Certain fractional integral operators and the generalized multi-index Mittag-Leffler functions

Transcription

1 Proc. Indian Acad. Sci. Math. Sci.) Vol. 125, No. 3, August 2015, pp c Indian Academy of Sciences Certain fractional integral operators the generalized multi-index Mittag-Leffler functions PRAVEEN AGARWAL 1,, SERGEI V ROGOSIN 2 JUAN J TRUJILLO 3 1 Department of Mathematics, An International College of Engineering, Jaipur , India 2 Department of Economics, Belarusian State University, Minsk, Belarus 3 Department de Analisis Matemático, Universidad de La Laguna, C/Astr. Fco. Sánchez s/n, La Laguna Tenerife, Spain * Corresponding Author. praveen.agarwal@anice.ac.in;goyal.praveen2011@gmail.com; rogosinsv@gmail.com; jtrujill@ullmat.es MS received 15 July 2013; revised 2 October 2014 Abstract. In this paper, we obtain formulas of fractional integration of Marichev Saigo Maeda type) of the generalized multi-index Mittag-Leffler functions E γ,κ [α j, β j ) m ; z generalizing 2m-parametric Mittag-Leffler functions studied by Saxena Nishimoto J. Fract. Calc ) 43 52). Some interesting special cases of our main results are considered too. Keywords. Marichev Saigo Maeda fractional integral operators; generalized multiindex Mittag-Leffler functions; Appell functions; generalized Wright function Mathematics Subject Classification. 33E12, 26A33, 33C Introduction The fractional calculus is nowadays one of the most rapidly growing subject of Mathematical Analysis in spite of the fact that is nearly 300 years old. Yet the giants of mathematics, G W Leibnitz L Euler, thought about the possibility to perform differentiation of non-integer order. The real birth far-reaching development of the fractional calculus is due to numerous attempts of mathematicians at the beginning of the 20th century. It is practically impossible to name all the important contributions made in construction of the early stages of the building of fractional calculus see [48 50). New era in the development of this branch of science began years ago due to numerous application of fractional-type models is continued up to now. One can mention a large list of areas of application, in particular, continuum mechanics [5, 37 including viscoelasticity [25, thermodynamics [11 anomalous diffusion [36), astrophysics [28, nuclear physics [47, nanophysics cosmic physics [51, 52, statistical mechanics [54, fractional order systems control [4, finance economics [3. Among the monographs developing the theory of fractional calculus presenting some applications we have to point out monographs by Diethelm [7, Kiryakova [18, Kilbas, Srivastava Trujillo [17, Miller Ross [30, Oldham Spanier [32, 291

2 292 Praveen Agarwal et al. Podlubny [34, of course the Bible of fractional calculus, monograph by Samko et al. [40. The interested reader can find in these books an extended list of publications on the theory applications of fractional calculus see also [50). A special role of the Mittag-Leffler functions in the fractional calculus has been discovered by many scientists from different view points. In 1899, 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 monogene On the analytic representation of a single-valued branch of a monogene function ) published mainly at Acta Mathematica. His research was connected with the solution of a problem of analytic continuation of complex functions represented by power series. The function which he used for the solution of this problem was named later as the Mittag-Leffler function. Following the line of Mittag-Leffler s consideration, several investigations related to this function, its generalizations related special functions have been done at the beginning of the 20th century see, e.g., review article [10 references therein). Probably for the first time, an interest to this function from the application appeared due to representation in terms of this function the solution of the Abel integral equation of the second order made by Hille Tamarkin [12. Nowadays this function its numerous generalizations are used in different fractional models see monographs listed above). A special role of the Mittag- Leffler function was pointed out by Kiryakova [23, 24, who included it into the class of special functions for fractional calculus. Moreover, based on the role of the Mittag-Leffler function in applications, Mainardi [25 called it the queen of fractional calculus. Due to this exceptional role of the collection of Mittag-Leffler functions, any new exact result involving these functions seem very interesting. This paper is devoted to the properties of the so-called Marichev Saigo Maeda generalized fractional operator, i.e. integral transform of the Mellin convolution type with the Appell or Horn) function F 3. This operator was introduced nearly 40 years ago by Marichev [26 studied in some recent papers, including the papers by Saigo Maeda [39, Saigo Saxena [44. The aim of our paper is to present formulas of the Marichev Saigo Maeda generalized fractional integration of the generalized multi-index Mittag-Leffler function E γ,κ [α j,β j ) m ;z. The latter function was introduced by Shukla Prajapati [45 in the case m = 1. In [41, this function was considred for arbitrary m N on the wider domain of parameters. Formulas of the Marichev Saigo Maeda generalized fractional integration of the multi-index Mittag-Leffler function of the Luchko Kiryakova Kilbas type were presented in a recent article [1 see also [6, 43, 46). 2. Mittag-Leffler function Marichev Saigo Maeda fractional operator In this section, we recall some known facts about the Mittag-Leffler function its generalizations, also about the Marichev Saigo Maeda generalized fractional integration. Let us begin with few notions facts related to the Mittag-Leffler function. In this presentation, we follow mainly the review article [10 see also [9). The Mittag-Leffler function E α z) with α>0 is named after the great Swedish mathematician G. M. Mittag-Leffler who introduced it at the beginning of the 20th century in a sequence of five notes. He defined in the form of series: E α z) = z n Ŵαn + 1). 2.1)

