Review Article Mittag-Leffler Functions and Their Applications

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1 Hindawi Publishing Corporation Journal of Applied Mathematics Volume 211, Article ID , 51 pages doi:1.1155/211/ Review Article Mittag-Leffler Functions and Their Applications H. J. Haubold, 1 A. M. Mathai, 2, 3 and R. K. Saxena 4 1 Office for Outer Space Affairs, United Nations, Vienna International Centre, P.O. Box 5, 14 Vienna, Austria 2 Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada H3A 2K6 3 Centre for Mathematical Sciences, Pala Campus, Arunapuram P.O., Pala, Kerala , India 4 Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur 3425, India Correspondence should be addressed to H. J. Haubold, hans.haubold@unvienna.org Received 15 December 21; Accepted 28 February 211 Academic Editor: Ch Tsitouras Copyright q 211 H. J. Haubold et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Motivated essentially by the success of the applications of the Mittag-Leffler functions in many areas of science and engineering, the authors present, in a unified manner, a detailed account or rather a brief survey of the Mittag-Leffler function, generalized Mittag-Leffler functions, Mittag- Leffler type functions, and their interesting and useful properties. Applications of G. M. Mittag- Leffler functions in certain areas of physical and applied sciences are also demonstrated. During the last two decades this function has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittag-Leffler, due to the vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences, and so forth. In this survey paper, nearly all types of Mittag-Leffler type functions existing in the literature are presented. An attempt is made to present nearly an exhaustive list of references concerning the Mittag-Leffler functions to make the reader familiar with the present trend of research in Mittag- Leffler type functions and their applications. 1. Introduction The special function z k E α z, α C, Rα >, z C 1.1 Γ1 αk k and its general form E α,β z k z k Γ β αk ), α,β C, Rα >, R β ) >, z C 1.2

2 2 Journal of Applied Mathematics with C being the set of complex numbers are called Mittag-Leffler functions 1, Section The former was introduced by Mittag-Leffler 2, 3, in connection with his method of summation of some divergent series. In his papers 2 4, he investigated certain properties of this function. Function defined by 1.2 first appeared in the work of Wiman 5, 6. The function 1.2 is studied, among others, by Wiman 5, 6, Agarwal 7, Humbert 8, and Humbert and Agarwal 9 and others. The main properties of these functions are given in the book by Erdélyi et al. 1, Section 18.1, and a more comprehensive and a detailed account of Mittag-Leffler functions is presented in Dzherbashyan 1, Chapter 2. In particular, functions 1.1 and 1.2 are entire functions of order ρ 1/α and type σ 1; see, for example, 1, page 118. The Mittag-Leffler function arises naturally in the solution of fractional order integral equations or fractional order differential equations, and especially in the investigations of the fractional generalization of the kinetic equation, random walks, Lévy flights, superdiffusive transport and in the study of complex systems. The ordinary and generalized Mittag-Leffler functions interpolate between a purely exponential law and power-law-like behavior of phenomena governed by ordinary kinetic equations and their fractional counterparts, see Lang 11, 12, Hilfer 13, 14, and Saxena 15. The Mittag-Leffler function is not given in the tables of Laplace transforms, where it naturally occurs in the derivation of the inverse Laplace transform of the functions of the type p α a bp β, where p is the Laplace transform parameter and a and b are constants. This function also occurs in the solution of certain boundary value problems involving fractional integrodifferential equations of Volterra type 16. During the various developments of fractional calculus in the last four decades this function has gained importance and popularity on account of its vast applications in the fields of science and engineering. Hille and Tamarkin 17 have presented a solution of the Abel-Volterra type equation in terms of Mittag-Leffler function. During the last 15 years the interest in Mittag-Leffler function and Mittag-Leffler type functions is considerably increased among engineers and scientists due to their vast potential of applications in several applied problems, such as fluid flow, rheology, diffusive transport akin to diffusion, electric networks, probability, and statistical distribution theory. For a detailed account of various properties, generalizations, and application of this function, the reader may refer to earlier important works of Blair 18, Torvik and Bagley 19, Caputo and Mainardi 2, Dzherbashyan 1, Gorenflo and Vessella 21, Gorenflo and Rutman 22, Kilbas and Saigo 23, Gorenflo and Luchko 24, Gorenflo and Mainardi 25, 26, Mainardi and Gorenflo 27, 28, Gorenflo et al. 29, Gorenflo et al. 3, Luchko 31, Luchko and Srivastava 32, Kilbas et al. 33, 34, Saxena and Saigo 35, Kiryakova 36, 37, Saxena et al. 38, Saxena et al , Saxena and Kalla 44, Mathaietal.45, Haubold and Mathai 46, Haubold et al. 47, Srivastava and Saxena 48, and others. This paper is organized as follows: Section 2 deals with special cases of E α z. Functional relations of Mittag-Leffler functions are presented in Section 3. Section 4 gives the basic properties. Section 5 is devoted to the derivation of recurrence relations for Mittag- Leffler functions. In Section 6, asymptotic expansions of the Mittag-Leffler functions are given. Integral representations of Mittag-Leffler functions are given in Section 7. Section 8 deals with the H-function and its special cases. The Melllin-Barnes integrals for the Mittag-Leffler functions are established in Section 9. Relations of Mittag-Leffler functions with Riemann-Liouville fractional calculus operators are derived in Section 1. Generalized Mittag-Leffler functions and some of their properties are given in Section 11. Laplace transform, Fourier transform, and fractional integrals and derivatives are discussed in Section 12. Section 13 is devoted to the application of Mittag-Leffler function in fractional

3 Journal of Applied Mathematics 3 kinetic equations. In Section 14, time-fractional diffusion equation is solved. Solution of space-fractional diffusion equation is discussed in Section 15. In Section 16, solution of a fractional reaction-diffusion equation is investigated in terms of the H-function. Section 17 is devoted to the application of generalized Mittag-Leffler functions in nonlinear waves. Recent generalizations of Mittag-Leffler functions are discussed in Section Some Special Cases We begin our study by giving the special cases of the Mittag-Leffler function E α z. i ii E z 1, z < 1, z E 1 z e z, 2.2 iii E 2 z cosh z ), z C, 2.3 iv E 2 z 2) cos z, z C, 2.4 v E 3 z 1 3 e z1/3 2e 1/2z1/3 cos 2 2 z1/3 ), z C, 2.5 vi E 4 z 1 2 cos z 1/4) cosh z 1/4), z C, 2.6 vii E 1/2 ±z 1/2) e z 1 erf ±z 1/2) e z erfc z 1/2), z C, 2.7 where erfc denotes the complimentary error function and the error function is defines as erfz 2 z exp t 2) dt, erfcz 1 erfz, z C. 2.8 π

