On a generalization of Mittag-Leffler function and its properties

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1 J. Math. Anal. Appl On a genealization of Mittag-Leffle function and its popeties A.K. Shukla, J.C. Pajapati Depatment of Mathematics, S.V. National Institute of Technology, Suat-3957 Gujaat, India Received 22 Septembe 26 Available online 3 Mach 27 Submitted by B.C. Bendt Abstact Let s and z be complex vaiables, Ɣs the Gamma function, and s ν Ɣs+ν Ɣs fo any complex ν the genealized Pochhamme symbol. The pincipal aim of the pape is to investigate the function z γ n z n Ɣαn + β n!, whee α, β, γ C;Reα >, Reβ >, Reγ > and, N. This is a genealization of the exponential function expz, the confluent hypegeometic function Φγ,α; z, the Mittag-Leffle function E α z, the Wiman s function E z and the function E γ z defined by Pabhaka. Fo the function z its vaious popeties including usual diffeentiation and integation, Laplace tansfoms, Eule Beta tansfoms, Mellin tansfoms, Whittake tansfoms, genealised hypegeometic seies fom, Mellin Banes integal epesentation with thei seveal special cases ae obtained and its elationship with Laguee polynomials, Fox H -function and Wight hypegeometic function is also established. 27 Elsevie Inc. All ights eseved. Keywods: Confluent hypegeometic function; Eule tansfom; Fox H -function; Laplace tansfom; Mellin tansfom; Mittag-Leffle function; Whittake tansfom; Wiman s function; Wight hypegeometic function * Coesponding autho. addesses: ajayshukla2@ediffmail.com A.K. Shukla, jyotinda8@ediffmail.com J.C. Pajapati X/$ see font matte 27 Elsevie Inc. All ights eseved. doi:.6/j.jmaa

2 798 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl Intoduction and peliminaies In 93, the Swedish mathematician Gosta Mittag-Leffle 6 intoduced the function E α z, defined as E α z z n Ɣαn +, whee z is a complex vaiable and Ɣs is a Gamma function, α. The Mittag-Leffle function is a diect genealisation of the exponential function to which it educes fo α. Fo <α< it intepolates between the pue exponential and a hypegeometic function z. Its impotance is ealised duing the last two decades due to its involvement in the poblems of physics, chemisty, biology, engineeing and applied sciences. Mittag-Leffle function natually occus as the solution of factional ode diffeential euation o factional ode integal euations. The genealisation of E α z was studied by Wiman 2 in 95 and he defined the function as E z z n Ɣαn + β. α, β C; Reα >, Reβ >,.2 which is known as Wiman s function o genealised Mittag-Leffle function as E α, z E α z. In 97, Pabhaka 7 intoduced the function E γ z in the fom of E γ z γ n z n Ɣαn + β n! whee γ n is the Pochhamme symbol Rainville 8 γ, γ n γγ + γ + 2 γ + n. α, β, γ C; Reα >, Reβ >, Reγ >,.3 The function E γ z is a most natual genealisation of the exponential function expz, Mittag- Leffle function E α z and Wiman s function E z. Goenflo et al. 2,3, Kilbas and Saigo 4,9 investigated seveal popeties and applications of..3. In continuation of this study, we investigate the function z which is defined fo α, β, γ C; Reα >, Reβ >, Reγ > and, N as z γ n z n Ɣαn + β n!,.4 whee γ n Ɣγ+n Ɣγ to n γ + The function denotes the genealized Pochhamme symbol which in paticula educes n if N. z conveges absolutely fo all z if <Re α + and fo z < if Re α +. It is an entie function of ode Re α..4 is a genealization of all above functions defined by Es...3. The following well-known facts ae pepaed fo studying vaious popeties of the function z.