3 Fractional integral operators Mittag-Leffler functions 293 It was noted by Mittag-Leffler himself that for all α, Re α>0 the series in 2.1) converges in the whole complex plane thus is an entire function of complex variable z). For special values of parameter α, the function E α z) coincides with some elementary special functions. In particular, E 1 z) = exp z. Hence, sometimes, the Mittag-Leffler function is called the generalized exponential. Anyway, the asymptotic behavior at infinity of this function differs from that of exponential function, namely, for all α, 0 < Reα) < 2,α = 1 there exists an angle of exponential growth, an angle at which the function is bounded. First generalization of the function E α z) was mentioned by Wiman [53: E α,β z) = z n Ŵαn + β). 2.2) For each α,β C, Reα) > 0, E α,β z) is an entire function. The function E α,β z) is reduced to the classical Mittag-Leffler function if we put β = 1. Further generalization of the function E α z) was proposed by Prabhakar [35: E γ α,β z) = γ) n z n Ŵαn + β), 2.3) where α,β C, Reα) > 0 γ) n is the Pochhammer symbol: { 1, n= 0, γ) n = γγ + 1)...γ + n 1), n N. Extended exposition on the theory applications of this function is given in [27. Evidently, the function E γ α,β z) is related to the classical Mittag-Leffler function E αz) two-parametric Mittag-Leffler function E α,β z): E 1 α,β z) = E α,βz); E 1 α,1 z) = E αz). Another generalization of the two-parametric Mittag-Leffler function is the so-called four-parametric function: z n E α1,β 1 ;α 2,β 2 z) = Ŵα 1 n + β 1 )Ŵα 2 n + β 2 ). 2.4) For positive α 1 > 0,α 2 > 0 real β 1 ;β 2 R, some fractional integral formulas were introduced by Dzrbashian [8. It is not hard to see that the convergence conditions for this function can be extended to all α 1,β 1 ;α 2,β 2 C, Reα 1 )>0, Reα 2 )>0. Besides, E α,β;0,1 z) = E α,β z). Generalizing the four-parametric Mittag-Leffler function, Al-Bassam Luchko [2 introduced the Mittag-Leffler type function: E αj,β j ) m z) = Eα j,β j ) m j=1 ;z) = z n m N), 2.5) m Ŵα j n + β j ) with 2m real parameters α j > 0;β j R j = 1,...,m) with complex z C. In [2 an explicit solution to Cauchy type problem for a fractional differential equation is given in terms of 2.5). The theory of this class of functions was developed in a series of articles by Kiryakova et al. [19, 20, see also [14). j=1

4 294 Praveen Agarwal et al. Another class of special functions that generalizes both the Prabhakar functions 2.3) multi-index Mittag-Leffler functions 2.5) with 2m indices is the class of 3m-parametric Mittag-Leffler functions, introduced by Paneva-Konovska for details, see [33), namely: E γ i),m α i ),β i ) z) = k=0 γ 1 ) k...γ m ) k Ŵα 1 k + β 1 )...Ŵα m k + β m ) z k k!) m. 2.6) The parameters are complex, i.e. α i,β i,γ i C, Reα i )>0 for all i = 1, 2,...,m, m>1 is an integer. Generalization of the Prabhakar type function was done by Shukla Prajapati [45: E γ,κ α;β z) = Eα,β;γ,κ;z) = γ) κn z n Ŵαn + β) k=0 n N), 2.7) under the following assumptions on parameters: κ = q, q 0, 1) N, min {Reβ); Reγ)} > 0. In [46, the existence of the function 2.7) for wider set of parameters was shown: {α,β,γ,κ C : Reα) > max{0; Reκ) 1}; Reκ) > 0}. The definition 2.7) was combined with 2.5) in [41 see also [42). As a result, the following definition of generalized multi-index Mittag-Leffler function appears: E γ,κ α j,β j ) m z) = E γ,κ α j,β j ) m j=1 ;z) = j=1 γ) κn z n m Ŵα j n + β j ) m N). 2.8) This function is defined for all α j,β j,γ,κ C, j = 1,..., m, such that m j=1 Reα j )> max{0; Reκ) 1}; Reκ) > 0. γ) κn denotes as before the Pochhammer symbol. Also, the function E γ,κ α j,β j ) m z) can be represented in terms of the generalized Wright function: E γ,κ α j,β j ) m z) = 1 [ γ,κ) Ŵγ) 1 m α j ),β j ) m 1 z. 2.9) We recall the definition of the generalized Wright) function see, for example [13): [ a1,a 1 ),...,a p,a p ) p p q b 1,B 1 ),...,b q,b q ) z j=1 = Ŵα j + A j k) z k q j=1 Ŵβ j + B j k) k!. 2.10) Our main results consist of application to generalized multi-index Mittag-Leffler function of the Marichev Saigo Maeda generalized fractional operators: α,α I,β,β,η f ) x) = xα Ŵη) x 0 k=0 x t) η1 t α F 3 α,α,β,β ;η; 1 t x, 1 x t ) ft)dt, Reη) > 0), 2.11)