4 4 Journal of Applied Mathematics viii For half-integer n/2 the function can be written explicitly as E n/2 z F n 1 : 1 n, 2 n,...,n 1 ) n ; z2 n n 2n1/2 z n! π 1 F 2n 1 1; n 2 2n, n 3 ) 2n,...,3n 2n ; z2 n n, 2.9 ix E 1,2 z ez 1 z, E 2,2z sinh ) z. 2.1 z 3. Functional Relations for the Mittag-Leffler Functions In this section, we discuss the Mittag-Leffler functions of rational order α m/n,withm, n N relatively prime. The differential and other properties of these functions are described in Erdélyi et al. 1 and Dzherbashyan 1. Theorem 3.1. The following results hold: d m dz m E mz m E m z m, 3.1 d m dz m E m/n z m/n) E m/n z m/n) n 1 z rm/n, Γ1 rm/n n 2, 3,..., 3.2 r1 r1 E m/n z 1 m 1 )) i2πr E 1/n z 1/m exp, 3.3 m m E 1/n z 1/n) n 1 e 1 z γ1 r/n,z, n 2, 3,..., 3.4 Γ1 r/n r1 where γa, z denotes the incomplete gamma function, defined by γa, z z e t t a 1 dt. 3.5 In order to establish the above formulas, we observe that 3.1 and 3.2 readily follow from definition 1.2. For proving formula 3.3, we recall the identity m 1 i2πkr m if k modm, exp m r if k / modm. 3.6

5 Journal of Applied Mathematics 5 By virtue of the results 1.1 and 3.6, wefindthat m 1 E α ze i2πr/m) me αm z m, m N 3.7 r which can be written as E α z 1 m 1 E α/m z 1/m e i2πr/m), m N 3.8 m r and result 3.3 now follows by taking α m/n. To prove relation 3.4, wesetm 1in3.1 and multiply it by exp z to obtain d e z E 1/n z 1/n) m 1 e z z r/m dz Γ1 r/m. r1 3.9 On integrating both sides of the above equation with respect to z and using the definition of incomplete gamma function 3.5, we obtain the desired result 3.4. An interesting case of 3.8 is given by E 2α z 2) 1 2 E αz E α z Basic Properties This section is based on the paper of Berberan-Santos 49. From1.1 and 1.2 it is not difficult to prove that E α x E 2α x 2) xe 2α,1α x 2), 4.1 E α ix E 2x x 2) ixe 2α,1α x 2), i It is shown in Berberan-Santos 49, page 631 that the following three equations can be used for the direct inversion of a function Ix to obtain its inverse Hk: Hk eck π 2eck π 2eck π RIc iω coskω IIc iω sinkωdω 4.3 RIc iω coskωdω, k > 4.4 IIc iω sinkωdω, k >. 4.5

6 6 Journal of Applied Mathematics With help of the results 4.2 and 4.4, it yields the following formula for the inverse Laplace transform Hk of the function E α x: H α k 2 π E 2α t 2) cosktdt, k >, α In particular, the following interesting results can be derived from the above result: H 1 k 2 π coshit cosktdt 2 π H 1/2 k 2 π H 1/4 k 2 π cost cosktdt δk 1, i 1, e t2 cosktdt 1 π e k2 /4, e t2 erfc t 2) cosktdt. 4.7 Another integral representation of H α k in terms of the Lévy one-sided stable distribution L α k was given by Pollard 5 in the form H α k 1 α k 11/α L α k 1/α). 4.8 The inverse Laplace transform of E α x β, denoted by H β αk with <α 1, is obtained as H β αk t α/β L α tl β kt α/β) dt, 4.9 where L α t is the one-sided Lévy probability density function. From Berberan-Santos 49, page 432 we have H α k 1 π E 2α ω 2) coskω ωe 2α,1α ω 2) sinkω dω, <α Expanding the above equation in a power series, it gives H α k 1 b n αk n, α< π n with b α E 2α t 2) dt. 4.12

7 Journal of Applied Mathematics 7 The Laplace transform of 4.11 is the asymptotic expansion of E α x as E α x 1 π n b n α, α< xn1 5. Recurrence Relations By virtue of definition 1.2, the following relations are obtained in the following form: Theorem 5.1. One has E α,β z ze α,αβ z 1 Γ β ), E α,β z βe α,β1 z αz d dz E α,β1z, d m dz m z β 1 E α,β z α z β m 1 E α,β m z α, R β m ) >, m N, d dz E α,βz E ) α,β 1z β 1 Eα,β z. αz 5.1 The above formulae are useful in computing the derivative of the Mittag-Leffler function E α,β z. The following theorem has been established by Saxena 15. Theorem 5.2. If Rα >, Rβ > and r N then there holds the formula r 1 z r z n E α,βrα z E α,β z Γ β nα ). n 5.2 Proof. We have from the right side of 5.2 r 1 z n E α,β z Γ β nα ) z n Γ β nα ). n nr 5.3 Put n r k or n k r. Then, z n nr Γ β nα ) k z kr Γ β rα kα ) z r E α,βrα z. 5.4 For r 2, 3, 4 we obtain the following corollaries.

8 8 Journal of Applied Mathematics Corollary 5.3. If Rα >, Rβ > then there holds the formula z 2 E α,β2α z E α,β z 1 Γ β ) z Γ α β ). 5.5 Corollary 5.4. If Rα >, Rβ > then there holds the formula z 3 E α,β3α z E α,β z 1 Γ β ) z Γ α β ) z 2 Γ 2α β ). 5.6 Corollary 5.5. If Rα >, Rβ > then there holds the formula z 4 E α,β4α z E α,β z 1 Γ β ) z Γ α β ) z 2 Γ 2α β ) z 3 Γ 3α β ). 5.7 Remark 5.6. For a generalization of result 5.2, see Saxena et al Asymptotic Expansions The asymptotic behavior of Mittag-Leffler functions plays a very important role in the interpretation of the solution of various problems of physics connected with fractional reaction, fractional relaxation, fractional diffusion, and fractional reaction-diffusion, and so forth, in complex systems. The asymptotic expansion of E α z is based on the integral representation of the Mittag-Leffler function in the form E α z 1 2πi Ω t α 1 expt t α dt, Rα >, α,z C, 6.1 z where the path of integration Ω is a loop starting and ending at and encircling the circular disk t z 1/α in the positive sense, arg t <πon Ω. The integrand has a branch point at t. The complex t-plane is cut along the negative real axis and in the cut plane the integrand is single-valued the principal branch of t α is taken in the cut plane. Equation 6.1 can be proved by expanding the integrand in powers of t and integrating term by term by making use of the well-known Hankel s integral for the reciprocal of the gamma function, namely 1 Γ β ) 1 e 2πi Ha ζ ζ dζ. β 6.2 The integral representation 6.1 can be used to obtain the asymptotic expansion of the Mittag-Leffler function at infinity 1. Accordingly, the following cases are mentioned.

