to the corresponding results due to Agarwal [11].

3. Fractional Calculus of the Generalized Gauss Hypergeometric Functions

Recently, fractional integral operators involving the various special functions have been investigated by several authors (see, e.g., [11, 13–26]; see also [27, 28]). Here, in this section, we will establish some fractional integral formulas for the generalized Gauss hypergeometric type functions 𝐹_𝑝^{(𝛼,𝛽;𝑚)}(𝑎,𝑏;𝑐;𝑧). To do this, we need to recall the following pair of Saigo hypergeometric fractional integral operators.

For 𝑥>0, 𝜇,𝜈,𝜂∈ℂ and ℜ(𝛼)>0, we have 𝐼^{𝜇,𝜈,𝜂}_0,𝑥𝑓(𝑡)(𝑥)=𝑥^{−𝜇−𝜈}Γ𝜇^𝑥₀(𝑥−𝑡)^𝜇−1×₂𝐹₁𝑡𝜇+𝜈,−𝜂;𝜇;1−𝑥𝑓(𝑡)𝑑𝑡,𝐽^{𝜇,𝜈,𝜂}_𝑥,∞𝑓(𝑡)(𝑥)=1Γ𝜇^∞_𝑥(𝑡−𝑥)^𝜇−1𝑡^{−𝜇−𝜈}×₂𝐹₁𝑥𝜇+𝜈,−𝜂;𝜇;1−𝑡𝑓(𝑡)𝑑𝑡, (25)(26) where the₂𝐹₁(⋅), a special case of the generalized hypergeometric function (10), is the Gauss hypergeometric function.

The operator 𝐼^{𝜇,𝜈,𝜂}_0,𝑥(⋅) contains both the Riemann-Liouville 𝑅^𝜇_0,𝑥(⋅) and the Erdélyi-Kober 𝐸^𝜇,𝜂_0,𝑥(⋅) fractional integral operators by means of the following relationships: 𝑅^𝜇_0,𝑥𝑓(𝑡)(𝑥)=𝐼^{𝜇,−𝜇,𝜂}_0,𝑥𝑓(𝑡𝑥)=1)(Γ𝜇^𝑥₀(𝑥−𝑡)^𝜇−1𝑓(𝑡)𝑑𝑡,𝐸^𝜇,𝜂_0,𝑥𝑓(𝑡)(𝑥)=𝐼^𝜇,0,𝜂_0,𝑥𝑓(𝑡)(𝑥)=𝑥^{−𝜇−𝜂}Γ𝜇^𝑥₀(𝑥−𝑡)^𝜇−1𝑡^𝜂𝑓(𝑡)𝑑𝑡. (27)(28) It is noted that the operator (26) unifies the Weyl type and the Erdélyi-Kober fractional integral operators as follows: 𝑊^𝜇_𝑥,∞𝑓(𝑡)(𝑥)=𝐽^{𝜇,−𝜇,𝜂}_𝑥,∞𝑓(𝑡𝑥)=1)(Γ𝜇^∞_𝑥(𝑡−𝑥)^𝜇−1𝑓(𝑡)𝑑𝑡,𝐾^𝜇,𝜂_𝑥,∞𝑓(𝑡)(𝑥)=𝐽^𝜇,0,𝜂_𝑥,∞𝑓(𝑡)(𝑥)=𝑥^𝜂Γ𝜇^∞_𝑥(𝑡−𝑥)^𝜇−1𝑡^{−𝜇−𝜂}𝑓(𝑡)𝑑𝑡. (29)(30)