5 Fractional integral operators Mittag-Leffler functions 295 I α,α,β,β,η = xα Ŵη) f ) x) t x) η1 t α F 3 α,α,β,β ;η; 1 x x t, 1 t ) ft)dt, x Reη) > 0). 2.12) These operators integral transforms) were introduced by Marichev [26 as Mellin type convolution operators with a special function F 3.) in the kernel. These operators were rediscovered studied by Saigo in [38 as a generalization of the so-called Saigo fractional integral operators, see [21. The properties of these operators were studied by Saigo Maeda [39, in particular, relations of operators with the Mellin transforms, hypergeometric operators or Saigo fractional integral operators), their decompositions acting properties in the McBride spaces F p;μ see [29). In 2.11) 2.12), the symbol F 3.) denotes the so-called 3rd Appell function known also as the Horn function) see p. 413 of [31): F 3 α,α ;β,β ;η;x;y) = m, α) m α ) n β) m β ) n x m y n max{ x, y } < 1). 2.13) η) m+n m!n! The properties of this function are discussed in pp of [31. In particular, its relation to the Gauss hypergeometric function is presented: F 3 α,η α;β,η β;η;x;y) = 2 F 1 α,β;η;x + y xy). It is known that the 3rd Appell function cannot be expressed as a product of two 2 F 1 functions, satisfy pairs of linear partial differential equations of the second order. 3. Left-sided fractional integration of generalized multiindex Mittag-Leffler functions Our results in this section are based on the preliminary assertions giving composition formula of fractional integral 2.11) with a power function. Lemma 3.1 p. 394 of [39). Let α,α,β,β,η C Reη) > 0, Reρ) > max { 0, Reα + α + β η), Reα β ) }. Then the following relation holds: I α,α,β,β,η x ρ1) x) = Ŵρ)Ŵρ + η α α β)ŵρ + β α ) Ŵρ + β )Ŵρ + η α α )Ŵρ + η α β) xρ+ηαα )

6 296 Praveen Agarwal et al. The value of the left-sided Marichev Saigo Maeda fractional integral 2.11) for the generalized multi-index Mittag-Leffler function 2.8) is given by the following theorem. Theorem 3.1. Let the parameters α,α,β,β,η,γ,σ,κ,ρ,α j,β j C be such that m Reγ) > 0, Reη) > 0, Re > max{0, Reκ) 1}, Reκ) > 0, j=1 α j Reσ) > 0, Reρ) > max[0, Reα + α + β γ),reα β ). Then for all x>0, the following relation is valid: I α,α,β,β,η) [ ) t ρ1 E γ,κ α j,β j at σ ) x) = xρ+ηαα 1 Ŵγ) [ γ,κ),ρ,σ),ρ + η α α β,σ),ρ + β α,σ) 4 m+3 ) m αj,β j 1,ρ+ β,σ),ρ + η α α,σ),ρ + η β α,σ) axσ. 3.2) Proof. For convenience, let the left-h side of the formula 3.2) be denoted by I. We apply 2.8) use the definition of the integral operator 2.11) the representation of 2.8) in terms of generalized Wright function 2.9). We use the series form definition of the generalized Wright function 2.9). Finally, we perform changing the order of integration summation find that I = = I α,α,β,β,γ) [ t ρ1 γ) κn a n Ŵα j n + β j ) n! γ) κn Ŵα j n + β j ) I α,α,β,β,γ) ) a n t σn x), n! { t ρ+σn1}) x). Due to the convergence conditions of Theorem 3.1, for any n N 0,we have Re ρ + σn) Re ρ) > max [ 0, Re α + α + β γ),re α β ). Therefore we can apply Lemma 3.1 use 3.1) with ρ replaced by ρ + σn): I = Ŵγ + κn) Ŵγ) Ŵ ρ + σn)ŵ ρ + σn+ η α α β ) m j=1 Ŵα j n + β j ) Ŵ ρ + σn+ β )Ŵ ρ + σn+ η α α ) Ŵ ρ + σn+ β α ) a n x ρ+σn+ηαα 1 Ŵ ρ + σn+ η α β) n! = xρ+ηαα 1 Ŵγ + κn) Ŵ ρ + σn) Ŵ ρ + σn+ η α α β ) Ŵγ) mj=1 Ŵα j n + β j ) Ŵ ρ + σn+ β ) Ŵ ρ + σn+ η α α ) Ŵ ρ + σn+ β α ) Ŵ ρ + σn+ η α β) a n x σn. 3.3) n! This, in accordance with 2.10), completes the proof.