3 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl Eule Beta tansfoms Sneddon : The Eule tansfom of the function fzis defined as B { fz: a,b } z a z b fzdz..5 Laplace tansfoms Sneddon : The Laplace tansfom of the function fzis defined as L { fz } e sz fzdz..6 Mellin tansfom Sneddon : The Mellin tansfom of the function fzis defined as M fz; s z s fzdz f s, Res >,.7 then fz M f s; x 2πi f sx s ds..8 Incomplete Gamma function Rainville 8: This is denoted by γα,zand is defined by γα,z z e t t α dt, Reα >..9 Confluent hypegeometic functions Rainville 8: This is also known as the Pochhamme Banes confluent hypegeometic function and is defined as Φa,b; z F a, b; z whee b o a negative intege is convegent fo all finite z. a n z n b n n!,.

4 8 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl Genealised Laguee polynomials Rainville 8: These ae also known as Sonine polynomials and ae defined as L α n x + α n n; F n! + α; in which n is a non-negative intege. x,. Wight genealised hypegeometic function Sivastava and Manocha : This is denoted by p Ψ and is defined as α,a,...,α p,a p ; z pψ β,b,...,β,b ; p i Ɣα i + A i n z n j Ɣβ j + B j n n! H,p p,+ z α,a,..., α p,a p,, β,b,..., β,b whee Hp, m,n z α,a,...,α p,a p β,b,...,β,b denotes the Fox H -function. Calitz used the following fomula: a + b m m m 2. Basic popeties of the function z.2,.3 a b m..4 As a conseuence of the definitions. to.4 the following esults hold: Theoem 2.. If α, β, γ C, Reα > ; N then z βeγ, + z + αz d dz Eγ, + z, 2.., α z Eγ α z z γ n+ z n Ɣαn + β n!, 2..2 in paticula, E γ α z Eγ α z zeγ z Poof. β + z + αz d dz γ n z n Ɣαn + β + n! β + z + αnγ n z n Ɣαn + β + n!

5 β + z + z, which is 2... Now, A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl , α z Eγ α z αn + β βγ n Ɣαn + β + z n z n n! γ n z n Ɣαn + β n! γ n z n Ɣαn + β n! γ n+ z n Ɣαn + β n!, which is the poof of Substitute in 2..2, which immediately leads to γ n z n Ɣαn + β n! Theoem 2.2. If α, β, γ, w C; Reα, Reβ, Reγ > and N then fo m N, d m z γ me γ +m, +mα z, 2.2. dz d dz m z β wz α z β m m z, Reβ m >, in paticula, d m z β E wz α z β m E m z dz and d m z β Φγ,β; wz Ɣβ dz Ɣβ m zβ m Φγ,β m; wz Poof. Fom.4, d m γ n z n dz Ɣαn + β n! γ n z n m Ɣαn + β n m! nm γ + m n z n γ m Ɣαn + β + αm n! γ m E γ +m, +αm z, which is the poof of Again using.4 and tem-by-tem diffeentiation unde the sign of summation which is possible in accodance with the unifom convegence of the seies in.4 in any compact set of C, we have

6 82 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl d m z β wz α dz γ n Ɣαn + β w n z αn+β m Ɣαn + β Ɣαn + β m n! z β m E m γ, wz α, which yields The elations and follow fom the case of by taking γ and α, espectively. Theoem 2.3. If α p with p, N elatively pime; β,γ C and N then d p p γ n Ɣ np dz p Eγ, p z,β + Ɣ np n + β Ɣ np p + z n p, n! 2.3. in paticula, E z e z γ n,z Ɣ n 2, 3, n Poof. d p p d p γ n z p n dz p Eγ, p z,β dz p Ɣ np + β n! n which gives Putting β γ in 2.3. it educes to d p dz p E p p z n p z Ɣ np p + n z np p Ɣ np p + + n n Ɣ np z np p p + + E p whee the above euation can be witten as d p dz p E p z np p z Ɣ np n Putting p, becomes + E p d dz E z n z Ɣ n + E z. n Multiplying both sides by e z, we get e z d dz E z e z E z e z Ɣ np z p n Ɣ np + p z, γ n Ɣ np + + β Ɣ np p + z n p p z n z n Ɣ n, n!,

