Int. J. Contemp. Math. Sciences, Vol. 7, 212, no. 15, 75-716 The -Mittag-Leffler Function Gustavo Abel Dorrego and Ruben Alejandro Cerutti Faculty of Exact Sciences National University of Nordeste. Avda. Libertad 554 34 Corrientes, Argentina rcerutti@exa.unne.edu.ar Abstract As it is now that the classical Mittag-Leffler function play an important role as solution of fractional order differential and integro-differential equations. We introduce the -Mittag-Leffler function, prove some of its properties and evaluate its Laplace transform. Mathematics Subject Classification: 33E12 Keywords: Mittag-Leffler function, -Gamma function, -Pochhammer symbol I Introduction and Preliminaries The repeated appearance of expressions of the form xx+...x+n 1 that may be interpreted as a generalization of the Pochhammer symbol x n = xx + 1x +2...x + n 1 and the direct relationship between x n and the classical gamma function Γx motivated the definitions introduced by Diaz and Pariguan in 27 see [2] of the Pochhammer -symbol x n, = xx + x +2...x +n 1, γ C, R and n N I.1 and the new gamma function, the -gamma Γ x, one parameter deformation of the classical gamma Γx that admit an integral representation given by Γ z = e t t z 1 dt, R,z C. I.2 The -gamma function Γ z is such that Γ z Γz as 1. cf. [2].
76 G. A. Dorrego and R. A. Cerutti Also, they provided explicit formulae that relate the -gamma function Γ x and the -beta B x, y; later, in 29 M. Mansour in [7] determined the -gamma function as the solution of certain functional equations. Recently in 21 some properties and inequalities of the -gamma, -beta and -zeta function has been studied see [5] and some limit for them by using asymptotic properties of the -gamma function see [6]. In fractional calculus it is well now the importance that have the Mittag- Leffler function definied by the serie E z = z n, >, I.3 Γn +1 which contain as particular case the exponential function e z = z = Γ+1 and admits a first generalization given by the two parameter Mittag-Leffler function defined by E,β z = cf.[4],p.4. From I.3 and I.4 it follows that z n, >,β >. I.4 Γn + β E,1 z =E z I.5 E γ,β In 1971, Prabhaar cf.[1] introduced the Mittag-Leffler type function z definied by E γ,β z = γ n z n Γn + β n! I.6 with, β and γ complex numbers; Re >, Reβ > and γ n the Pochhammer symbol given by γ n = γγ + 1γ +2...γ + n 1 = Γγ + n Γγ I.7 where Γz is the classical Gamma function defined by the following integral Γz = e t t z 1 dt, Rez >. I.8
The -Mittag-Leffler function 77 I.6 is also nown as the three parameter Mittag-Leffer function cf.[11]. For further development of this wor we need to remember elements of fractional calculus as derivatives and integrals of arbitrary orders. Also remember the action of integral transforms such as Laplace and Euler transformation on fractional operators. Definition 1 The Riemann-Liouville integral of order ν of a function f is given by I ν ft := 1 t t τ ν 1 fτdτ, t R +,ν C, Reν >. I.9 Γν cf.[4],p.69. Definition 2 The Riemann-Liouville derivative of order ν C, Reν > of a function f is given by n d D ν ft := I n ν ft, n =[Reν] + 1,t> I.1 dt cf.[4]. Definition 3 Let f : R + R an exponential order function and piecewise continuous, then the Laplace transform of f is L{ft}s := e st ftdt. I.11 The integral exist for Res >. Cf. [1] p. 4. Proposition 1 Laplace transform of the Riemann-Liouville fractional integral. Let C,Re >, then L{I ft}s = L{ft}s s, cf.[9]. I.12 Proposition 2 Laplace transform of the Riemann-Liouville fractional derivative. Let C,m 1 <Re m, then: L{D ft}s =s L{ft}s m 1 j= s j D j 1 f I.13