7 Fractional integral operators Mittag-Leffler functions 297 Corollaries of Theorem 3.1 For γ = κ = 1 in 3.2), Theorem 3.1 yields the following result: COROLLARY 3.1 Let α,α,β,β,η,σ,ρ,α j,β j C,x>0, m Reη) > 0, Reσ) > 0, Re j=1 α j > 0, j = 1,m Reρ) > max[0, Reα + α + β γ),reα β ). Then the following result holds: I α,α,β,β,η) [ ) t ρ1 E 1,1 α j ),β j ) atσ ) x) = x ρ+ηαα 1 [ ρ,σ),ρ + η α α β,σ),ρ + β α,σ), 3 m+3 ) m αj,β j 1,ρ+ β,σ),ρ + η α α,σ),ρ + η β α,σ) If we set m = 1 in Theorem 3.1, we obtain the following integral formula: COROLLARY 3.2 Let α,α,β,β,η,γ,κ,σ,ρ,α,β C,x>0,κ 0, 1) N, Reη) > 0, Reγ) > 0, Reμ) > max{0, Reκ) 1}, axσ. 3.4) Reρ) > max [ 0, Reα + α + β γ),reα β ). Then the following result holds: [ ) t ρ1 E γ,κ α,β atσ ) x) = xρ+ηαα 1 [ Ŵγ) γ,κ),ρ,σ),ρ + η α α 4 β,σ),ρ + β α,σ), 4 α,β),ρ + β,σ),ρ + η α α,σ),ρ + η β α,σ) I α,α,β,β,η) Further, setting κ = 1 in equation 3.5) yields the following result: COROLLARY 3.3 Let α,α,β,β,η,γ,σ,ρ,α,β C,x >0, Reη) > 0, Reγ) > 0, Reα) > 0 axσ. 3.5) Reρ) > max[0, Reα + α + β γ),reα β ).

8 298 Praveen Agarwal et al. Then the following result holds: I α,α,β,β,η) [ ) t ρ1 E γ,1 α,β atσ ) x) = xρ+ηαα 1 Ŵγ) 4 4 [ γ, 1),ρ,σ),ρ + η α α β,σ),ρ + β α,σ), α,β),ρ + β,σ),ρ + η α α,σ),ρ + η β α,σ) axσ. 3.6) If we set γ = 1 in equation 3.6), then we get the representation involving the Mittag- Leffler function. COROLLARY 3.4 Let α,α,β,β,η,σ,ρ,α,β C,x >0, Reη) > 0, Reα) > 0 Reρ) > max[0, Reα + α + β γ),reα β ). Then the following result holds: [ ) t ρ1 E 1,1 α,β atσ ) x) = x ρ+ηαα 1 I α,α,β,β,η) 3 4 [ ρ,σ),ρ + η α α β,σ),ρ + β α,σ), α,β),ρ + β,σ),ρ + η α α,σ),ρ + η β α,σ) axσ. 3.7) 4. Right-sided fractional integration of generalized multiindex Mittag-Leffler functions Our results in this section are based on the preliminary assertions giving composition formula of fractional integral 2.12) with a power function. Lemma 4.1 p. 394 of [39). Let α,α,β,β,η C Reη) > 0, Reρ) < 1 + min { Re β),reα + α η), Reα + β η) }. Then the following relation holds: x ρ1) x) I α,α,β,β,η = Ŵ1 ρ β)ŵ1 ρ η + α + α )Ŵ1 ρ + α + β η) Ŵ1 ρ)ŵ1 ρ + α + α + β x ρ+ηαα 1. η)ŵ1 ρ + α β) 4.1) The value of the right-sided Marichev Saigo Maeda fractional integral 2.12) for the generalized multi-index Mittag-Leffler function 2.8) is given by the following theorem.