7 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl witing above euation as d dz e z E z e z n which can also be witten as e z E z Ɣ n n z n Ɣ n, z e z z n dz and applying.9 gives Theoem 2.4. If α, β, γ C; Reα >, Reβ >, Reγ >, Reδ > and N then Ɣδ u β u δ zu α du +δ z If α, β, γ, δ, λ C; Reα >, Reβ >, Reγ >, Reδ > and N then Ɣδ x t x s δ s t β λs t α ds x t δ+β E+δ γ, λx t α If α, β, γ, δ, μ, ν, λ C; Reα >, Reβ >, Reγ >, Reδ >, Reμ >, Reν > and then x t v x t μ E γ, wx t α Eα,v δ, wt α dt x μ+v Eα,μ+v γ +δ, wx α If α, β, γ C; Reα >, Reβ >, Reγ > and N then z In paticula, and z z t β wt α dt z β E+ γ, wz α t β E wt α dt z β E + wz α t β Φγ,β; wtdt zβ Φγ,β + ; wz β

8 84 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl Poof. Ɣδ u β u δ zu α du Ɣδ γ n z n Bαn + β,δ Ɣαn + β n! γ n z n Ɣαn + β + δ n! +δ z, which is the poof of Change the vaiable fom s to u x t s t. Then the LHS of becomes Ɣδ x t δ u δ x t β u β x t x tδ+β Ɣδ γ n Ɣαn + β simplification of the above euation yields Conside x λx t α n Bαn + β,δ; n! t ν x t μ wx t α Eα,ν δ, wt α dt x γ n λ n x t αn u αn du Ɣαn + β n! γ n δ k w n+k t αk+v x t αn+μ dt Ɣαn + μ Ɣαk + v n!k! k x μ+v γ n δ k w n+k Ɣαn + μ Ɣαk + v n!k! xαn+αk Bαk + v,αn + μ k x μ+v γ n δ k wx α n+k Ɣαk + v + αn + μn!k! k n n γ x μ+v n k δ k wx α n. k Ɣαn + μ + v n! k Substituting and using.4, the above euation becomes Kilbas et al. 5 x μ+v γ + δ n wx α n Ɣαn + μ + v n! x μ+v Eα,μ+v γ +δ, wx α, which is the poof of 2.4.3, and z t β wt α dt γ n w n Ɣαn + βn! z t αn+β dt z β + wz α,

9 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl which is the poof of follows fom with γ and putting α in leads to Genealized hypegeometic function epesentation of z Using.4, taking α k N and N we have k,β z γ n z n Ɣkn + β n! Ɣβ Ɣβ γ n z n β kn n! γ +i i n Ɣβ F k Ɣβ F k kj β+j z k k n n! k n γ +,..., ; β k, β+ β+k k,..., k ; γ, γ + Δ; γ; z Δk; β; k k z k k, 3. whee Δ; γis a -tuple γ, γ + γ +,..., ; Δk; β is a k-tuple β k, β+ k Convegence citeia fo genealized hypegeometic function F k : β+k,..., k. i If k, the function F k conveges fo all finite z. ii If k +, the function F k conveges fo z < and diveges fo z >. iii If >k+, the function F k is divegent fo z. iv If k +, the function F k is absolutely convegent on the cicle z if k Re j β + j k i γ + i >. 4. Mellin Banes integal epesentation of z Theoem 4.. Let α R + ; γ,δ Cγ and N. Then the function z is epesented by the Mellin Banes integal as z ƔsƔγ s z s ds, 4.. 2πiƔγ Ɣβ αs L whee agz <π; the contou of integation beginning at i and ending at +i, and indented to sepaate the poles of the integand at s n fo all n N to the left fom those at s γ +n fo all n N to the ight. Poof. We shall evaluate the integal on the RHS of 4.. as the sum of the esidues at the poles s,, 2,...Wehave