9 Journal of Applied Mathematics 9 i If <α<2andμ is a real number such that πα 2 <μ<minπ, πα, 6.3 then for N N, N / 1 there holds the following asymptotic expansion: E α z 1 α z1 β/α exp z 1/α) N 1 1 Γ1 αr z r O as z, arg z μ and r1 N 1 E α z Γ1 αr r1 1 z r O 1 z N 1 1 z N 1 as z, μ arg z π. ii When α 2 then there holds the following asymptotic expansion:, 6.4, 6.5 E α z 1 2nπi N z 1/n exp exp )z 1/α 1 1 α α Γ1 αr z r O 1 z N 1 n r1 6.6 as z, arg z απ/2 and where the first sum is taken over all integers n such that argz 2πn απ The asymptotic expansion of E α,β z is based on the integral representation of the Mittag-Leffler function E α,β z in the form E α,β z 1 2πi Ω t α β expt t α dt, Rα >, R β ) >, z,α,β C, 6.8 z which is an extension of 6.1 with the same path. As in the previous case, the Mittag-Leffler function has the following asymptotic estimates. iii If <α<2andμis a real number such that πα 2 then there holds the following asymptotic expansion: <μ<minπ, πα, 6.9 E α,β z 1 α z1 β/α exp z 1/α) N 1 Γ β αr ) 1 z r O 1 z N 1 r1 6.1

10 1 Journal of Applied Mathematics as z, arg z μ and N 1 E α,β z Γ β αr ) 1 z r O r1 1 z N 1, 6.11 as z, μ arg z π. iv When α 2 then there holds the following asymptotic expansion: E α,β z 1 ) 2nπi 1 β N z 1/n exp exp z 1/α 1 α α Γ β αr ) 1 z r O n r1 1 z N 1, 6.12 as z, arg z απ/2 and where the first sum is taken over all integers n such that argz 2πn απ Integral Representations In this section several integrals associated with Mittag-Leffler functions are presented, which can be easily established by the application by means of beta and gamma function formulas and other techniques, see Erdélyi et al. 1, Gorenflo et al. 29, 51, 52, e ζ E α ζ α zdζ 1, z < 1, α C, Rα >, 1 z e x x β 1 E α,β x α zdx 1 1 z, z < 1, α,β C, Rα >, R β ) >, x x ζ β 1 E α ζ α dζ Γ β ) x β E α,β1 x α, R β ) >, e sζ E α ζ α dζ e sζ ζ mαβ 1 E m α,β ±aζα dζ m!s α β sα 1, Rs >, 1 sα s α a m1, Rs >, Rα >, R β ) >, 7.1

11 Journal of Applied Mathematics 11 where α, β C and E α x α 2 π sin απ 2 ) E m dm α,β z dz m E α,βz, ζ α 1 cosxζ dζ, α C, Rα >, 1 2ζ α cosαπ/2 ζ2α 7.2 E α x 1 π sinαπ E α x 1 1 x1/α 2α π ζ α 1 e ζx1/α ζ dζ, α C, Rα >, ζ α cosαπ ζ2α ζ α cosαπ arctan e ζx1/α ζ dζ, α C, Rα >. 7.4 sinαπ Note 1. Equation 7.3 can be employed to compute the numerical coefficients of the leading term of the asymptotic expansion of E α x. Equation 7.4 yields b α α π Γα sin2απ ζ α 1 1 2ζ 2α cos2απ ζ 4α dζ, α < From Berberan-Santos 49 and Gorenflo et al. 29, 51 the following results hold: E α x 2x π E 2α t 2 ) dt, α 1, α C. 7.6 x 2 t2 In particular, the following cases are of importance: E 1 x 2x coshit dt exp x, π x 2 t2 E 1/2 x 2x exp t 2) dt e x2 erfcx, π x 2 t 2 E 1/4 x 2x e t2 erfc t 2) dt. π x 2 t Note 2. Some new properties of the Mittag-Leffler functions are recently obtained by Gupta and Debnath 53.

12 12 Journal of Applied Mathematics 8. The H-Function and Its Special Cases The H-function is defined by means of a Mellin-Barnes type integral in the following manner 54: Hp,q m,n z Hp,q m,n z a p,a p b q,b q 1 Θζz ζ dζ, 2πi Ω Hp,q m,n z a 1,A 1,...,a p,a p b 1,B 1,...,b q,b q 8.1 where i 1 and Θs { m j1 Γ b j sb j ) }{ n j1 Γ 1 a j sa j ) } { q jm1 Γ ) }{ p 1 b j sb j jn1 Γ )}, 8.2 a j sa j and an empty product is interpreted as unity, m, n, p, q N with n p, 1 m q, A i,b j R, a i,b j C, i 1,...,p; j 1,...,qsuch that A i bj k ) / B j a i λ 1, k,λ N ; i 1,...,n; j 1,...,m, 8.3 where we employ the usual notations: N, 1, 2,..., R,, R,,andC being the complex number field. The contour Ω is the infinite contour which separates all the poles of Γb j sb j, j 1,...,mfrom all the poles of Γ1 a i sa i, i 1,...,n. The contour Ω could be ΩL or ΩL or ΩL iγ, where L is a loop starting at encircling all the poles of Γb j sb j, j 1,...,mand ending at. L is a loop starting at, encircling all the poles of Γ1 a i sa i,i 1,...,n and ending at. L iγ is the infinite semicircle starting at γ i and going to γ i. A detailed and comprehensive account of the H-function is available from the monographs of Mathai and Saxena 54, Prudnikov et al. 55 and Kilbas and Saigo 56. The relation connecting the Wright s function p ψ q z and the H-function is given for the first time in the monograph of Mathai and Saxena 54, page 11, equation as pψ q a1,a 1,..., a p,a p ) b 1,B 1,..., b q,b q ) z H 1,p p,q1 z 1 a 1 A 1,...,1 a p,a p,1,1 b 1,B 1,...,1 b q,b q, 8.4 where p ψ q z is the Wright s generalized hypergeometric function 57, 58; alsoseeerdélyi et al. 59, Section 4.1, defined by means of the series representation in the form pψ q z { p j1 Γ a j A j r )} zr { q j1 Γ )} b j B j r r!, 8.5 r

13 Journal of Applied Mathematics 13 where z C, a i,b j C, A i,b j R, A i /, B j / ; i 1,...,p; j 1,...,q, q p B j A i Δ> 1. j1 i1 8.6 The Mellin-Barnes contour integral for the generalized Wright function is given by pψ q ap,a p ) bq,b q ) z 1 2πi Ω Γs p j1 Γ a j A j s ) q j1 Γ b j sb j ) z s ds, 8.7 where the path of integration separates all the poles of Γs at the points s ν, ν N lying to the left and all the poles of p j1 Γa j sa j, j 1,...,p at the points s A j ν j /A j, ν j N, j 1,...,plying to the right. If Ωγ i,γi, then the above representation is valid if either of the conditions are satisfied: i Δ < 1, arg z 1 Δπ <, z/, ii Δ1, 1 Δγ 1 2 < Rδ, arg z, z /, ) q p p q δ b j a j j1 j1 This result was proved by Kilbas et al. 6. The generalized Wright function includes many special functions besides the Mittag- Leffler functions defined by equations 1.1 and 1.2. It is interesting to observe that for A i B j 1, i 1,...,p; j 1,...,q, 8.5 reduces to a generalized hypergeometric function pf q z. Thus ap, 1 ) p pψ q bq, 1 ) j1 z Γ ) a j q j1 Γ ) p F q a1,...,a p ; b 1,...,b q ; z ), b j 8.1 where a j / v, j 1,...,p, v, 1,..., b j / λ, j 1,...,q, λ, 1,..., p q or p q 1, z < 1. Wright 61 introduced a special case of 8.5 in the form φa, b; z ψ 1 b, a z 1 z r Γar b r!, 8.11 r

