ON EXTENSION OF MITTAG-LEFFLER FUNCTION

Available athttp://pvamu.edu/aamAppl. Appl. Math.ISSN: 1932-9466Applications andApplied Mathematics:An International Journal(AAM)Vol. 11, Issue 1 (June 2016), pp. 307 - 316ON EXTENSION OF MITTAG-LEFFLER FUNCTIONEkta Mittal1, Rupakshi Mishra Pandey2 and Sunil Joshi31Department of MathematicsThe IIS UniversityJaipur, Rajasthan, Indiaekta.jaipur@gmail.com2Department of MathematicsAmity University Uttar PradeshNoida, Indiarup ashi@yahoo.com3Department of Mathematics and StatisticsManipal UniversityJaipur, Rajasthan, India.sunil.joshi@jaipur.manipal.eduReceived: May 5, 2015; Accepted: March 21, 2016AbstractIn this paper, we study the extended Mittag -Leffler function by using generalized beta functionand obtain various differential properties, integral representations. Further, we discuss Mellintransform of these functions in terms of generalized Wright hyper geometric function andevaluate Laplace transform, and Whittaker transform in terms of extended beta function. Finally,several interesting special cases of extended Mittag -Leffler functions have also be given.Keywords: Beta Functions; Mellin Transform; Laplace Transform; Whittaker Transform;Extended Riemann Liouville Fractional derivative operatorMSC 2010 No. : 33E12, 44A10307

308Ekta Mittal et al.1. IntroductionThe Mittag-Leffler function occurs naturally in the solution of fractional order and integralequation. The importance of such functions in physics and engineering is steadily increasing.Some application of the Mittag-Leffler is carried out in the Study of Kinetic Equation, Study ofLorenz System, Random Walk, Levy Flights and Complex System and also in applied problemssuch as fluid flow, electric network, probability and statistical distribution theory. Haubold et al.(2011) studied various properties of Mittag-Leffler function.In 1903, the Swedish mathematician Gosta Mittag-Leffler introduced the function.Eα z zn, n 0 Γ n 1 (1)where z is a complex variable, is a Gamma function, and α 0 . It is a direct generalization ofthe exponential function for 1 , and for 0 α 1, it interpolates between exponential andhypergeometric function 1/(1-z). The generalization of E(zα )(1905) as:E α,β z was further studied by Wimanzn n , , C, Re 0, Re 0 ,n 0 Γ(2)γPrabhakar (1971) introduced the function E α,β(z ) in as:γE α,β z where γ n z n, , , γ C, Re 0, Re 0, Re γ 0 , n 0 Γ n n! (3)is the Pochhamer symbol by Rainville (1960) such thatΓ γ n γ n Γ γ , γ 0 1, γ n γ γ 1 γ 2 . γ n - 1 , for n 1.γ,qShukla and Prajapati (2007) introduced the extended Mittag-Leffler function E α,β z which isdefined as follows:γ,qE α,β z γ qnzn n 0 Γ n β n!, , , γ C, Re 0, Re 0, Re γ 0, q 0,1 N ,(4)

AAM: Intern. J., Vol. 11, Issue 1 (June 2016)309γ,qE α,β z Converge absolutely z 1, if q Re 1, z and for Beta function definedas:1 1B , uα -1 1 u du 0Γ Γ , Re α 0, Re β 0.Γ (5)Laplace transform of the function f (z) is defined as: L f z ; s e - sz f z dz .(6)0Mellin-transform and its inverse Transform of the function f (z) is defined as:M f z ; s f* s z s-1 f z dz,0f z M -1 f * s ; z Re s 0 ,1f * s z -s ds .2πi L (7)(8)Whittaker transforms (Whittaker and Watson (1996)) is defined as: 1 1 Γ μ v Γ - μ v 2 2 ,-t/2v-1 0 e t Wλ,μ t dt Γ 1 - λ v where Re μ v -(9)1and w λ,μ t is the Whittaker confluent hyper geometric function.2Wright generalized hypergeometric function is defined as:p ψq a , A ,., a , A ; b , B ,., b , B ; z pp11qq 1 1 P n 0 Γ ai Ai n i 0q Γ b j B jj 0n .zn.n!The classical Riemann-Liouville fractional derivative of order is usually defined by1zDzμ f z f t z t Γ μ 0 μ-1dt, Re μ 0 ,(10)