9 Fractional integral operators Mittag-Leffler functions 299 Theorem 4.1. Let the parameters α,α,β,β,η,γ,σ,κ,ρ,α j,β j C be such that m Reγ) > 0, Reη) > 0, Re > max{0, Reκ) 1}, Reκ) > 0, j=1 α j Reσ) > 0, Reρ) < 1 + min[reβ),reα + α η), Reα + β η). Then for all x>0, the following relation is valid: I α,α,β,β,η) [ t ρ1 E γ,κ a )) α j,β j t σ x) = xρ+ηαα 1 Ŵγ) [ γ,κ),1ρβ,σ),1ρη+αβ,σ),1ρη+α+α,σ) 4 m+3 ) m αj,β j 1,1ρ,σ),1ρη+α+α +β,σ),1ρβ+α,σ) axσ. 4.2) Proof. For convenience, let the left-h side of formula 4.2) be denoted byj. We apply 2.8) use definition of the integral operator 2.12) the representation of 2.8) in terms of the generalized Wright function 2.9). We then use the series form definition of the generalized Wright function 2.9). Finally, we perform changing the order of integration summation find that J = = I α,α,β,β,γ) [ t ρ1 γ) κn a n Ŵα j n + β j ) n! γ) κn Ŵα j n + β j ) I α,α,β,β,γ) ) a n t σn x) n! { t ρσn1}) x). Due to the convergence conditions of Theorem 4.1, for any n N 0,wehaveReρσn1) Re ρ 1) < 1 min [ Re β),re α + α η), Re α + β η). Therefore we can apply Lemma 4.1 use 4.1) with ρ replaced by ρ σn): J= Ŵγ +κn) Ŵγ) Ŵ 1ρη+σn+α+α ) Ŵ 1ρσnη+α+β ) m j=1 Ŵα j n+β j ) Ŵ 1ρ +σn) Ŵ 1ρ +σnη+α+α +β ) Ŵ 1 ρ + σnβ) a n x ρσn+ηαα 1 Ŵ 1 ρ + σn+ α β) n! = xρ+ηαα 1 Ŵγ + κn) Ŵγ) mj=1 Ŵα j n + β j ) Ŵ 1 ρ η + α + α + σn ) Ŵ 1 ρ + σn η + α β ) Ŵ 1 ρ + σn)ŵ 1 ρ + σn η + α + α + β ) Ŵ 1 ρ + σn β) Ŵ 1 ρ + σn+ α β) a n x σn. 4.3) n! This, in accordance with 2.10), completes the proof.

10 300 Praveen Agarwal et al. Corollaries of Theorem 4.1. For γ = κ = 1 in 4.2), Theorem 4.1 yields the following result: COROLLARY 4.1 Let α,α,β,β,η,σ,ρ,α j,β j C,x >0, m Reη) > 0, Re > max{0, Reκ) 1}, Reκ) > 0, j=1 α j Reσ) > 0, Reρ) < 1 + min[reβ),reα + α η), Reα + β η). Then the following result holds: I α,α,β,β,η) [ t ρ1 E 1,1 α j,β j ) m a t σ )) x) = x ρ+ηαα m [ 1ρβ,σ),1ρη+αβ,σ),1ρη+α + α,σ), α j,β j ) m 1 1ρ,σ),1ρη + α+α +β,σ),1ρβ+α,σ) If we set m = 1 in Theorem 4.1, we obtain the following corollary: COROLLARY 4.2 Let α,α,β,β,η,γ,κ,σ,ρ,α,β C,x>0,κ 0, 1) N, a x σ. 4.4) Reγ) > 0, Reη) > 0, Re α) > max{0, Reκ) 1} Reρ) < 1 + min[reβ),reα + α η), Reα + β η). Then the following result holds: I α,α,β,β,η) [ t ρ1 E γ,κ a )) α,β t σ x) = xρ+ηαα 1 Ŵγ) 4 4 [ γ,κ),1ρβ,σ),1ρη+αβ,σ),1ρη+α+α,σ), α,β),1ρ,σ),1ρη+α+α +β,σ),1ρβ+α,σ) Further, if we set κ = 1 in equation 4.5), it yields the following result: COROLLARY 4.3 Let α,α,β,β,η,γ,σ,ρ,α,β C,x >0, Reγ) > 0, Reη) > 0, Reα) > 0 a x σ. 4.5) Reρ) < 1 + min[reβ),reα + α η), Reα + β η).