We also use the following image formulas which are easy consequences of the operators (25) and (26) (see [24, 26]): 𝐼^{𝜇,𝜈,𝜂}_0,𝑥𝑡^𝜆−1(𝑥)=Γ(𝜆)Γ𝜆−𝜈+𝜂Γ()Γ𝑥𝜆−𝜈𝜆+𝜇+𝜂^{𝜆−𝜈−1}𝜆>0,𝜆−𝜈+𝜂>0,𝐽^{𝜇,𝜈,𝜂}_𝑥,∞𝑡^𝜆−1(𝑥)=Γ(𝜈−𝜆+1)Γ𝜂−𝜆+1Γ()Γ𝑥1−𝜆𝜈+𝜇−𝜆+𝜂+1^{𝜆−𝜈−1}𝛽−𝜆+1>0,𝜂−𝜆+1>0. (31)(32)

The Saigo fractional integrations of generalized Gauss hypergeometric type functions (8) are given by the following results.

Theorem 4. Let 𝑥>0, ℜ(𝑝)≥0, and 𝜇,𝜈,𝜂,𝜌,𝑒∈ℂ be parameters such that ℜ𝜇>0,ℜ𝜌>0,ℜ𝜌>max0,ℜ𝜈−𝜂. (33) Then, the following fractional integral formula holds: 𝐼^{𝜇,𝜈,𝜂}_0,𝑥𝑡^𝜌−1𝐹_𝑝^{(𝛼,𝛽;𝑚)}(𝑎,𝑏;𝑐;𝑒𝑡)(𝑥)=𝑥^{𝜌−𝜈−1}Γ𝜌Γ𝜌−𝜈+𝜂ΓΓ×𝜌+𝜇+𝜂𝜌−𝜈₂𝐹^{(𝛼,𝛽;𝑚)}_𝑝,2.𝑎,𝑏,𝜌,𝜌−𝜈+𝜂;𝑒𝑥𝑐,𝜌−𝜈,𝜌+𝜇+𝜂; (34)

Proof. For convenience, we denote the left-hand side of the result (34) by ℐ. Using (8), and then changing the order of integration and summation, which is valid under the conditions of Theorem 4, we find ℐ=𝐼^{𝜇,𝜈,𝜂}_0,𝑡𝑡^∞𝜌−1_𝑛=0𝑎_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()()𝑏,𝑐−𝑏𝑒𝑡^𝑛𝑥)=𝑛!_∞𝑛=0(𝑎_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()𝑒𝑏,𝑐−𝑏^𝑛𝐼𝑛!^{𝜇,𝜈,𝜂0,𝑡}𝑡^{𝜌+𝑛−1}(𝑥). (35) Now, making use of the result (31), we obtain ℐ=𝑥^{∞𝜌−𝜈−1}_𝑛=0𝑎_𝑛𝐵_𝑝^{(𝛼,𝛽;𝑚)}(𝑏+𝑛,𝑐−𝑏)𝐵()×ΓΓ𝑏,𝑐−𝑏𝜌+𝑛𝜌−𝜈+𝜂+𝑛ΓΓ()𝜌−𝜈+𝑛𝜌+𝜇+𝜂+𝑛𝑒𝑥^𝑛.𝑛! (36) This, in view of (8), gives the desired result (34).

Theorem 5. Let 𝑥>0, ℜ(𝑝)≥0, and 𝜇,𝜈,𝜂,𝜌,𝑒∈ℂ be parameters satisfying the following inequalities: ℜ𝜇>0,ℜ𝜌>0,ℜ𝜌<1+minℜ𝜂,ℜ(𝜈). (37) Then, the following fractional integral formula holds: 𝐽^{𝜇,𝜈,𝜂}_𝑥,∞𝑡^𝜌−1𝐹_𝑝^{(𝛼,𝛽;𝑚)}𝑒𝑎,𝑏;𝑐;𝑡(𝑥)=𝑥^{𝜌−𝜈−1}Γ1−𝜌+𝜈Γ1−𝜌+𝜂ΓΓ×1−𝜌1−𝜌−𝜂+𝜈+𝜇₂𝐹^{(𝛼,𝛽;𝑚)}_𝑝,2𝑒𝑎,𝑏,1−𝜌+𝜈,1−𝜌+𝜂;𝑥.𝑐,1−𝜌,1−𝜌+𝜇+𝜈−𝜂; (38)

Proof. As in the proof of Theorem 4, taking the operator (26) and the result (32) into account, one can easily prove (38). Therefore, we omit the details of the proof.

