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

and Applied Analysis 3 Proof. Using (13) and applying (8) to the Euler (beta) transform of (12), we get


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][3][4][5][6][7] and, for a very recent work, see also [8]).
The above-mentioned detailed and systematic investigation was indeed motivated largely by a demonstrated potential for applications of the more generalized Gauss hypergeometric function  (,;)  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 2  1 , which will be introduced in Sections 2 and 3, respectively.We also consider some interesting special cases of our main results.

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  (,;)  (, ; ; ) defined by (8).
where the beta transform of () is defined as (see [12]) Further, it is assumed that the involved Euler (beta) transforms  (,;)  exist.
Abstract and Applied Analysis 3 Proof.Using (13) and applying (8) to the Euler (beta) transform of ( 12), we get By changing the order of integration and summation and using beta integral, we obtain which, upon using (8), yields our desired result (12).
where the Laplace transform of () is defined as (see [12]) and assume that both sides of (16) exist.
Proof.Using (17) and applying (8), we get By changing the order of integration and summation and using Laplace transform, we get which, upon using (8), yields our desired result (16).
Proof.Applying (8) and setting  = ] in the left-hand side of (20), we get By changing the order of integration and summation, we obtain Next, we can use the following integral formula involving the Whittaker function: Then, after a little simplification, (22) becomes the following form: 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  = 1 immediately reduce to the corresponding results due to Agarwal [11].

The operator 𝐼
,], 0, (⋅) contains both the Riemann-Liouville   0, (⋅) and the Erdélyi-Kober  , 0, (⋅) fractional integral operators by means of the following relationships: It is noted that the operator (26) unifies the Weyl type and the Erdélyi-Kober fractional integral operators as follows: We also use the following image formulas which are easy consequences of the operators ( 25) and ( 26) (see [24,26]): The Saigo fractional integrations of generalized Gauss hypergeometric type functions (8) are given by the following results.
Theorem 4. Let  > 0, R() ≥ 0, and , ], , ,  ∈ C be parameters such that Then, the following fractional integral formula holds: 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 Now, making use of the result (31), we obtain This, in view of ( 8), gives the desired result (34).
Theorem 5. Let  > 0, R() ≥ 0, and , ], , ,  ∈ C be parameters satisfying the following inequalities: Then, the following fractional integral formula holds: 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 ] = 0 in Theorems 4 and 5 and employing the relations ( 28) and (30) yield certain interesting results asserted by the following corollaries.

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  =  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  ()   (, ; ; ) asserted by the following corollaries.
Corollary 10.Let  > 0, R(p) ≥ 0, and , ], , ,  ∈ C be parameters such that Then, one has  6) yield some known fractional integral formulas due to Agarwal [11].Also setting  = 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  = 1 and  = 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 2  1 .
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  (,;)  (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.