310Ekta Mittal et al.where the integration path is a line from 0 to z in the complex t-plane. For thecase m 1 Re μ m m 1,2,3,. , it is defined byd m μ-mD zμ f z D f z dz m zz dm 1-μ m- 1 f t z - t dt . mdz Γ -μ m 0 The extended Riemann-Liouville fractional derivative operator was defined by Özarslan andÖzergin (2010) as follows:-μ-1 -pz 2 1 zDzμ,p f z f t z t exp dt , Γ -μ 0tz t (11)Re μ 0,Re p 0 and for m 1 Re μ m, m 1,2,3,. .dmDzμ,p f z m D zμ-m f z dz dmdz mz -pz 2 1-μ m-1ftz texp dt , t z t Γ -μ m 0(12)where the path of integration is a line from 0 to z in the complex t-plane. For the case p 0, weobtain the classical Reimann-Liouville fraction derivative operator.2. Main ResultThe extended Mittag-Leffler function can be given as:E α, γ,cp ; q z ; p B p γ nq, c - γ c nqzn. B γ, c - γ Γ αn β n! ,n 0 (13)p 0 , Re c 0 , Re γ 0 , q Re α 1,where10 -p y -1 exp u 1 - u du , Re p 0, Re x 0,Re y 0.B p x,y u x -1 1- u(14)The above Mittag-Leffler function can be derived by using the following relations which aregiven by Chaudhary et al. (2004), Chaudhary and Zubair (2001):

AAM: Intern. J., Vol. 11, Issue 1 (June 2016)311 γ nq B γ nq, c - γ ,B γ, c - γ c nqγ,qE α,β z γ qn c nq z n. n 0 Γ αn β c nq(15)n!Now, we state some theorems on this function3. Fractional Derivative of Extended Mittag-Leffler functionTheorem 1.Let p 0 , Re μ 0 , Re λ 0 , Re α 0, Re β 0, where q Re α 1, thenD zλ- μ,p z λ-1 E α, μβ ; q z q ; p z ; q B λ, μ - λ E α, λ,μ z q ; pβΓ μ - λ μ-1 ,(16)Proof:Taking L.H.S. and using the extended Riemann-Liouville fractional derivativeμ,p z λ-1 E μ ;q z q ; p Dλ zα,β z -p z 2 1-λ μ-1λ-1 E μ ;q t q ; ptztexp t z t dt ,α,βΓ μ - λ 0 on putting t zu and using Equation (4) , we obtain μ kq z qk u qk -p z u-1 1 λ-1-λ μ-1u1uexpdu . u 1- u Γ αk βΓ μ - λ 0 k 0 k! After changing the order of summation and integration, we get the desired result.Theorem 2. dnE γ, c ;dz n α, βq z; p γ qn E α,γ β n q,n qc n q z; p .(17)

312Ekta Mittal et al.Proof:Taking first derivative with respect to z of Equation (14), we getd γ, c ; qE z; p dz α, β n 0 n 0 B p γ n 1 q,c - γ c n 1 qzn.B γ,c - γ Γα n 1 β n!B p γ q nq,c - γ c q c q nqΓ α n 1 β B γ,c - γ .znn! γ q E α, γβ q; αc q ; q z; p .Further, applying the process n times, we get the result.Corollary 1.For the extended Mittag-Leffler function, the following differentiation formula holds:d n β-1 γ; c ; qγ; c ; qz E α, βλ z α ; p z β - n -1 E α, β - n λ z α ; p , n dz in Equation (17). We put z λz α and multiply by z β -1 . Then taking z derivative n times, weget required result.Similarly, another interesting derivative formula for the extended Mittag-Leffler function isE α, γ, βc ; q z; p β E α, γ,βc ;1q z; p α zdE α, γ,βc ;1q z; p ,dz(18)4. Integral Representation of Extended Mittag-Leffler functionTheorem 3.For Extended MittagLeffler function, we have(Y,C);qEα,β( Z ; p) 11B γ,c - γ 0t γ -1c-γ - 1 1- t e-pt 1-t E α,c; βq t q z dt ,where p 0 , Re c 0 , Re γ 0 , Re α 0 , Re β 0 , q Re α 1.(19)