14 14 Journal of Applied Mathematics which widely occurs in problems of fractional diffusion. It has been shown by Saxena et al. 41, also see Kiryakova 62,that E α,β z 1 ψ 1 1, 1 β, α ) z H 1,1 1,2 z,1,1,1 β,α If we further take β 1in8.12 we find that E α,1 z E α z 1 ψ 1 1, 1 1,α z H 1,1 1,2 z,1,1,,α, 8.13 where α C, Rα >. Remark 8.1. A series of papers are devoted to the application of the Wright function in partial differential equation of fractional order extending the classical diffusion and wave equations. Mainardi 63 has obtained the result for a fractional diffusion wave equation in terms of the fractional Green function involving the Wright function. The scale-variant solutions of some partial differential equations of fractional order were obtained in terms of special cases of the generalized Wright function by Buckwar and Luchko 64 and Luchko and Gorenflo Mellin-Barnes Integrals for Mittag-Leffler Functions These integrals can be obtained from identities 8.12 and Lemma 9.1. If Rα >, Rβ > and z C the following representations are obtained: E α z 1 γi ΓsΓ1 s 2πi γ i Γ1 αs z s ds, E α,β z 1 γi ΓsΓ1 s 2πi γ i Γ β αs ) z s ds, 9.1 where the path of integration separates all the poles of Γs at the points s ν, ν, 1,...from those of Γ1 s at the points s 1 v, v, 1,... On evaluating the residues at the poles of the gamma function Γ1 s we obtain the following analytic continuation formulas for the Mittag-Leffler functions: E α z 1 γi ΓsΓ1 s 2πi γ i Γ1 αs z s ds E α,β z 1 γi ΓsΓ1 s 2πi γ i Γ β αs ) z s ds k1 k1 z k Γ1 αk, z k Γ β αk ). 9.2

15 Journal of Applied Mathematics Relation with Riemann-Liouville Fractional Calculus Operators In this section, we present the relations of Mittag-Leffler functions with the left- and rightsided operators of Riemann-Liouville fractional calculus, which are defined I α φ )x 1 Γα I α x 1 Γα x ) ) d α1 D α φ x I 1 {α} dx φ x ) 1 d α1 x Γ1 {α} dx D α φ ) x x x t α 1 φtdt, Rα >, 1.1 t x α 1 φtdt, Rα >, 1.2 d ) α1 I 1 {α} φ x dx 1 d ) α1 Γ1 {α} dx x t α 1 φtdt, Rα > x t x {α} φtdt, Rα >, where α means the maximal integer not exceeding α and {α} is the fractional part of α. Note 3. The fractional integrals 1.1 and 1.2 are connected by the relation 66, page 118 I α φ 1 t )x x α 1 I α t α 1 φt ) 1 x ). 1.5 Theorem 1.1. Let Rα > and Rβ > then there holds the formulas I α t α 1 E α,β at α )x xβ 1 a E α,β ax α 1 Γ β ), a/, ) I α E α at α x 1 a E αax α 1, a/ 1.6 which by virtue of definitions 1.1 and 1.2 can be written as ) I α t β 1 E α,β at α x x αβ 1 E α,αβ ax α, ) I α E α at α x x α E α,α1 ax α. 1.7

16 16 Journal of Applied Mathematics Theorem 1.2. Let Rα > and Rβ > then there holds the formulas I t α α β E α,β at α )) x xα β E α,β ax α ) 1 a Γ β ), a/, I α t α 1 E α at α )) x 1 a xα 1 E α ax α ) 1, a/. 1.8 Theorem 1.3. Let < Rα < 1 and Rβ >Rα then there holds the formulas D α t β 1 E α,β at α )x xβ α 1 Γ β α ) axβ 1 E α,β ax α, ) D α E α at α x α x Γ1 α ae αax α. 1.9 Theorem 1.4. Let Rα > and Rβ > Rα 1 then there holds the formula D α t α β E α,β at α )) x β x Γ β α ) ax α β E α,β ax α ) Generalized Mittag-Leffler Type Functions By means of the series representation a generalization of 1.1 and 1.2 is introduced by Prabhakar 67 as E γ α,β z n γ ) n n!γ β αn ), α,β,γ C, Rα >, R β ) >, 11.1 where ) γ n γ γ 1 ) γ n 1 ) Γ γ n ) Γ γ ) 11.2 whenever Γγ is defined, γ 1,γ/. It is an entire function of order ρ Rα 1 and type σ 1/ρ{Rα} Rα ρ. It is a special case of Wright s generalized hypergeometric function, Wright 57, 68 as well as the H-function 54. For various properties of this function with applications, see Prabhakar 67. Some special cases of this function are enumerated below E α z E 1 α,1 z, E α,β z E 1 α,β z, αγe γ1 α,β z 1 αγ β ) E γ α,β z Eγ α,β 1 z, 11.3 φ α, β; z ) Γ β ) E α 1,β z,

17 Journal of Applied Mathematics 17 where φα, β; z is the Kummer s confluent hypergeometric function. E γ z has the α,β following representations in terms of the Wright s function and H-function: ) E γ 1 γ,1 z α,β Γ γ ) 1 ψ 1 ) ; z β, α 1 Γ γ )H1,1 1,2 z 1 γ,1,1,1 β,α 1 ci 2πωΓ γ ) ΓsΓ γ s ) c i Γ β αs ) z s ds, R γ ) >, 11.4 where 1 ψ 1 and H 1,1 1,2 are, respectively, Wright generalized hypergeometric function and the H-function. In the Mellin-Barnes integral representation, ω 1andthecin the contour is such that <c<rγ, and it is assumed that the poles of Γs and Γγ s are separated by the contour. The following two theorems are given by Kilbas et al. 34. Theorem If α, β, γ, a C, Rα >, Rβ >n, Rγ > then for n N the following results hold: d n dz n z β 1 E γ α,β azα z β n 1 E γ α,β n azα In particular, d n dz n z β 1 E α,β az α z β n 1 E α,β n az α, d n dz n z β 1 φ γ,β; az ) Γ β ) Γ β n )zβ n 1 φ γ; β n; az ) Theorem If α, β, γ, a, ν, σ C, Rα >, Rβ >, Rγ >, Rν >, Rσ > then, x x t β 1 E γ α,β ax t α t ν 1 E σ α,νat α dt x βν 1 E γσ α,βν axα The proof of 11.7 can be developed with the help of the Laplace transform formula L x β 1 E γ α,β axα s s β 1 as α) γ, 11.8 where α, β, γ, a C, Rα >, Rβ >, Rγ >, s C, Rs >, as α < 1. For γ 1, 11.8 reduces to L x β 1 E α,β ax α s s β 1 as α)