10 86 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl πi L ƔsƔγ s z s ds Ɣβ αs Re s s n lim s n ƔsƔγ s z s Ɣβ αs πs + n sin πs Ɣγ s Ɣ s Ɣβ αs z s n Ɣγ + n Ɣ + n Ɣβ + αn zn Ɣγ γ n z n Ɣαn + β n! ƔγEγ, z. 5. Integal tansfoms of z In this section, we discussed some useful integal tansfoms like Eule tansfoms, Laplace tansfoms, Mellin tansfoms, Whittake tansfoms. Theoem 5. Eule Beta tansfoms. z a z b xz σ dz Ɣb γ,, a, σ ; x Ɣγ 2 Ψ 2, 5.. β, α, a + b,σ; whee a,b,,γ,σ C; Rea >, Reb >, Reα >, Reβ >, Reγ >, Reσ > and N. Poof. z a z b xz σ dz which is the poof of 5... Remak. Putting γ in 5.., we get γ n x n Bσn + a,b Ɣαn + β n! Ɣb γ,, a, σ ; x Ɣγ 2 Ψ 2, β, α, a + b,σ; z a z b E xz σ,, a, σ ; x dz Ɣb 2 Ψ β, α, a + b,σ; If a β and α σ then 5.. educes to z β z b σ,β xz σ dz Ɣb σ,β+b x. 5..3

11 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl with α β γ gives and z a z b exp xz σ a, σ,, ; x dz Ɣb 2 Ψ 2,, a + b,σ; 5..4 z a z β x z α dz Ɣa α,a+β x Theoem 5.2 Laplace tansfoms. z a e sz xz σ dz s a Ɣγ 2 Ψ γ,, a, σ ; x s σ, 5.2. β, α; whee a,σ,,γ C; Reα >, Reβ >, Reγ >, Rea >, Res >, Reσ > and x s σ <. Poof. z a e sz xz σ dz which is the poof of s a Ɣγ e sz z a s a Ɣγ 2 Ψ Remak. Special cases of 5.2. ae given below. Substituting a β,, σ α in 5.2., we get x n z σn γ n Ɣαn + βn! dz Ɣγ + nɣa + σn x Ɣβ + αnn! s σ γ,, a, σ ; x s σ, β, α; z β e sz E γ xz σ dz s a xs σ γ Taking a β, σ α, γ, x ±y, 5.2. educes to e sz z β E ±yz α ±y dz s β s α s β ±y s α, n whee ±y s α <, sα β s α y, whee Res > y α n

12 88 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl Putting β y in 5.2.3, we get L { E ±z α }, Res >, s s α and using 5.2.3, we have e st t β E k ±yt α dt, whee E k dk ±yz denotes E dy k ±yz,k,, 2,..., dk dy k e st t β E ±yt α dt dk s α β dy k s α y ±k k!s α β s α y k+, whee Res > y α Putting α β 2 in 5.2.4, we get e st t k 2 E ±y tdt ±k k! 2, 2 s y k+, whee Res > y Theoem 5.3 Mellin tansfoms. t s wtdt ƔsƔγ s w s ƔγƔβ αs, 5.3. whee α, β, γ, s C; Reα >, Reβ >, Reγ >, Res > and N. Poof. Putting z wt in 4.., we get wt ƔsƔγ s wt s ds 2πiƔγ Ɣβ αs whee 2πiƔγ L L f st s ds, f ƔsƔγ s s w s ƔγƔβ αs, using.7,.8 and 5.3.2, which immediately leads to To obtain Whittake tansfom, we use the following integal:

