Int Journal of Math Analysis, Vol 7, 213, no 11, 535-542 The --Exponential Function Luciano L Luque and Rubén A Cerutti Faculty of Exact Sciences National University of Nordeste Av Libertad 554 34 Corrientes, Argentina lucianoluque@hotmailcom rcerutti@exaunneeduar Abstract A new function of Mittag-Leffler function type, the --Exponential function, es introduced some properties are studied and some integral representations are given Mathematics Subject Classification: 33E12 Keywords: -Fractional Calculus, -Exponential Function I Introduction and Preliminaries It is well now the important role played by the Mittag-Leffler function in the solution of the fractional differential equations A particular and also important case in the function expressed in terms of the Mittag-Leffler function defined by z β 1 E,β λz,z C {} I1, β, λ C, and Re > cf4 Where E,β z denote the Mittag-Leffler function given by the series E,β z z + β ;, β C; Re > I2 The special case for β of the function defined in I1 give a function denoted by e λz and called the -Exponential function, is defined by e az z 1 E, az I3
536 L L Luque and R A Cerutti where z C {}, Re >, and a C According I1we have e z z 1 a z Γ +1 I4 where Re > Hence e az the following properties: lim z is analytic with respect to z C {} and verifies z 1 e az 1 Γ I5 where Re > For n N y a C n d e az dz z n 1 E, n az ; I6 Also, for Res >, a C, and as < 1; the Laplace transform is L{e az } s 1 s a I7 cf4 Diaz and Pariguan cf1 defined, in 27, the Pochhammer -symbol given by x n, xx + x +2x +n 1, I8 where γ C, R and n N; and a new Gamma function, the -gamma Γ x, one parameter deformation of classical gamma function Γx that admit the following integral representation Γ z e t t z 1 dt, R,z C I9 The -gamma function Γ z is such that Γ z Γz as 1 Later Dorrego and Cerutti cf2 introduced the -Mittag-Leffler function given by following definition Definition 1 Let R;, β, γ C; Re >, Reβ >, the -Mittag- Leffler function is defined by the following series E γ,,β z n γ n, z n Γ n + β n! I1 where γ n, is the -Pochhammer symbol given in I8 and Γ x is the - gamma function given in I9cf2
The --exponential function 537 The following relations are valid Lemma 1 If, β, γ C, Re >, Reβ > and j N Then n d E γ,,b dz z γ j,e γ+n,,j+β z I11 Relations between classical Gamma function and the -Gamma function are given in the following Proposition 1 Let γ C,, s R and n N Then the following identity holds s n γ Γ s γ I12 s n, As particular case Γ γ γ γ 1 Γ I13 cf2 Proposition 2 For any a C, >and x < 1, the following identity holds n a n, x n 1 x a n! I14 cf1 II Main Result In this section we introduce a new Mittag-Leffler type function the -- Exponential function in the context of the -calculus; consider some of their basic properties and evaluated its Laplace transform Definition 2 Let, γ C, Re >, Reγ >, Res >, > y z C {}, the --Exponential function is defined by e az γ, z 1 E γ,, az a II1 where E γ,, az a is the -Mittag-Leffler function given in I1
538 L L Luque and R A Cerutti Proposition 3 Let a,, γ C, Re >, Reγ >, > then lim z 1 e az 1 γ, z Γ lim z 1 eγ, az γ, lim z z Γ! + γ n, a n z n Γ n 1 n +1 n! 1 Γ II2 Lemma 2 Let, γ C, Re > y n N Then e az γ, az 1 1 e az γ, + az 1 γ 1, z 1 E γ+,,2 az e az 1 γ, a 1 / w1 / E γ K,,w 1 w 1 / E γ a 1 / K,, w w By Using the Lemma 1 we have } {{ } 1 w 1 1 E γ,, w+w1 w 1 w 1 1 E γ,, w+w1 az a z 1 Replacing and in II3 we have w }{{} E γ K,,w γ 1, E γ+,,2 w II3 e az 1 γ, 1 w 1 a 1 / 1 E γ,, w+w1 γ1, E γ+,,2 w a a z 1 a 1 / z 1 a 1 / az 1 az 1 1 w 1 1 E γ,, w+w1 γ1, E γ+ 1 1 1 w 1 1 E γ,, w+a z 1 w1 a1 / w 1 1 x E γ,, w + az 1 γ 1, a 1 / e az γ, + az 1 γ 1, z 1 E γ+,,2 az a z 1,,2 w γ1, E γ+,,2 w w 1 Eγ+ a1 /,,2 w
