Some fractional integral formulas for the Mittag-Leffler type function with four parameters

Abstract In this paper we present some results from the theory of fractional integration operators (of Marichev- Saigo-Maeda type) involving the Mittag-Leffler type function with four parameters ζ , γ, Eμ, ν[z] which has been recently introduced by Garg et al. Some interesting special cases are given to fractional integration operators involving some Special functions.


Introduction and Preliminaries
The fractional calculus and its various applications have become a very popular subject between mathematicians and engineers. New era in the development of this branch of science began 40-50 years ago due to numerous application of fractional-type models and is continued up to now (see [54] and [55]). One can mention a large list of areas of application, in particular, continuum mechanics [8,39] (including viscoelasticity [27], thermodynamics [17] and anomalous diffusion [38]), astrophysics [30], nuclear physics [53], nanophysics and cosmic physics [57,58], statistical mechanics [60], fractional order systems and control [7], finance and economics [5], solutions of differential equations [4].
Among the monographs developing the theory of fractional calculus and presenting some applications we have to point out monographs by Diethelm [11], Gorenflo and Mainardi [15], Kiryakova [21], Kilbas, Srivastava and Trujillo [20], Miller and Ross [32], Oldham and Spanier [35], Podlubny [36], and of course the Bible of fractional calculus, monograph by Samko, Kilbas and Marichev [43]. Interested reader can find in these books an extended list of publications on the theory and applications of fractional calculus (see also [56]).
Recently, Mittag-Leffler functions show its close relation to Fractional Calculus and especially to fractional problems which come from applications. This new era of research attract many scientists from different point of view (see [2,6,9,12,16,18,19,22,23,37,40,44,46,52]). In 1899 G. Mittag-Leffler began the publication of a series of articles under the common title Sur la representation analytique d'une branche uniforme d'une fonction monogµene (On the analytic representation of a single-valued branch of a monogene function) published mainly at Acta Mathematica. Nowadays this function and its numerous generalizations are involved in the different fractional models (see monographs listed above). Motivated by the above works Kiryakova [25,26] for the first time pointed out the special role of the Mittag-Leffler function and included it into the class of Special Functions for Fractional Calculus. Moreover, based on the role of the Mittag-Leffler function in application, Mainardi called it The Queen of Fractional Calculus (see [27]).
Here, our investigation are based on the so-called Marichev-Saigo-Maeda type generalized fractional operator, i.e. integral transform of the Mellin convolution type with the Appell (or Horn) function F 3 developed by Marichev [28] and studied in some recent papers, including the papers by Agarwal et al [2], Choi and Agarwal [10], Saigo and Maeda [42], Saigo and Saxena [45]. The aim of our paper is to present formulas of the Marichev-Saigo-Maeda generalized fractional integration of the generalized Mittag-Leffler type function with four parameters ; E ; OEz which has been recently introduced by Garg et al. [13], and study its various properties, which mainly motivated our present investigation. Throughout this paper, let C, R, R C , Z C 0 , N be the sets of complex numbers, real and positive real numbers, nonpositive integers, and positive integers, respectively, and N 0 WD N [ f0g.

Definitions and earlier works
For the present investigation, we consider the following definitions and earlier works.
In recent years, the Mittag-Leffler function and its various generalizations have become a very popular subject of mathematics and its applications. Among the large number of works regarding the Mittag-Leffler function, for a remarkably clear, insightful, and systematic exposition of the investigations carried out by various authors in the field of mathematical analysis and its applications, the interested reader should refer also to a survey-cum-expository Book by Gorenflo et al., which contains a fairly comprehensive bibliography of as many as 170 further references on the subject.