11 Fractional integral operators Mittag-Leffler functions 301 Then the following result holds: I α,α,β,β,η) [ t ρ1 E γ,1 α,β a )) t σ x) = xρ+ηαα 1 Ŵγ) 4 4 [ γ, 1),1ρβ,σ),1ρη+αβ,σ),1ρη+α+α,σ), α,β),1ρ,σ),1ρη+α+α +β,σ),1ρβ+α,σ) a If we set γ = 1 in equation 4.6), then we get the representation involving the Mittag- Leffler function. COROLLARY 4.4 Let α,α,β,β,η,σ,ρ,α,β C,x >0, Reη) > 0, Reα) > 0 x σ. 4.6) Reρ) < 1 + min[reβ),reα + α η), Reα + β η). Then the following result holds: I α,α,β,β,η) [ t ρ1 E 1,1 a )) α,β t σ x) = x ρ+ηαα [ 1ρ β,σ),1ρ η+αβ,σ),1ρ η+α+α,σ), α,β),1ρ,σ),1ρ η+α+α +β,σ),1ρ β +α,σ) 5. Another approach a x σ. 4.7) In this section, we briefly consider another variant of the derivation of results in the preceding sections. This approach is due to representation of the generalized multi-index function E γ,κ α j,β j ) m z) in terms of the Fox H -function with special values of parameters find its Mellin Laplace transform. The Fox H -function is a general multi-parametric function containing as concrete cases, many known special functions. The theory of H - function is rather cumbersome. It is presented in the recent monographs [15, 28. Lemma 5.1. Let α j,β j,γ,κ C, j = 1,...,m be such that m j=1 Re α j > max {0; Re κ 1};κ > 0, Re γ > 0. Then the following representation holds: E γ,κ α j,β j ) m z) = 1 2πiŴγ) L Ŵs)Ŵγ κs) β j α j s) z)s dsz = 0), 5.1) wherelis a vertical line φi,φ+i ),0<φ<Re γ κ, the single-valued branch of the function z) s is chosen in the stard way in a complex plane cut along the positive semi-axes. Proof. It suffices to calculate the integral in the right-h side. From the conditions of lemma, it follows that the line L separate the poles of Ŵs) Ŵγ κs),

12 302 Praveen Agarwal et al. i.e. the following families of points {p : p N 0 } { γ κ + p κ : p N 0}. Therefore Since 1 2πiŴγ) L = 1 Ŵγ) Re s s =p Ŵs)Ŵγ κs) Ŵβ j α j s) z)s ds p=0 Re s s =p 1)p Ŵs) =, p! [ Ŵs)Ŵγ κs) Ŵβ j α j s) z)s. 5.2) Re s s =p Ŵs)Ŵγ κs) Ŵβ j α j s) z)s = 1)p p! Ŵγ + κp) Ŵβ j + α j p) z)p. It completes the proof as γ) κp = Ŵγ+κp) Ŵγ). Remark 5.1. Note that the conditions in the lemma κ>0, Reγ) > 0 are sufficient to separate the poles of the functions Ŵs) Ŵγ κs). This kind of separation appear also if Im γ κ = 0, Im γ κ Im κ<0orimγ κ = 0, Re γ κ > 0. In these cases, the representation 5.1) takes place too, but the contour L can be more compound. The above formula 5.1) is nothing but the Mellin Barnes representation of the generalized multiindex function E γ,κ α j,β j ) m z). Hence the following representation holds see, e.g. [15): E γ,κ α j,β j ) m z) = 1 [ Ŵγ) H 1,1 1,m+1 z 1 γ,κ). 0, 1),1 β 1,α 1 ),...,1 β m,α m ) 5.3) Therefore, the results presented in this paper are easily converted in terms of the Fox H-function after some suitable parametric replacement. Lemma 5.2. Let α j,β j,γ,κ,s C, j = 1,...,m be such that m j=1 Re α j > max{0; Re κ 1};κ >0, Re γ>0. Then the following Mellin transform holds: M[E γ,κ Ŵs)Ŵγ κs) α j,β j ) m wz);s= w s Ŵγ) m j=1 β j α j s) 5.4) Proof. Putting z =wt in 5.1), we get E γ,κ 1 Ŵs)Ŵγ κs) α j,β j ) m wt) = 2πiŴγ) mj=1 L β j α j s) wt)s dsz = 0), = 1 fs)t s ds, 5.5) 2πi L