Setting 𝜈=0 in Theorems 4 and 5 and employing the relations (28) and (30) yield certain interesting results asserted by the following corollaries.

Corollary 6. Let 𝑥>0, ℜ(𝑝)≥0, and 𝜇,𝜂,𝜌,𝑒∈ℂ be parameters such that ℜ(𝜇)>0,ℜ(𝜌)>0, and ℜ(𝜌)>ℜ(−𝜂). Then, the right-side Erdélyi-Kober fractional integrals of the generalized Gauss hypergeometric type functions are given by 𝐸^𝜇,𝜂_0,𝑥𝑡^𝜌−1𝐹_𝑝^{(𝛼,𝛽;𝑚)}(𝑎,𝑏;𝑐;𝑒𝑡)(𝑥)=𝑥^𝜌−1Γ𝜌−𝜂Γ×𝜌+𝜇+𝜂₁𝐹^{(𝛼,𝛽;𝑚)}_𝑝,1.𝑎,𝑏,𝜌−𝜂;𝑒𝑥𝑐,𝜌+𝜇+𝜂; (39)

Corollary 7. Let 𝑥>0, ℜ(𝑝)≥0, and 𝜇,𝜂,𝜌,𝑒∈ℂ be parameters satisfying the following inequalities ℜ(𝜇)>0, ℜ(𝜌)>0, and ℜ(𝜌)<1+ℜ(𝜂). Then, one has 𝐾^𝜇,𝜂_𝑥,∞𝑡^𝜌−1𝐹_𝑝^{(𝛼,𝛽;𝑚)}𝑒𝑎,𝑏;𝑐;𝑡(𝑥)=𝑥^𝜌−1Γ1−𝜌+𝜂Γ×1−𝜌−𝜂+𝜇₁𝐹^{(𝛼,𝛽;𝑚)}_𝑝,1𝑒𝑎,𝑏,1−𝜌+𝜂;𝑥.𝑐,1−𝜌+𝜇−𝜂; (40)

Further, if we replace 𝜈 with −𝜇 in Theorems 4 and 5 and make use of the relations (27) and (29), we obtain other Riemann-Liouville and Weyl fractional integrals of the generalized Gauss hypergeometric type function 𝐹_𝑝^{(𝛼,𝛽;𝑚)} given by the following corollaries.

Corollary 8. Let 𝑥>0, ℜ(𝑝)≥0, and 𝜇,𝜌∈ℂ be parameters such that ℜ(𝜇)>0 and ℜ(𝜌)>0. Then, one gets 𝑅^𝜇_0,𝑥𝑡^𝜌−1𝐹_𝑝^{(𝛼,𝛽;𝑚)}(𝑎,𝑏;𝑐;𝑒𝑡)(𝑥)=𝑥^{𝜌+𝜇−1}Γ𝜌Γ𝜌+𝜇¹𝐹^{(𝛼,𝛽;𝑚)}_𝑝,1.𝑎,𝑏,𝜌;𝑒𝑥𝑐,𝜌+𝜇; (41)

Corollary 9. Let 𝑥>0, ℜ(𝑝)≥0, and 𝜇,𝜌∈ℂ be parameters satisfying the following inequalities ℜ(𝜇)>0 and ℜ(𝜌)>0. Then, one obtains 𝑊^𝜇_𝑥,∞𝑡^𝜌−1𝐹_𝑝^{(𝛼,𝛽;𝑚)}𝑒𝑎,𝑏;𝑐;𝑡(𝑥)=𝑥^{𝜌+𝜇−1}Γ1−𝜌−𝜇Γ1−𝜌¹𝐹^{(𝛼,𝛽;𝑚)}_𝑝,1𝑒𝑎,𝑏,1−𝜌−𝜇;𝑥.𝑐,1−𝜌; (42)