18 18 Journal of Applied Mathematics Generalization of the above two results is given by Saxena 15 ) L t ρ 1 E δ β,γ atα s s ρ δ, 1, ρ, α Γδ 2 ψ 1 ) ; γ,β a s α, 11.1 where Rβ >, Rγ >, Rs >, Rρ >,s> a 1/Rα, Rδ >. Relations connecting the function defined by 11.1 and the Riemann-Liouville fractional integrals and derivatives are given by Saxena and Saigo 35 in the form of nine theorems. Some of the interesting theorems are given below. Theorem Let α>, β>, γ>, and a R. LetI α Liouville fractional integral. Then there holds the formula be the left-sided operator of Riemann- I α t γ 1 E δ β,γ at β)) x x αγ 1 E δ β,αγ ax β) Theorem Let α>, β>, γ>, and a R. LetI α be the right-sided operator of Riemann- Liouville fractional integral. Then there holds the formula I α t α γ E δ β,γ at β)) x x γ E δ β,αγ ax β) Theorem Let α>, β>, γ>, and a R. LetD α be the left-sided operator of Riemann- Liouville fractional derivative. Then there holds the formula D α t γ 1 E δ β,γ at β)) x x γ α 1 E δ β,γ α ax β) Theorem Let α>, β>, γ α {α} > 1, and a R. LetD α be the right-sided operator of Riemann-Liouville fractional derivative. Then there holds the formula D α t α γ E δ β,γ at β)) x x γ E δ β,γ α ax β) In a series of papers by Luchko and Yakubovich 69, 7, Luckho and Srivastava 32, Al-Bassam and Luchko 71, Hadid and Luchko 72, Gorenflo and Luchko 24, Gorenflo et al. 73, 74, Luchko and Gorenflo 75, the operational method was developed to solve in closed forms certain classes of differential equations of fractional order and also integral equations. Solutions of the equations and problems considered are obtained in terms of generalized Mittag-Leffler functions. The exact solution of certain differential equation of fractional order is given by Luchko and Srivastava 32 in terms of function 11.1 by using operational method. In other papers, the solutions are established in terms of the following functions of Mittag-Leffler type: if z, ρ, β j C, Rα j >, j 1,...,mand m N then, ) αj ) ) ρ E ρ,β j 1,m ; z k m j1 Γ ) zk a j k β j k! k

19 Journal of Applied Mathematics 19 For m 1, reduces to The Mellin-Barnes integral for this function is given by αj ) ) E ρ,β j 1,m ; z 1 γi 2πiΓ ρ ) γ i ΓsΓ ρ s ) m j1 Γ β j α j s ) z s ds, 1 Γ ρ )H1,1 1,m1 z 1 ρ,1,1,1 β j,α j,,j1,...,m i where, γ<rρ, Rρ >, and the contour separates the poles of Γs from those of Γρ s. Rα j >, j 1,...,m,arg z <π. The Laplace transform of the function defined by is given by where Rs >. ) αj ) L E ρ,β j )1,m ; t 1 ρ, 1, 1, 1 s sγ ρ ) 1 2 ψ m ), αj,β j s 1,m Remark In a recent paper, Kilbas et al. 33 obtained a closed form solution of a fractional generalization of a free electron equation of the form τ Dτ α aτ λ t δ aτ te b ρ,δ1 iνtρ dt βτ σ E γ ρ,σ1 iνtρ, τ 1, i 1, where b, λ C, ν, β R, α>, ρ>, α> 1, ρ> 1, δ> 1, and E b is the generalized ρ,δ1 Mittag-Leffler function given by 11.1,andατ is the unknown function to be determined. Remark The solution of fractional differential equations by the operational methods are also obtained in terms of certain multivariate Mittag-Leffler functions defined below: The multivariate Mittag-Leffler function of n complex variables z 1,...,z n with complex parameters a 1,...,a n,b C is defined as E a1,...,a n,bz 1,...,z n k L 1 L n k L 1,...,L n k L 1,...,L n ) n j1 zl j Γ b ± ) n j1 a jl j j in terms of the multinomial coefficients ) k k! L 1,...,L n L 1! L n!, k,l j, N,j1,...,m Another generalization of the Mittag-Leffler function 1.2 was introduced by Kilbas and Saigo 23, 76 in terms of a special function of the form E α,m,β z c k z k, c 1, c k k k 1 i Γ α im β ) 1 ) Γ α im β 1 ) 1 ), k N {, 1, 2,...}, 11.21

20 2 Journal of Applied Mathematics where an empty product is to be interpreted as unity; α, β C are complex numbers and m R, Rα >, m>, αim β / Z {, 1, 2,...}, i, 1, 2,...and for m 1 the above defined function reduces to a constant multiple of the Mittag-Leffler function, namely E α,1,β z Γ αβ 1 ) E α,αβ1 z, where Rα > andαi β / Z. It is an entire function of z of order Rα 1 and type σ 1/m, see Gorenflo et al. 3. Certain properties of this function associated with Riemann-Liouville fractional integrals and derivatives are obtained and exact solutions of certain integral equations of Abel-Volterra type are derived by their applications 23, 76, 77. Its recurrence relations, connection with hypergeometric functions and differential formulas are obtained by Gorenflo et al. 3, also see, Gorenflo and Mainardi 25. Inorderto present the applications of Mittag-Leffler functions we give definitions of Laplace transform, Fourier transform, Riemann-Liouville fractional calculus operators, Caputo operator and Weyl fractional operators in the next section. 12. Laplace and Fourier Transforms, Fractional Calculus Operators We will need the definitions of the well-known Laplace and Fourier transforms of a function Nx, t and their inverses, which are useful in deriving the solution of fractional differential equations governing certain physical problems. The Laplace transform of a function Nx, t with respect to t is defined by L{Nx, t} Ñx, s e st Nx, tdt, t >, x R, 12.1 where Rs > and its inverse transform with respect to s is given by L 1{ } Ñx, s L 1{ } Ñx, s; t 1 2πi ci c i e st Ñx, sds The Fourier transform of a function Nx, t with respect to x is defined by F{Nx, t} F k, t e ikx Nx, tdx The inverse Fourier transform with respect to k is given by the formula F 1 {F k, t} 1 e ikx F k, tdk. 2π 12.4 From Mathai and Saxena 54 and Prudnikov et al. 55, page 355, it follows that the Laplace transform of the H-function is given by { L t ρ 1 Hp,q m,n } zt σ a p,a p b q,b q s ρ H m,n1 p1,q zs σ 1 ρ,σ,a p,a p, 12.5 b q,b q

21 Journal of Applied Mathematics 21 where Rs >, Rρ σmin 1 j m b j /B j >, σ>, arg z 1 n < 2 πω, Ω >, Ω A i i1 p m A i B j in1 j1 q B j. jm By virtue of the cancelation law for the H-function 54 it can be readily seen that { L 1 s ρ H m,n p,q } zs σ a p,a p t ρ 1 H m,n b q,b q p1,q zt σ a p,a p,ρ,σ, 12.7 b q,b q where σ >, Rs >, Rρ σmax 1 i n 1 a i /A i >, arg z < 1/2πΩ 1, Ω 1 >, Ω 1 Ω ρ. In view of the results 2 J 1/2 x πx cos x 12.8 the cosine transform of the H-function 54, page 49 is given by t ρ 1 coskthp,q m,n at μ a p,a p dt b q,b q π ) ) ) k μ 1 bq,b q, 1 ρ /2,μ/2 k ρ Hm1,n q1,p2 a ) ) ) ) ρ, μ, a ap,a p, 1 ρ /2,μ/2, 12.9 where Rρ μmin 1 j m b j /B j >, Rρ μmax 1 j n a j 1/A j <, arg a < 1/2πΩ, Ω > ; Ω m j1 B j q jm1 B j n j1 A j p jn1 A j,andk>. The definitions of fractional integrals used in the analysis are defined below. The Riemann-Liouville fractional integral of order ν is defined by 78, page 45 D ν t fx, t 1 Γν t t u ν 1 fx, udu, 12.1 where Rν >. Following Samko et al. 16, page 37 we define the Riemann-Liouville fractional derivative for α> in the form d n Dt α fx, t 1 Γn α dt n t t u n α 1 fx, udu, n α 1, where α means the integral part of the number α. From Erdélyi et al. 79, page 182 we have L { D ν t fx, t } s ν Fx, s, 12.12