13 Fractional integral operators Mittag-Leffler functions 303 where fs)= Ŵs)Ŵγ κs) w s Ŵγ) m j=1 β j α j s) by using the well-known definition of the Mellin transform of fz)see p. 46, eqs 2.1) 2.2) of [28): then M[fz);s= 0 fs)= M 1 [ fs);t= 1 2πi L Equation 5.5) immediately leads to 5.6). z s1 fz)dz = fs), Rs >0, fs)t s ds. Lemma 5.3. Let α j,β j,γ,κ,s C, j = 1,...,m be such that m j=1 Re α j > max{0; Re κ 1};κ >0, Re γ> 0, Re s>0 1 s < 1. Then the following Laplace transform holds: [ L E γ,κ α j,β j ) m t);s = 1 sŵγ) 2 m [ γ,κ),1, 1) α j ),β j ) m ) s We are not giving its proof here due to lack of space. The fractional integration of Marichev Saigo Maeda type) of the generalized multiindex Mittag-Leffler function E γ,κ [α j,β j ) m ;z Fox H-function) established in this paper will be useful for researchers in various disciplines of applied sciences engineering physics. We are also trying to find certain possible applications of those results presented here to some other research areas, for example, recently Kilbas et al. [16 applied fractional integral operator on the product of generalized Mittag-Leffler functions to obtain a closed form solution of a fractional generalization of a free electron equation. Acknowledgements The authors take this opportunity to express their deepest thanks to the worthy referee for his/her valuable comments suggestions that helped improve this paper in its present form. The work of the second author is partially supported by IMAPS EU PEOPLE IAPP Project PIAP-GA HYDROFRAC. References [1 Agarwal P, Chnad M Jain S, Certain integrals involving generalized Mittag-Leffler functions, Proc. Nat. Acad. Sci. India Sect. A under printing) [2 Al-Bassam M A Luchko Y F, On generalized fractional calculus it application to the solution of integro-differential equations, J. Fract. Calc ) [3 Baleanu D, Diethelm K, Scalas E Trujillo J J, Fractional Calculus: Models Numerical Methods 2012) N. Jersey, London, Singapore: World Scientific Publishers) [4 Caponetto R, Dongola G, Fortuna L Petráš I, Fractional Order Systems: Modeling Control Applications 2010) Singapore: World Scientific Publ. Co Inc)

14 304 Praveen Agarwal et al. [5 Caputo M Mainardi F, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento Ser. II) ) [6 Choi J, Agarwal P, Certain integral transform fractional integral formulas for the generalized Gauss hypergeometric functions, Abstr. Appl. Anal ) ,7 pages; available online at [7 Diethelm K, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type 2010) Berlin: Springer) Springer Lecture Notes in Mathematics No 2004 [8 Dzrbashjan M M, On the integral transforms generated by the generalized Mittag-Leffler function, Izv. AN Arm. SSR 133) 1960) [9 Gorenflo R Mainardi F, Fractional calculus: integral differential equations of fractional order, in: A Carpinteri F Mainardi eds) Fractals Fractional Calculus in Continuum Mechanics 1997) Wien: Springer Verlag) [10 Haubold H J, Mathai A M, Saxena R K, Mittag-Leffler functions their applications, J. Appl. Math ) , 51 pages; available online at /2011/ [11 Hilfer R ed) Applications of Fractional Calculus in Physics 2000) New Jersey, London, Hong Kong, Word Scientific Publishing Co.) [12 Hille E Tamarkin J D, On the theory of linear integral equations, Ann. Math ) [13 Kilbas A A, Fractional calculus of the generalized Wright function, Fract. Calc. Appl. Anal. 82) 2005) [14 Kilbas A A, Koroleva A A Rogosin S V, Multi-parametric Mittag-Leffler functions their extension, Fract. Calc. Appl. Anal. 162) 2013) [15 Kilbas A A Saigo M, H -Transform. Theory Applications 2004) Boca Raton- London- New York-Washington, D.C.: Chapman Hall/CRC) [16 Kilbas A A, Saigo M Saxena R K, Solution of Volterra integro-differential equations with generalized Mittag-Leffler function in the kernels, J. Integral Equations Appl. 144) 2002) [17 Kilbas A A, Srivastava H M, Trujillo J J, Theory Applications of Fractional Differential Equations North-Holl Mathematics Studies ) Elsevier, Amsterdam, etc) [18 Kiryakova V, Generalized Fractional Calculus Applications 1994) Harlow: Longman) [19 Kiryakova V, Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators Laplace type integral transforms, Fract. Calc. Appl. Anal. 24) 1999) [20 Kiryakova V, Multiple multi-index) Mittag-Leffler functions relations to generalized fractional calculus, J. Comput. Appl. Math ) [21 Kiryakova V, On two Saigo s fractional integral operators in the class of univalent functions, Fract. Calc. Appl. Anal. 92) 2006) [22 Kiryakova V, Some special functions related to fractional calculus fractional noninteger) order control systems equations, Facta Universitatis Sci. J. of University of Nis), Series: Automatic Control Robotics 71) 2008) [23 Kiryakova V S, The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Comp. Math. Appl. 593) 2010) [24 Kiryakova V S, The multi-index Mittag-Leffler function as an important class of special functions of fractional calculus, Comp. Math. Appl. 595) 2010) [25 Mainardi F, Fractional Calculus Waves in Linear Viscoelasticity 2010) London: Imperial College Press) [26 Marichev O I, Volterra equation of Mellin convolution type with a Horn function in the kernel, Izv. AN BSSR Ser. Fiz.-Mat. Nauk. No )