Definition 2.2. The H -function is defined in terms of a Mellin-Barnes integral in the following manner ([29]):
zˇ. a 1 ;˛1/ ; ; .a p ;˛p/ .b 1 ;ˇ1/ ; ; where m; n; p; q are integers such that 0 Ä m Ä q, 0 Ä n Ä p, and for parameters a i ; b i 2 C and for parameters i ;ˇj 2 R C .i D 1; : : : ; pI j D 1; : : : ; q/ with the contour L suitably chosen, and an empty product, if it occurs, is taken to be unity. The theory of the H-function are well explained in the book of Srivastava, Gupta and Goyal ( [50], Ch.1) (see also [30]).
where the coefficients A 1 ; : : : ; A p 2 R C and B 1 ; : : : Here, in this paper, our main results are obtained by applying the ; E ; OEz to the fractional integration operators (of Marichev-Saigo-Maeda type) given in (7) and (8), respectively . So we continue to recall the following definitions.
These operators (integral transforms) were introduced by Marichev [28] as Mellin type convolution operators with a special function F 3 .:/ in the kernel. These operators were rediscovered and studied by Saigo in [41] as generalization of so-called Saigo fractional integral operators, see [24]. The properties of these operators were studied by Saigo and Maeda [42], in particular, relations of operators with the Mellin transforms, hypergeometric operators (or Saigo fractional integral operators), their decompositions and acting properties in the McBride spaces F pI (see [31]). In (7), (8)   .˛/ m .˛0/ n .ˇ/ m .ˇ0/ n .Á/ mCn mŠnŠ x m y n max fjxj; jyjg < 1/: The properties of this function are discussed in [34, p. 412-415]. In particular, its relation to the Gauss hypergeometric function is presented: F 3 .˛; Á ˛;ˇ; Á ˇI ÁI xI y/ D 2 F 1 .˛;ˇI ÁI x C y xy/: Moreover, it is easily observed that and F 3 .0;˛0;ˇ;ˇ0I ÁI xI y/ D F 3 .˛;˛0;ˇ0I ÁI xI y/ D 2 F 1 .˛0;ˇ0I ÁI x/: It is known that the 3rd Appell function cannot be expressed as a product of two 2 F 1 functions, and satisfy pairs of linear partial differential equations of the second order.

Left-sided fractional integration of generalized Mittag-Leffler functions with four parameters
Our results in this Section are based on the preliminary assertions giving composition formula of fractional integral (7) with a power function.
The value of the left-sided Marichev-Saigo-Maeda fractional integral (7) for the generalized Mittag-Leffler function (1) is given by the following theorem.
Proof. For convenience, let the left-hand side of the formula (14) be denoted by I. We apply (1) and use definition of the integral operator (7) and the representation of (1) in terms of generalized Wright function (5). We use then series form definition of the generalized Wright function (5). Finally, we change the order of integration and summation and find Due to the convergence conditions of Theorem 3.2, for any n 2 N 0 ; we have < . C n/ < . / > max OE0; < .˛C˛0 Cˇ /; < .˛0 ˇ0/ Therefore we can apply Lemma 3.1 and use (13) with replaced by . C n/: . C n/ . n C / . C n/ . C n C Á ˛ ˛0 ˇ/ . C n Cˇ0/ . C n C Á ˛ ˛0/ . C n Cˇ0 ˛0/ .1 C n/ . C n C Á ˛0 ˇ/ c n x n nŠ : This, in accordance with (5), completes the proof.

Right-sided fractional integration of generalized Mittag-Leffler functions with four parameters
In this Section, our results are based on the preliminary assertions giving composition formula of fractional integral (8) with a power function.
Proof. For convenience, let the left-hand side of the formula (18) be denoted by J . We apply (1) and use definition of the integral operator (8) and the representation of (1) in terms of generalized Wright function (5). We use then series form definition of the generalized Wright function (5). Finally, we change the order of integration and summation and find . / n . n C / c n I .˛;˛0;ˇ;ˇ0; / n t n 1 oÁ .x/: Due to the convergence conditions of Theorem 4.2, for any n 2 N 0 ; we have < . n 1/ Ä < . 1/ < 1 min OE< . ˇ/; < .˛C˛0 Á/; < .˛Cˇ0 Á/ Therefore we can apply Lemma 4.1 and use (17) with replaced by . n/: This, in accordance with (5), completes the proof.
Remark 4.4. It is easily seen that setting ! 0 in equation (18) with some suitable parametric replacements in the resulting identities yields the corresponding known integral formulas in Agarwal et al. [1].
It is noted that if we setˇD ˛andˇD 0 (21) and (22) yields the Erdélyi-Kober fractional integral operators E˛; Á 0C and K˛; Á ; the Riemann-Liouville fractional integral operator R0 C ; and the Weyl fractional integral operatorW˛ : Therefore the results presented here are easily shown to be converted to those corresponding to the above well known fractional operators.
We conclude our present investigation by remarking further that several further consequences of Theorems 3.2 and 3.2 and Corollaries 3.3-5.4 can easily be derived by using some known and new relationships between Mittag-Leffler type function with four parameters ; E ; OEz, which is an elegant unification of various special functions (see [13]), and Fox H -function as given in Definition 2.2, after some suitable parametric replacements, which are more simpler fractional integration operators (of Marichev-Saigo-Maeda type), can be deduced from Theorems 3.