22 22 Journal of Applied Mathematics where Fx, s is the Laplace transform of fx, t with respect to t, Rs >, Rν >. The Laplace transform of the fractional derivative defined by is given by Oldham and Spanier 8, page 134, L { D α t fx, t} s α Fx, s n s k 1 Dt α k fx, t t, n 1 <α n, k1 In certain boundary value problems arising in the theory of visco elasticity and in the hereditary solid mechanics the following fractional derivative of order α> is introduced by Caputo 81 in the form D α t fx, t 1 Γm α t m fx, t, if α m, tm t τ m α 1 f m x, tdt, m 1 <α m, Rα >, m N where m / t m f is the mth partial derivative of the function fx, t with respect to t. The Laplace transform of this derivative is given by Podlubny 82 in the form L { m 1 Dt α ft; s} s α Fs s α k 1 f k, m 1 <α m k The above formula is very useful in deriving the solution of differintegral equations of fractional order governing certain physical problems of reaction and diffusion. Making use of definitions 12.1 and it readily follows that for ft t ρ we obtain D ν t t ρ Γ ρ 1 ) Γ ρ ν 1 )tρν, Rν >, R ρ ) > 1; t>, D ν t tρ Γ ρ 1 ) Γ ρ ν 1 )tρ ν, Rν <, R ρ ) > 1; t> On taking ρ in12.17 we find that D ν t 1 1 Γ1 ν t ν, t >, Rν < From the above result, we infer that the Riemann-Liouville derivative of unity is not zero. We also need the Weyl fractional operator defined by D μ x 1 t Γ n μ ) dn dt n t u n μ 1 fudu, 12.19

23 Journal of Applied Mathematics 23 where n μ1 is the integer part of μ>. Its Fourier transform is 83, page 59, A.11 { } F Dxfx μ ik μ f k, 12.2 where we define the Fourier transform as h q ) hx exp iqx ) dx Following the convention initiated by Compte 84 we suppress the imaginary unit in Fourier space by adopting a slightly modified form of the above result in our investigations 83, page 59, A.12 { } F Dxfx μ k μ f k, instead of We now proceed to discuss the various applications of Mittag-Leffler functions in applied sciences. In order to discuss the application of Mittag-Leffler function in kinetic equations, we derive the solution of two kinetic equations in the next section. 13. Application in Kinetic Equations Theorem If Rν >, then the solution of the integral equation Nt N c ν D ν t Nt 13.1 is given by Nt N E ν c ν t ν, 13.2 where E ν t is the Mittag-Leffler function defined in 1.1. Proof. Applying Laplace transform to both sides of 13.1 and using it gives Ñs L{Nt; s} N s 1 1 ) s ν c By virtue of the relation L 1{ s ρ} tρ 1 Γ ρ ), R ρ ) >, Rs >, s C, 13.4

24 24 Journal of Applied Mathematics it is seen that L 1 N s 1 1 ) s ν 1 c N 1 k c νk L 1{ s νk 1} k N 1 k c νk t νk Γ1 νk N E ν c ν t ν. k This completes the proof of Theorem Remark If we apply the operator D ν t from the left to 13.1 and make use of the formula D ν t 1 1 Γ1 ν t ν, t >, Rν < 1, 13.7 we obtain the fractional diffusion equation t ν Dt ν Nt N Γ1 ν cν Nt, t >, Rν < whose solution is also given by Remark We note that Haubold and Mathai 46 have given the solution of 13.1 in terms of the series given by The solution in terms of the Mittag-Leffler function is given in Saxena et al. 39. Alternate Procedure We now present an alternate method similar to that followed by Al-Saqabi and Tuan 85 for solving some differintegral equations, also, see Saxena and Kalla 44 for details. Applying the operator c ν m Dt mν to both sides of 13.1 we find that c ν m D mν t Nt c ν m1 D νm1 t Nt N Dt mν 1, m, 1, 2, Summing up the above expression with respect to m fromto, itgives c ν m Dt mν Nt c ν m1 D νm1 t Nt N c ν m Dt mν m m m which can be written as c ν m Dt mν Nt c ν m Dt νm Nt N c ν m Dt mν m m1 m

25 Journal of Applied Mathematics 25 Simplifying the above equation by using the result D ν t t μ 1 Γ μ ) Γ μ ν )tμν 1, where min{rν, Rμ} >, we obtain Nt N c ν m t mν Γ1 mν m for m, 1, 2,... Rewriting the series on the right in terms of the Mittag-Leffler function, it yields the desired result The next theorem can be proved in a similar manner. Theorem If min{rν, Rμ} >, then the solution of the integral equation is given by Nt N t μ 1 c ν D ν t Nt Nt N Γ μ ) t μ 1 E ν,μ c ν t ν, where E ν,μ t is the generalized Mittag-Leffler function defined in 1.2. Proof. Applying Laplace transform to both sides of and using 1.11, itgives Ñs {Nt; s} N Γ μ ) s μ 1 ) s ν c Using relation 13.4, it is seen that { L 1 N Γ μ ) s 1 μ ) s ν 1 } c N 1 k c νk L 1{ s μ νk} k N Γ μ ) t μ 1 1 k c νk t νk Γ μ νk ) k N Γ μ ) t μ 1 E ν,μ c ν t ν. This completes the proof of Theorem Alternate Procedure We now give an alternate method similar to that followed by Al-Saqabi and Tuan 85 for solving the differintegral equations. Applying the operator c ν m Dt mν to both sides of 13.14, wefindthat c ν m D mν t Nt c ν m1 D νm1 t Nt N c ν m Dt mν t μ

26 26 Journal of Applied Mathematics for m, 1, 2,... Summing up the above expression with respect to m from to, itgives c ν m Dt mν Nt c ν m1 D νm1 t Nt N c ν m Dt mν t μ m m m which can be written as c ν m Dt mν Nt c ν m Dt mν Nt N c ν m Dt mν t μ m m1 m Simplifying by using result we obtain Nt N Γ μ ) c ν m t μmν 1 Γ ); mν μ m, 1, 2, m Rewriting the series on the right of in terms of the generalized Mittag-Leffler function, it yields the desired result Next we present a general theorem given by Saxena et al. 4. Theorem If c>, Rν >, then for the solution of the integral equation Nt N ft c ν D ν t Nt where ft is any integrable function on the finite interval,b, there exists the formula Nt cn t H 1,1 1,2 c ν t τ ν 1/ν,1 fτdτ, 1/ν,1, 1,ν where H 1,1 1,2 is the H-function defined by 8.1. The proof can be developed by identifying the Laplace transform of Nc ν Dt ν Nt as an H-function and then using the convolution property for the Laplace transform. In what follows, E δ will be employed to denote the generalized Mittag-Leffler function, defined β,γ by Note 4. For an alternate derivation of this theorem see Saxena and Kalla 44. Next we will discuss time-fractional diffusion. 14. Application to Time-Fractional Diffusion Theorem Consider the following time-fractional diffusion equation: α 2 Nx, t D Nx, t, <α<1, 14.1 tα x2