15 Fractional integral operators Mittag-Leffler functions 305 [27 Mathai A M Haubold H J, Special Functions for Applied Scientists 2008) New York: Springer) [28 Mathai A M, Saxena R K Haubold H J, TheH -function: Theory applications 2010) Dordrecht: Springer) [29 McBride A C, Fractional Calculus Integral Transforms of Generalized Functions, Research Notes in Math ) London: Pitman) [30 Miller K S Ross B, An Introduction to the Fractional Calculus Fractional Differential Equations 1993) New York: John Wiley Sons) [31 NIST Hbook of Mathematical Functions, edited by Frank W J Olver, D W Lozier, R F Boisvert, C W Clark. Gaithersburg, Maryl, National Institute of Stards Technology, New York, Cambridge University Press, xv pages a CD 2010) [32 Oldham K B Spanier J, The Fractional Calculus 1974) New York: Academic Press) [33 Paneva-Konovska J, On the multi-index 3m-parametric) Mittag-Leffler functions, fractional calculus relations series convergences, Central European Journal of Physics 1110) 2013) ; available online at s [34 Podlubny I, Fractional Differential Equations 1999) New York: Academic Press) [35 Prabhakar T R, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J ) 7 15 [36 Prakasa Rao B L S, Statistical inference for fractional diffusion processes 2010) Chichester: John Wiley & Sons Ltd.) [37 Rabotnov Yu N, Elements of Hereditary Solid Mechanics 1980) Moscow: MIR) [38 Saigo M, On generalized fractional calculus operators, in: Recent Advances in Applied Mathematics Proc. Internat. Workshop held at Kuwait Univ.) 1996) Kuwait: Kuwait Univ.) pp [39 Saigo M Maeda N, More generalization of fractional calculus, in: Transform methods special functions, Varna 1996 Proc. 2nd Intern. Workshop) eds) P Rusev, I Dimovski, V Kiryakova 1998) Sofia: IMI-BAS) pp [40 Samko S G, Kilbas A A Marichev O I, Fractional Integrals Derivatives: Theory Applications 1993) New York London: Gordon Breach Science Publishers) [41 Saxena R K Nishimoto K, N-Fractional calculus of generalized Mittag-Leffler functions, J. Fract. Calc ) [42 Saxena R K, Pogány T K, Ram J Daiya J, Dirichlet averages of generalized multiindex Mittag-Leffler functions, Armen. J. Math. 34) 2010) [43 Saxena R K, Ram J Vishnoi M, Fractional integration fractional differentiation of generalized Mittag-Leffler functions, J. Indian Acad. Math. 321) 2010) [44 Saxena R K Saigo M, Generalized fractional calculus of the H -function associated with the Appell function, J. Fract. Calc ) [45 Shukla A K Prajapati J C, On a generalization of Mittag-Leffler function its properties, J. Math. Anal. Appl ) [46 Srivastava H M Tomovski Ž, Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 2111) 2009) [47 Tarasov V E, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields Media 2010) Beijing: Springer, Heidelberg; Higher Education Press) [48 Tenreiro Machado J A, Kiryakova V Mainardi F, A poster about the old history of fractional calculus, Fract. Calc. Appl. Anal. 134) 2010) [49 Tenreiro Machado J A, Kiryakova V Mainardi F, A poster about the recent history of fractional calculus, Fract. Calc. Appl. Anal. 133) 2010)

16 306 Praveen Agarwal et al. [50 Tenreiro Machado J A, Kiryakova V, Mainardi F, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simulat. 163) 2011) ; available online at [51 Uchaikin V V, Fractional derivatives for physicists engineers, Vol. I, Background theory 2013) Beijing: Springer, Berlin Higher Education Press) [52 Uchaikin V V, Fractional derivatives for physicists engineers, Vol. II, Applications 2013) Beijing: Springer, Berlin Higher Education Press) [53 Wiman A, Über den Fundamentalsatz in der Theorie der Funktionen E α x), Acta Math ) [54 Zaslavsky G M, Hamiltonian Chaos Fractional Dynamics 2005) Oxford: Oxford University Press) COMMUNICATING EDITOR: B V Rajarama Bhat