AAM: Intern. J., Vol. 11, Issue 1 (June 2016)313Proof:Using Equation (14) in Equation (13), we get γ,c ;qE α,β k 01c-γ -1 t γ kq-1 1- t e0-pt 1- t c kq1zk. dt.B γ,c - γ Γ αk β k!On interchanging order of summation and integration in the above equation, we get1 0t γ -1c-γ - 1 1- t e-k c kqtqz .dt , k 0 B γ,c - γ Γ αk β k!p t 1-t (20)and using Equation (4) in Equation (20), we get the desired result.Corollary 2. γ,c ;qE α,β z; p 1B γ,c - γ u γ -1 0 u 1 c -p 1 u 2 c;q E α,βu exp. The above result can be prove by putting t u q z; p du. 1 u (21)uin Theorem 3.1 uCorollary 3. γ,c ;qE α,β z; p π / 2 2γ-1-p 2 sin θ cos 2 c-γ 1θ exp. 22B γ,c - γ 0 sin θcos θ1 c;q2q .E α,β sin θz dθ. (22) Taking t sin 2 θ in theorem 3, we get above trigonometric form of Extended Beta function.5. Integral transform of extended Mittag-Leffler functionTheorem 4.The Mellin transform of the extended Mittag-Leffler function is given by γ,c ;qM E α,β z; p ; s c,q , γ s,q ΓsΓ c s - γ ; z .2Ψ 2 Γγ Γ c - γ β,α , c 2s,q (23)

314Ekta Mittal et al.Proof:We start with γ,c ;q γ,c ;qM E α,β z; p ; s p s-1E α,β z; p dp, (24)0and using Equation (19) in Equation (24), we get γ,c ;qM E α,β z; p ; s 1 B γ,c - γ 0 1p s-1 0t γ -1 Now, changing the order of integration and putting u c-γ - 1 1- t pt(1 - t)e-p t 1- t c;q E α,β t q z dt dp. , we get γ,c ;qM E α,β z; p ; s 1 c kq t q z dt .Γsγ s- 1 1- t c s-γ - 1 . t. k!B γ, c - γ 0k 0 Γ αk β kFinally, using Beta function, [Equation (5)] in the above case, we get the desired result.Corollary 4.Taking s 1 in Equation (23), we get γ,c ; q 0 E α,β c,q , γ 1,q Γ c 1- γ ; z . z; p dp Γγ Γ c - γ 2Ψ 2 β,α , c 2,q (25)Corollary 5.Taking Inverse Mellin transform γ,c ;qE α,β z; p dp 112πi ΓγΓc - γγ i γ-i (26) c,q , γ s,q ; z .p -s ds, β,α , c 2s,q ΓsΓ c s - γ 2Ψ 2

AAM: Intern. J., Vol. 11, Issue 1 (June 2016)315Similarly, Laplace and Whittaker transform [4] of Extended Mittag-Leffler function are asfollows: γ,c ;q γ,c ;qL z b-1E α,β z q ; p ; s z b-1e -sz E α,β z q ; p dz 0 Γb B p γ kq,c - γ c kq b k 1. k,k!s b k sB γ,c - γ 0(27)and γ,c ;q 0 e -ft / 2 t ε-1Wλ,Ψ f t E α,β w t η dt 1 1 Γ ε ηk ψ k Γ ε ηk - ψ B p γ kq,c - γ c kq2 2 1 ω ε f . η .B γ,c - γ Γ αk β k! f Γ ε ηk - λ 1 k 0 (28)Using the definition of extended Mittag-Leffler function [Equation (14)] in middle term ofEquation (27), and changing the order of integration and summation, and using LaplaceTransform Γn 0 e -st t n-1dt s n,we get the required result. Taking L.H.S. of Equation (28) and put ft v and using definition ofExtended Mittag-Leffler function changing order of summation and integration and usingEquation (9) we get the desired result.6. Special Cases(i)If we put q 1, in above Theorems 1, 3, and 4, it reduces to the result given recently byÖzarslan and Yilmaz (2014).(ii)If we put c q 1, andfunction (1971) .p 0,in the above Theorems 1, 3, and 4, it reduces to Prabhakar