13 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl e t/2 t v W λ,μ t dt Ɣ 2 + μ + v Ɣ 2 μ + v whee Rev ± μ > 2. Theoem 5.4 Whittake tansfoms. Ɣ λ + v t ρ e 2 pt W λ,μ pt wt δ dt p ρ γ,, Ɣγ 3 Ψ 2 ± μ + ρ,δ; 2 β, α, λ + ρ,δ;, w p δ, 5.4. whee α, β, γ, ρ, δ C; Reα >, Reβ >, Reγ >, Reρ >, Reσ > and N. Poof. Substituting pt v in LHS of 5.4. educes to v ρ e v 2 Wλ,μ v p p ρ Ɣγ p ρ Ɣγ p ρ Ɣγ 3 Ψ 2 Ɣγ + n w n Ɣβ + αn p δ n! γ n w n v δn Ɣαn + βn! p p dv β, α, λ + ρ,δ; which is the poof of v δn+ρ e v 2 Wλ,μ v dv Ɣγ + n w n Ɣ 2 + μ + ρ + δn Ɣ 2 μ + ρ + δn Ɣβ + αnn! p δ Ɣ λ + δn + ρ γ,, w 2 ± μ + ρ,δ; p δ, 6. Relationship with some known special functions genealised Laguee polynomials, Fox H -function, Wight hypegeometic function 6.. Relationship with genealised Laguee polynomials Putting α k, β μ +, γ m, N with m and eplacing z by z k in.5, we get E m, m k,μ+ z k m n z kn Ɣkn + μ + n! m n m! m n! Ɣkn + μ + m Ɣm + Ɣkm + μ + Ɣm + n m n! z kn n! Ɣkm + μ + z kn Ɣkn + μ + n! Ɣkm + μ + Zμ m z, k, 6..

14 8 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl whee Z μ m z, k is a polynomial of degee m in zk. In paticula, Z m μ z, L μ m z, so that E m Ɣm + k,μ+ z Ɣkm + μ + Lμ m z Relationship with Fox H -function Using 4.., we get z ƔsƔγ s z s ds 2πiƔγ Ɣβ αs L Ɣγ H,,2 z γ,. 6.2.,, β,α 6.3. Relationship with Wight hypegeometic function If, then.4 can be witten as z Ɣγ + n z n Ɣγ Ɣβ + αn n! ; 6.3. fom.2 and 6.3., we get Acknowledgment Ɣγ Ψ γ, ; z β, α; Authos ae thankful to the efeee fo valuable suggestions Refeences L. Calitz, Some expansion and convolution fomulas elated to Mac Mohan s maste theoems, SIAM J. Math. Anal R. Goenflo, A.A. Kilbas, S.V. Rogosin, On the genealised Mittag-Leffle type function, Integal Tansfoms Spec. Funct R. Goenflo, F. Mainadi, On Mittag-Leffle function in factional evaluation pocesses, J. Comput. Appl. Math A.A. Kilbas, M. Saigo, On Mittag-Leffle type function, factional calculus opeatos and solution of integal euations, Integal Tansfoms Spec. Funct A.A. Kilbas, M. Saigo, R.K. Saxena, Genealised Mittag-Leffle function and genealised factional calculus opeatos, Integal Tansfoms Spec. Funct G.M. Mittag-Leffle, Su la nouvelle fonction E α x, C. R. Acad. Sci. Pais T.R. Pabhaka, A singula integal euation with a genealized Mittag-Leffle function in the kenel, Yokohama Math. J E.D. Rainville, Special Functions, Macmillan, New Yok, M. Saigo, A.A. Kilbas, On Mittag-Leffle type function and applications, Integal Tansfoms Spec. Funct I.N. Sneddon, The Use of Integal Tansfoms, Tata McGaw Hill, New Delhi, 979.

15 A.K. Shukla, J.C. Pajapati / J. Math. Anal. Appl H.M. Sivastava, H.L. Manocha, A Teatise on Geneating Functions, John Wiley and Sons/Ellis Howood, New Yok/Chicheste, A. Wiman, Übe den fundamental Satz in de Theoie de Funktionen E α x, Acta Math