78 G. A. Dorrego and R. A. Cerutti II Main results In this section we introduce a new Mittag-Leffler type function in the context of the -calculus. Consider some of their properties and the action of Riemann-Liouville operators on it. Definition 4 Let R;, β, γ C; Re >,Reβ >, the -Mittag- Leffler function is defined by the following serie E γ,,β z = γ n, z n Γ n + β n! II.1 where γ n, is the -Pochhammer symbol given in I.1 and Γ x is the - gamma function given in I.2. It may be observed that E γ,,β z is such that Eγ,,β z Eγ,β z as 1, since γ n, γ n,γ z Γz and the convergence of the series in II.1 is uniform on compact subsets. For some particular choice of the parameters γ,, and β we can obtain certain classical functions: E1,1,1 1 z =ez II.2 E 1 1,,1z =E z = with convergence radius r = 1. z n Γ1 = 1 1 z, II.3 E 1 1,,1 z =E z II.4 E 1 1,,β z =E,βz II.5 E γ 1,,β z =Eγ,β z II.6 Relations between the classical Pochhammer symbol and -Pochhammer symbol and between classical Gamma function and the -Gamma function are given in the following
The -Mittag-Leffler function 79 Proposition 3 Let γ C,,s R and n N. Then the following identity holds s n γ γ n,s = II.7 s As particular case Γ s γ = n, s γ s 1 Γ γ s II.8 and γ n, = n γ Γ γ = γ γ 1 Γ n II.9 II.1 Proof. The identities II.7 and II.9 are deduced from I.7 and I.1, and for the Gamma function from I.8 and I.2 Moreover, from Definition II.1 and Proposition 3 it may be obtained the following functional relation between the three parameter Mittag-Leffler function and the -Mittag-Leffler function given by E γ,,β z β γ =1 E 1, β z II.11 or equivalent β 1 E γ,,β 1 az =E γ az, a R. II.12, β Lemma 1 Let, β, γ C, Re >,Reβ >. Then E γ,,β z =βeγ,,β+ z+z d dz Eγ,,β+ z Proof. Starting for the right member of II.13, we have βe γ,,β+ z+z d dz Eγ,,β+ z = = βγ n, z n Γ n n + β + n! + nγ n, z n Γ n n + β + n! = n n + βγ n, z n Γ n + β + n! = n II.13 n + βγ n, z n n + βγ n + βn! = Eγ,,β z
71 G. A. Dorrego and R. A. Cerutti Here we use property of the -Gamma function namely Γ x+ =xγ x, cf. [2],p.183 Lemma 2 Let, β, γ C, Re >,Reβ >. Then E γ,,β z Eγ,,β z =z1 E γ,,+β z II.14 Proof. Using II.9 and II.11 we have E γ,,β z Eγ,,β z = β 1 [ E γ, β 1 γ z E 1 z [ n γ Γ n, β n γ 1 z ] = 1 β n + β n n! 1 = 1 β z n γ n n 1 Γ n + β n 1 n! 1 = 1 β z n+1 γ n +1 n Γ n + + β n n + 1! +β 1 z n γ 2 = z n Γ n + +β n! = z 1 1 β E γ, +β = z 1 E γ,,+β z 1 z n ] Lemma 3 If, β, γ C, Re >,Reβ > and j N. Then j d E γ,,β dz z =γ j,e γ+j,,j+β z II.15 Proof. Using the relation γ n+j, =γ j, γ + j n, which can be proved with the help of II.9 we can write:
The -Mittag-Leffler function 711 j j d d E γ,,β dz z = γ n, z n dz Γ n n + β n! γ n, z n j = Γ n + βn j! = n j n = γ j, γ n+j, z n Γ n + j+β n! n = γ j, E γ+j,,j+β z γ + j n, z n Γ n + j + β n! Lemma 4 Let, β C,and γ be complex number Re >,Reβ > and R. Then x + y n E n+,2,n+β xy = x + y xy r E r+,,2r+β II.16 Proof. Taing into account that n + r, r! r= = r n + r r! = r n n n +1 r r! = r n n Γn + r +1 r Γn +1r! = r n n r +1 n n! r+ = r n n n n! = r n r + n, n! Starting from the left hand member we have: r= x + y n xy r n + r, Γ 2r + n + βr! = = xy r r= r= xy r E r+,,2r+β r + x+y n n, Γ n +2r + βn! x + y
712 G. A. Dorrego and R. A. Cerutti It is well now that the function Et, ν, a =t ν E 1,ν+1 at, cf[8]. II.17 play an important role in the solution of fractional differential equations. Now we will introduce an -analogous of it given by the following Definition 5 Let, β and γ be complex numbers that Re >,Reβ > and Reγ >, >. Let Et,,, β =t β 1 E γ,,β 1 t II.18 Proposition 4 Let, β C, and ν be complex number, Re >, Reβ >, Reν >, >, the Riemann-Liouville integral of order ν of the function Et,,, β is given by I ν Et,,, βx = ν x β +ν 1 E γ,,β+ν x II.19 Proof. If in II.12 we tae a = 1 and mae the following substitution δ = γ, ρ =, μ = β II.2 we have μ 1 E δ,ρ,μ ρ 1 t ρ =E δ ρ,μ tρ, II.21 multiplying both sides by t μ 1 it μ 1 t μ 1 E δ,ρ,μ ρ 1 t ρ =t μ 1 E δ ρ,μ tρ II.22 Now, applying the integral operator Riemann-Liouville fractional order ν both member, is μ 1 I ν [t μ 1 E,ρ,μ δ ρ 1 t ρ ]x = I ν [t μ 1 Eρ,μ δ tρ ]x II.23 = x μ+ν 1 Eρ,ν+μx δ ρ, Cf.[3], II.24