27 Journal of Applied Mathematics 27 where D is the diffusion constant and Nx, t δx is the Dirac delta function and lim x ± Nx, t. Then its fundamental solution is given by Nx, t 1 x H1, 1,1 x 2 Dt α 1,α 1, Proof. In order to find a closed form representation of the solution in terms of the H-function, we use the method of joint Laplace-Fourier transform, defined by Ñ k, s e stikx Nx, tdx dt, 14.3 where, according to the convention followed, will denote the Laplace transform and the Fourier transform. Applying the Laplace transform with respect to time variable t, Fourier transform with respect to space variable x, and using the given condition, we find that s α Ñ k, s s α 1 Dk 2 Ñ k, s, 14.4 and then Ñ k, s s α 1 s α Dk Inverting the Laplace transform, it yields { N k, t L 1 s α 1 s α Dk 2 } E α Dk 2 t α), 14.6 where E α is the Mittag-Leffler function defined by 1.1. In order to invert the Fourier transform, we will make use of the integral coskte α,β at 2) dt π β,α k H1, k2 1,1, 14.7 a 1,2 where Rα >, Rβ >, k>, a>, and the formula 1 e ikx fkdk 1 2π π fk coskxdk Then it yields the required solution as Nx, t 1 x H2,1 3,3 x 2 Dt α 1,1,1,α,1,1 1,2,1,1,1,1 1 x H1, 1,1 x 2 Dt α 1,α 1,

28 28 Journal of Applied Mathematics Note 5. For α 1, 14.9 reduces to the Gaussian density 1 Nx, t 2πDt 1/2 exp Fractional-space-diffusion will be discussed in the next section. ) x Dt 15. Application to Fractional-Space Diffusion Theorem Consider the following fractional-space-diffusion equation: α Nx, t D Nx, t, <α<1, 15.1 t xα where D is the diffusion constant, α / x α is the operator defined by 12.19, and Nx, t δx is the Dirac delta function and lim x ± Nx, t. Then its fundamental solution is given by Nx, t 1 a x H1,1 2,2 x Dt 1/α 1,1/α,1,1/2 1,1,1,1/ The proof can be developed on similar lines to that of the theorem of the preceding section. 16. Application to Fractional Reaction-Diffusion Model In the same way, we can establish the following theorem, which gives the fundamental solution of the reaction-diffusion model given below. Theorem Consider the following reaction-diffusion model: β t β Nx, t η D α xnx, t, <β with the initial condition Nx, t δx, lim x ± Nx, t, whereη is a diffusion constant and δx is the Dirac delta function. Then for the solution of 16.1 there holds the formula Nx, t 1 a x H2,1 3,3 x ) ηt β 1/α 1,1/α,1,β/α,1,1/2 1,1,1,1/α,1,1/ For details of the proof, the reader is referred to the original paper by Saxena et al. 43. Corollary For the solution of the fraction reaction-diffusion equation t Nx, t η D α xnx, t, 16.3

29 Journal of Applied Mathematics 29 with initial condition Nx, t δx there holds the formula where α>. Nx, t 1 α x H1,1 2,2 x 1,1/α,1,1/2 ) 1/α, 16.4 ηt 1,1,1,1/2 Note 6. It may be noted that 16.4 is a closed form representation of a Lévy stable law. It is interesting to note that as α 2 the classical Gaussian solution is recovered since Nx, t 1 2 x H1,1 2,2 17. Application to Nonlinear Waves x 1,1/2,1,1/2 ) 1/α ηt 1,1,1,1/2 2k1 1 1 k x 2π 1/2 x k! k 2 ηt ) 1/α 4π ηt ) ) 2/α 1/2 exp x 2 4 ηt ). 2/α 16.5 It will be shown in this section that by the application of the inverse Laplace transforms of certain algebraic functions derived in Saxena et al. 43, we can establish the following theorem for nonlinear waves. Theorem Consider the fractional reaction-diffusion equation: D α t Nx, t a D β t Nx, t ν2 D γ xnx, t ζ 2 Nx, t φx, t 17.1 for x R, t>, α 1, β 1 with initial conditions Nx, fx, lim x ± Nx, t for x R, whereν 2 is a diffusion constant, ζ is a constant which describes the nonlinearity in the system, and φx, t is nonlinear function which belongs to the area of reaction-diffusion, then there holds the following formula for the solution of 17.1 Nx, t a r t α βr f k exp kx 2π r E r1 α,α βr1 btα t α β E r1 α,α βr11 btα dk r a r 2π t ζ αα βr 1 φk, t ζ exp ikx 17.2 E r1 α,αα βr bζα dkdζ, where α>βand E δ β,γ is the generalized Mittag-Leffler function, defined by 11.1, and b ν2 k γ ζ 2.

30 3 Journal of Applied Mathematics Proof. Applying the Laplace transform with respect to the time variable t and using the boundary conditions, we find that s α Ñx, s s α 1 fx as β Ñx, s as β 1 fx ν 2 D γ xñx, s ζ2 Ñx, s fx, s If we apply the Fourier transform with respect to the space variable x to 17.3 it yields s α Ñ k, s s α 1 f k as β Ñ k, s as β 1 f k ν 2 k γ Ñ k, s ζ 2 Ñ k, s f k, s Solving for Ñ it gives s α 1 Ñ as β 1) f k f k, s k, s, 17.5 s α as β b where b ν 2 k γ ζ 2. For inverting 17.5 it is convenient to first invert the Laplace transform and then the Fourier transform. Inverting the Laplace transform with the help of the result Saxena et al. 41, equation 28 { } L 1 s ρ 1 s α as β b ; t t α ρ a r t α βr E r1 α,αα βr ρ1 btα, 17.6 r where Rα >, Rβ >, Rρ >, as β /s α b < 1 and provided that the series in 17.6 is convergent, it yields N k, t a r t α βr f k r E r1 α,α βr1 btα t α β E r1 α,α βr11 btα t r a r φ k, t ζζ αα βr 1 E r1 α,α βrα bζα dζ. Finally, the inverse Fourier transform gives the desired solution Generalized Mittag-Leffler Type Functions 17.7 The multi-index m-tuple Mittag-Leffler function is defined in Kiryakova 86 by means of the power series E 1/ρi,μ i z φ k z k z k m j1 Γ ). μ j k/ρ j k k 18.1 Here m>1 is an integer, ρ 1,...,ρ m and μ 1,...,μ m are arbitrary real parameters.

31 Journal of Applied Mathematics 31 The following theorem is proved by Kiryakova 86, page 244 which shows that the multiindex Mittag-Leffler function is an entire function and also gives its asymptotic estimate, order, and type. Theorem For arbitrary sets of indices ρ i >, <μ i <, i 1,...,m the multiindex Mittag-Leffler function defined by 18.1 is an entire function of order m 1 1 ρ, that is, ρ i1 i 1 ρ 1 1 ρ 1 ρ m 18.2 and type σ ρ1 ρ ) ρ/ρ1 ρm ρ ) ρ/ρm Furthermore, for every positive ɛ, the asymptotic estimate E 1/ρi,μ i z < exp σ ɛ z ρ), 18.4 holds for z r ɛ, r ɛ sufficiently large. It is interesting to note that for m 2, 18.2 reduces to the generalized Mittag-Leffler function considered by Dzherbashyan 87 denoted by φ ρ1,ρ 2 z; μ 1,μ 2 and defined in the following form 62, Appendix: E 1/ρ1,1/ρ 2 ;μ 1,μ 2 z φ ρ1,ρ 2 z; μ1,μ 2 ) k z k Γ μ 1 k/ρ 1 ) Γ μ2 k/ρ 2 ), 18.5 and shown to be an entire function of order ρ ρ 1ρ 2 ρ 1 ρ 2 and type σ ρ1 ) ρ2 /ρ 1 ) ρ1 /ρ ρ ρ 2 ρ 1 Relations between multiindex Mittag-Leffler function with H-function, generalized Wright function and other special functions are given by Kiryakova; for details, see the original papers Kiryakova 86, 88. Saxena et al. 38 investigated the relations between the multiindex Mittag-Leffler function and the Riemann-Liouville fractional integrals and derivatives. The results derived are of general nature and give rise to a number of known as well as unknown results in the theory of generalized Mittag-Leffler functions, which serve as a backbone for the fractional calculus. Two interesting theorems established by Saxena et al. 38 are described below. Theorem Let α>, ρ i >, μ i >, i 1,...,m, and further, let I α be the left-sided Riemann- Liouville fractional integral. Then there holds the relation )) ) I α t ρi 1 E 1/ρi,μ i at 1/ρ i x x αρi 1 E 1/ρi,μ 1 α,μ 2,...,μ m at 1/ρ i. 18.7