4. Concluding Remarks

We can also present a large number of special cases of our main fractional integral formulas in Theorems 4 and 5. Here, we illustrate two more formulas. Setting 𝛼=𝛽 in (34) and (38), respectively, and using the known formula due to Lee et al. (see, [5, page 197, Equation (6.1)]), we obtain certain interesting (presumably) new fractional integral formulas involving the extended hypergeometric function 𝐹_𝑝^(𝑚)(𝑎,𝑏;𝑐;𝑧) asserted by the following corollaries.

Corollary 10. Let 𝑥>0, ℜ(𝑝)≥0, and 𝜇,𝜈,𝜂,𝜌,𝑒∈ℂ be parameters such that ℜ𝜇>0,ℜ𝜌>0,ℜ𝜌>max0,ℜ𝜈−𝜂. (43) Then, one has 𝐼^{𝜇,𝜈,𝜂}_0,𝑥𝑡^𝜌−1𝐹^𝑚_𝑝(𝑎,𝑏;𝑐;𝑒𝑡)(𝑥)=𝑥^{𝜌−𝜈−1}Γ𝜌Γ𝜌−𝜈+𝜂ΓΓ×𝜌+𝜇+𝜂𝜌−𝜈₂𝐹^𝑚_𝑝,2.𝑎,𝑏,𝜌,𝜌−𝜈+𝜂;𝑒𝑥𝑐,𝜌−𝜈,𝜌+𝜇+𝜂; (44)

Corollary 11. Let 𝑥>0, ℜ(𝑝)≥0, and 𝜇,𝜈,𝜂,𝑒,𝜌∈ℂ be parameters satisfying the following inequalities: ℜ𝜇>0,ℜ𝜌>0,ℜ𝜌<1+minℜ𝜂,ℜ(𝜈). (45) Then, one gets 𝐽^{𝜇,𝜈,𝜂}_𝑥,∞𝑡^𝜌−1𝐹_𝑝𝑒𝑎,𝑏;𝑐;𝑡(𝑥)=𝑥^{𝜌−𝛽−1}Γ1−𝜌+𝜈Γ1−𝜌+𝜂ΓΓ×1−𝜌1−𝜌−𝜂+𝜈+𝜇₂𝐹_𝑝,2𝑒𝑎,𝑏,1−𝜌+𝜈,1−𝜌+𝜂;𝑥.𝑐,1−𝜌,1−𝜌+𝜇+𝜈−𝜂; (46)

Setting 𝑚=1 and 𝛼=𝛽 in Theorems 4 and 5 and making use of the relation (6) yield some known fractional integral formulas due to Agarwal [11]. Also setting 𝑚=1 and making use of the relation (7) give the known integral transforms and fractional integral formulas due to Agarwal [11]. Further, if we set 𝑚=1 and 𝑝=0 in Theorems 1 to 5 or make use of the result (8), we obtain various integral transforms and fractional integral formulas for the Gauss hypergeometric function ₂𝐹₁.

The generalized Gauss hypergeometric type functions defined by (8) possess the advantage that most of the known and widely investigated special functions are expressible in terms of the generalized Gauss hypergeometric functions 𝐹_𝑝^{(𝛼,𝛽;𝑚)} (see [9]). We may also emphasize that results derived in this paper are of general character and can specialize to give further interesting and potentially useful formulas involving integral transform and fractional calculus. Finally, it is expected that the results presented here with potential special cases can find some applications in probability theory and to the solutions of fractional differential and integral equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their deep gratitude for the reviewers’ helpful comments. This research was, in part, supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology of the Republic of Korea (Grant no. 2010-0011005). This work was supported by Dongguk University Research Fund.