The -Mittag-Leffler function 713 namely I ν [t β 1 E γ,,β 1 t ]x = 1 β x β +ν 1 E γ Then, taing into account II.12 it finally, β+ν x II.25 I ν Et,,, βx = ν x β +ν 1 E γ,,β+ν 1 x. II.26 Following a procedure entirely analogous, but applying the derivative operator Riemann-Liouville fractional in II.23 we can prove the following Proposition 5 Let, β, C and ν be complex number, Re >, Reβ >,n 1 <Reν n, n N. Let D ν be the Riemann-Liouville fractional derivative of order ν. Then hold D ν Et,,, βx = ν x β x ν 1 E γ,,β ν II.27 Lemma 5 Let, β, γ C, Re >,Reβ > and j N. Then j [ d z β 1 E γ,,β dz z ] = j z β j 1 E γ,,β j z II.28 Proof. It is sufficient tae ν = j in Proposition 5 Proposition 6 Let, β, γ C, Re >, Reβ >, Reγ >, Res >, and as / < 1. Then hold L{z β 1 E γ,,β 1 az}s = s β s γ 2 1 β s a γ II.29 Proof. From II.12, multiplying both members by z β 1, we have: β 1 z β 1 E γ,,β 1 az =z β 1 E γ az, β II.3 Taing the Laplace transform to both sides of II.3 result: β 1 L{z β 1 E γ,,β 1 az}s =L{z β 1 E γ az}s, β II.31
714 G. A. Dorrego and R. A. Cerutti and doing again the substitution δ = γ, ρ =, μ = β II.32 its results μ 1 L{z μ 1 E δ,ρ,μ ρ 1 az}s =L{z μ 1 E δ ρ,μ az}s II.33 But the second members of II.33 is equal to: namely s μ 1 as ρ δ, as ρ < 1, Cf.[3],eq.11.8, pp.17 II.34 μ 1 L{z μ 1 E δ,ρ,μ ρ 1 az}s = s ρδ s μ s ρ a δ II.35 Then using II.32 arrive to L{z β 1 E γ,,β 1 az}s = s β s γ 2 1 β s a γ, as / < 1. II.36 Proposition 7 Beta transform If, β, γ, δ C; Re >, Reβ >, Reγ >, Reδ > and R then: 1 Γ δ 1 Proof. Starting the first member μ β 1 1 μ δ 1 E γ,,β zμ dμ = E γ,,β+δ z II.37 1 Γ δ 1 μ β 1 1 μ δ 1 E γ,,β zμ dμ = = [ γn, z n 1 1 ] μ β 1 1 μ δ 1 μ n dμ Γ n n + β Γ δ = [ γn, z n 1 1 ] μ n+ β 1 1 μ δ 1 dμ Γ n δ Γ n + β and applying II.1 = [ γ n, z n 2 + β + δ Γ δ Γ n + β n 1 ] μ n+ β 1 1 μ δ 1 dμ
The -Mittag-Leffler function 715 = [ γ n, z n 1 + β + δ Γ δ Γ n + β B n + β ; δ ]. n By using the well now relations between Gamma and Beta functions we have: 1 1 μ β 1 1 μ δ 1 E γ γ n, z n 1 + β + δ,,β Γ δ zμ dμ = Γ n + β + δ n = γ n, z n Γ n n + β + δ = E γ,,β+δ z References [1] J. Dettman. Applied Complex Variables. Dover Publications, INC. New Yor. 197. [2] R. Diaz and E. Pariguan. On hypergeometric functions and -Pochammer symbol. Divulgaciones Matematicas Vol.15 2. 27. [3] H. J. Haubold, A. M. Mathai and R. K. Saxena. Mittag-Leffler functions and their applications. Journal of Applied Math. 211. [4] A. Kilbas, H. Srivastava and J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier. 26. [5] Ch. Koologiannai. Propierties and Inequalities of generalized -Gamma, Beta and Zeta Functions. Int. J. Contemp. Math. Science, vol 5. 21. [6] V. Krasniqi. A limit for the -Gamma and -Beta Function. Int. Math. Forum, 5. N33. 21. [7] M. Mansour. Determinig the -generalized Gamma Function Γx by Functional Equations. Int. J. Contemp. Math. Science. Vol 4. N21. 29. [8] K. Miller, B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley. 1993. [9] I. Podlubny. Fractional Differential Equations. An introduction to Fractional Derivatives. Academic Press. 1999. [1] T. R. Prabhaar. A singular integral equation with a generalized Mittag- Leffler function in the ernel. Yoohama Math. J., 19. 1971.
716 G. A. Dorrego and R. A. Cerutti [11] A. Soubhia, R. Camargo, E. de Oliveira, J. Vaz. Theorem for series in the three-parametrer Mittag-Leffler function. Fractional Calculus and Applied Analysis. Vol 13. N1. 21. Received: October, 211