The --exponential function 539 Lemma 3 Let, γ C, Re > and n N Then n n j n e az γ, j+1, 1 z 1 n { L l1 n j! n j γ p, E γ+p,,p+1 p 1!p n j! w q1 w q pq } z where: p 1,p 2,, p n j is solution of the equation p 1 +2p 2 ++n jp n j n and L is its number of solutions, p p 1 + p 2 + + p n j q! Maing the change z w a we have: e az γ, z 1 E γ,, az a w 1 E γ K,, a w Applying the Leibniz rule for the derivatives and the chain rule for higher derivatives of a composition of two function, and taing into account Lemma 1, we have n { L l1 { L l1 e az n w γ, a n n j n n j 1 E γ,, w j w 1 n j E γ,, a w z 1 n j+1, 1 p n j! E γ,, p 1!p n j! w w z p 1 w z 2 n n z 1 n j+1, 1 j n j! γ p, E γ+p,,p+1 p 1!p n j! w w z p 1 p2 w p2 z 2 w n j pn j } z n j! w n j pn j } z n j! where: p 1,p 2,, p n j is solution of the equation p 1 +2p 2 ++n jp n j n and L is its number of solutions, p p 1 + p 2 + + p n j
54 L L Luque and R A Cerutti Proposition 4 Let a,, γ C, Re >, Reγ >, Res >, > az < 1 y as / < 1 then L { e az γ, } s s γ 2 1 s s a γ Taing Laplace Transform, applying I13, and then I14 we obtain III L { } e az γ, s n 1 s s γ n, a n Γ n + 1n! γ n, a n n s γ 2 n n!s n+1 1 a γ e sz z n+1 1 dz Some Integral representations of e az γ, In this section we exhibit two integral representations of the -a-exponential function and finally in Corollary 1 is obtained as an integral expression of another function fractional --Exponential Theorem 1 Let a,, γ C, Re >, Reγ >, Res >, > then Consider, e az γ, 1 1 t 1 1 t 1 e atz γ, dt e atz γ, dt Applying II1and substituting u t 1/, we get n z 1 γ n, az n 1 u n+1 1 1 udu Γn + 1 n! n1 Taing into account Beta function definition, we arrived at z 1 n1 γ n, Γn + 1 az n n! n e az γ,
The --exponential function 541 Theorem 2 Let a,, γ C, Re >, Reγ >, Res >, >, then Now consider, γ, 1 t 1 e a1 t z γ, dt III1 e az Applying II1, we have 1 t 1 e a1 t z γ, dt z 1 γ n, az 1 t n+1 1 dt Γn + 1 n! n1 Now, taing into account Beta function definition, we get z 1 n1 n γ n, az Γn + 1 n! n 1 t n+1 1 dt e az γ, Taing into account the following Definition 3 The Riemann-Liouville integral of order of a function f is given by I ft 1 Γ we obtain t x t 1 ftdt, t R +, C,Re > III2 Corollary 1 Let a,, γ C, Re >, Reγ >, >, then e az γ, Γ x t a x a z x a I a eγ, x III3 Applying the change t a + ɛx a to Teorem 2, we have e az γ, x a 1 x t x a 1 Γ e Γ x a Γ x a I a e x x t a x a z γ, a x t a x a z γ, 1 x a dt x t 1 e x x t a x a z γ, 1 x a dt
542 L L Luque and R A Cerutti References 1 Diaz, R; Pariguan, E On hypergeometric functions and -Pochhammer symbol Divulgaciones Matematicas Vol15 227 2 G Dorrego; R Cerutti The -Mittag-Leffler Function IntJ Contemp Math Sciences, no 15,75-76, Vol 7,212 3 HJ Haubold, AM Mathai and R K Saxena Mittag-Leffler functions and their applications Journal of Applied Math211 4 H Kilbas; H Srivastava and J Trujillo, Theory and Applications of Fractional Differential Equations Elsevier26 5 K Miller, B Ross An Introduction to the fractional Calculus and Fractional Differential Equation Wiley 1993 6 I Podlubny Fractional Differential Equations An introduction to Fractional Derivatives Academic Press 1999 7 TR Prabhaar A singular integral equation with a generalizer Mittag- Leffler function in the ernel Yoohama Math J,191971 Received: October, 212