32 32 Journal of Applied Mathematics Theorem Let α>, ρ i >, μ i >, i 1,...,m, and further, let I α be the right-sided Riemann- Liouville fractional integral. Then there holds the relation )) ) I α t ρi α E 1/ρi,μ i at 1/ρ i x x μ i E 1/ρi,μ 1 α,μ 2,...,μ m at 1/ρ i Another generalization of the Mittag-Leffler function is recently given by Sharma 89 in terms of the M-series defined by p Mq α p Mq α a1,...,a p ; b 1,...,b q ; α; z ) a 1 r a p )r ) b 1 r b q r Γαr, p q 1. 1zr r 18.9 Remark According to Saxena 9, them-series discussed by Sharma 89 is not a new special function. It is, in disguise, a special case of the generalized Wright function p ψ q z, which was introduced by Wright 57, as shown below a1, 1,..., a p, 1 ), 1, 1 κ p1 ψ q1 b 1, 1,..., b q, 1 ), 1,α ; z a 1 r a p )r 1 r zr ) b 1 r b q Γαr 1 r! r a 1 r a p )r ) b 1 r b q Γαr 1zr r p M α q a1,...,a p ; b 1,...,b q ; α; z ), r r 18.1 where q j1 Γ b j ) κ p j1 Γ ). a j Fractional integration and fractional differentiation of the M-series are discussed by Sharma 89. The two results proved in Sharma 89 for the function defined by 18.9 are reproduced below. For Rν > I ν p Mq α a1,...,a p ; b 1,...,b q ; α; z )) x z ν Γ1 ν p1 M α q1 a1,...,a p, 1; b 1,...,b q, 1 ν; α; z ), and for Rν < D ν p Mq α a1,...,a p ; b 1,...,b q ; α; z )) x z ν Γ1 ν p1 M α q1 a1,...,a p, 1; b 1,...,b q, 1 ν; α; z )

33 Journal of Applied Mathematics Mittag-Leffler Distributions and Processes Mittag-Leffler Statistical Distribution and Its Properties A statistical distribution in terms of the Mittag-Leffler function E α y was defined by Pillai 91 in terms of the distribution function or cumulative density function as follows: G y y ) 1 Eα y α ) k1 1 k1 y αk, Γ1 αk <α 1, y> 19.1 and G y y fory. Differentiating on both sides with respect to y we obtain the density function fy as follows: f y ) d dy G ) y y d dy k1 k1 1 k1 y αk Γ1 αk 1 k1 αky αk 1 k1 1 k1 y αk Γ1 αk 1 k y ααk 1 Γα αk k Γ1 αk 19.2 by replacing k by k 1 y α 1 E α,α y α ), <α 1, y>, 19.3 where E α,β x is the generalized Mittag-Leffler function. It is straightforward to observe that for the density in 19.3 the distribution function is that in The Laplace transform of the density in 19.3 is the following: L f t e tx fxdx e tx x α 1 E α,α x α dx 1 t α 1, t α < Note that 19.4 is a special case of the general class of Laplace transforms discussed in 92, Section From19.4 one can also note that there is a structural representation in terms of positive Lévy distribution. A positive Lévy random variable u >, with parameter α is such that the Laplace transform of the density of u> is given by e tα.thatis, E e tu e tα, 19.5 where E denotes the expected value of or the statistical expectation of. When α 1 the random variable is degenerate with the density function 1, for x 1, f 1 x, elsewhere. 19.6

34 34 Journal of Applied Mathematics Consider an exponential random variable with density function e x, for x< f 1 x, elsewhere L f1 t 1 t and with the Laplace transform L f1 t. Theorem Let y> be a Lévy random variable with Laplace transform as in 19.5 and let x and y be independently distributed. Then u yx 1/α is distributed as a Mittag-Leffler random variable with Laplace transform as in Proof. For establishing this result we will make use of a basic result on conditional expectations, which will be stated as a lemma. Lemma For two random variables x and y having a joint distribution, Ex E E x y ) 19.8 whenever all the expected values exist, where the inside expectation is taken in the conditional space of x given y and the outside expectation is taken in the marginal space of y. Now by applying 19.8 we have the following: let the density of u be denoted by gu. Then the Laplace transform of g is given by E e tx1/αy x e tα x But the right side of 19.9 is in the form of a Laplace transform of the density of x with parameter t α. Hence the expected value of the right side is L g t 1 t α which establishes the result. From 19.9 one property is obvious. Suppose that we consider an arbitrary random variable y with the Laplace transform of the form L g t e φt whenever the expected value exists, where φt be such that φ tx 1/α) xφt, lim t φt Then from 19.9 we have E e tx1/αy x e xφt

35 Journal of Applied Mathematics 35 Now, let x be an arbitrary positive random variable having Laplace transform, denoted by L x t where L x t ψt. Then from 19.1 we have L g t ψ φt For example, if y is a random variable whose density has the Laplace transform, denoted by L y t φt, withφtx 1/α xφt, andifx is a real random variable having the gamma density, f x x xβ 1 e x/δ δ β Γ ), x, β>, δ> β and f x x elsewhere, and if x and y are statistically independently distributed and if u yx 1/α then the Laplace transform of the density of u, denoted by L u t is given by L u t 1 δ { φt } β Note 7. Since we did not put any restriction on the nature of the random variables, except that the expected values exist, the result in holds whether the variables are continuous, discrete or mixed. Note 8. Observe that for the result in to hold we need only the conditional Laplace transform of y given x be of the form in and the marginal Laplace transform of x be ψt. Then the result in will hold. Thus statistical independence of x and y is not a basic requirement for the result in to hold. Thus from we may write a particular case as z yx 1/α, where x is distributed as in 19.7, andy as in 19.5 then z will be distributed as in 19.4 or 19.1 when x and y are assumed to be independently distributed. Note 9. The representation of the Mittag-Leffler variable as well as the properties described in Jayakumar 93, page 1432 and in Jayakumar and Suresh 94, page 53 are to be rewritten and corrected because the exponential variable and Lévy variable seem to be interchanged there. By taking the natural logarithms on both sides of we have 1 ln x ln y ln z. α Then the first moment of ln z is available from by computing Eln x and Eln y.but Eln x is available from the following procedure: E e t ln x E e ln x t E x t x t e x dx Γ1 t for R1 t > 19.19