A GENERALIZATION OF MITTAG-LEFFLER FUNCTION AND INTEGRAL OPERATOR ASSOCIATED WITH FRACTIONAL CALCULUS

Journl of Frctionl Clculus nd Applictions, Vol. 3. July 212, No. 5, pp. 1-13. ISSN: 29-5858. http://www.fcj.webs.com/ A GENERALIZATION OF MITTAG-LEFFLER FUNCTION AND INTEGRAL OPERATOR ASSOCIATED WITH FRACTIONAL CALCULUS TARIQ O. SALIM, AHMAD W. FARAJ Abstrct. This pper is devoted for the study of new generlized function of Mittg-Leffler type. Its vrious properties including differentition, Lplce trnsform, Bet trnsform, Mellin trnsform, Whittker trnsform, generlized hypergeometric series form, Mellin-Brnes integrl representtion nd its reltionship with Fox s H-function nd Wright hypergeometric function re investigted nd estblished. Further properties of generlized Mittg- Leffler function ssocited with frctionl differentil nd integrl opertors re considered. Also n integrl opertor ssocited with frctionl clculus opertors is studied 1. Introduction The Swedish mthemticin Mittg-Leffler 5 introduced the function E α z defined s E α z Γαn + 1 where z C nd Γs is the Gmm function; α. The Mittg-Leffler function is direct generliztion of expz in which α 1. Mittg - Leffler function nturlly occurs s the solution of frctionl order differentil eqution or frctionl order integrl equtions. A generliztion of E α z ws studied by Wimn 14 where he defined the function E α,β z s E α,β z Γαn + β α, β C; Rα >, Rβ > which is lso known s Mittg-Leffler function or Wimn s function. Prbhkr 6 introduced the function E γ α,β z in the formsee lso Kilbs et l. 4 1 2 2 Mthemtics Subject Clssifiction. 33E12, 65R1, 26A33. Key words nd phrses. Generlized Mittg-Leffler function; frctionl clculus opertors; integrl trnsforms; integrl opertors. Submitted Jn. 3, 211. Published July 1, 211. 1

2 TARIQ O. SALIM, AHMAD W. FARAJ JFCA-212/3 E γ α,β z γ n Γαn + β n! α, β, γ C; Rα >, Rβ >, Rγ > Shukl nd Prjpti 1 see lso Srivstv nd Tomovski 13 defined nd investigted the function E γ,q α,β z s E γ,q α,β z γ qn Γαn + β n! where α, β, γ C; Rα >, Rβ >, Rγ >, q, 1 N nd γ qn Γγ+qn Γγ denotes the generlized Pochhmmer symbol which in prticulr reduces to q qn q γ+r 1 r1 q if q N n A new generliztion of Mittg-Leffler function ws defined by Slim 8 s E γ,δ α,β z γ n 3 4 5 Γαn + β δ n where α, β, γ, δ C; Rα >, Rβ >, Rγ >, Rδ > In this pper, we introduce new generliztion of Mittg-Leffler function defined s z γ qn 6 Γαn + β δ pn where α, β, γ, δ C; min{rα, Rβ, Rγ, Rδ} > ; p, q > nd q Rα + p 7 Eqution 6 is generliztion of equtions 1-5. Setting p q 1, it reduces to Eq. 5 defined by Slim 8. Setting δ p 1, it reduces to Eq. 4 defined by Shukl nd Prjpti 1, in ddition of tht if q 1, then we get Eq. 3 defined by Prbhkr 6. On putting γ δ p q 1 in 6 it reduces to Wimn s function, moreover if β 1, Mittg-Leffler function E α z will be the result. Some recurrence reltions, derivtion formuls, Lplce trnsform, Bet trnsform, Mellin-Brnes integrl of z will be estblished, lso its reltionship to Fox s H-function nd Wright hypergeometric function will be estblished. The integrl opertor defined by,w, + x x x t β 1 wx tα φtdt 8 which contins the generlized Mittg-Leffler function 6 in its kernel is investigted nd its boundedness is proved under certin conditions. Theorems of composition of frctionl clculus opertors I λ φ x 1 Γλ x x t λ 1 φtdt λ C, Rλ > 9

JFCA-212/3 A GENERALIZATION OF MITTAG-LEFFLER FUNCTION 3 nd D λ φ n d x I n λ φx n Rλ + 1 1 with integrl opertors defined in 8 re given nd proved. As mtter of fct if w, q 1 nd p 1, then the integrl opertor corresponds essentilly to the Riemnn-Liouville frctionl integrl opertor defined in 9. The generlized frctionl derivtive opertor D u,v φ known s Hilfer s frctionl derivtive see Hilfer 2 is written s D u,v φ x I v1 u d I 1 v1 u φ x 11 D u,v yields the clssicl Riemnn-Liouville frctionl derivtive D u when v ; lso if v 1 it reduces to Cputo frctionl derivtive. Throughout this pper, we need the following well-known fcts nd rules. Bet trnsform Sneddon 11 B{fz;, b} 1 Lplce trnsform Sneddon 11 z 1 1 z b 1 fzdz, R >, Rb > 12 L{fz; s} e sz fzdz, Rs > 13 Convolution theorem of Lplce trnsform Finney et l. 1 t L f g s L{ Mellin trnsform Sneddon 11 ft ξfξdξ} L f sl g s; L{ tn 1 Γn ; s} 1 s n, n > 14 M{fx; s} f s nd the inverse Mellin trnsform is given by fz M 1 {f s; z} 1 2πi c+i c i Confluent hypergeometric function Rinville 7 Φ, b, z 1 F 1, b, z z s 1 fzdz 15 z s f sds, c R 16 n b n n! 17

4 TARIQ O. SALIM, AHMAD W. FARAJ JFCA-212/3 Wright generlized hypergeometric function Srivstv nd Mnoch 12. p 1, A 1,..., p, A p Γ i + A i n pψ q b 1, B 1,..., b q, B q ; z i1 18 q n! Γb j + B j n Fox s H-function Kilbs nd Sigo 3 Hp,q m,n z 1, α 1,..., p, α p b 1, β 1,..., b q, β q m 1 Γb j + β j s n Γ1 j α j s j1 j1 z 2πi q p s ds 19 L Γ1 b j β j s Γ j + α j s jm+1 jn+1 The generlized hypergeometric function Rinville 7 p α i n i1 pf q α 1,..., α p ; β 1,..., β q ; z q zn n! β j n j1 j1 Whittker trnsform Whittker nd Wtson 15 e t/2 t v 1 W λ,µ tdt Γ 1 2 + µ + vγ 1 2 µ + v Γ1 λ + v where Rµ ± v > 1/2 nd W λ,µ t is the Whittker confluent hypergeometric function. Fubini s theorem Dirichlet formul Smko et l. 9 d b x x fx, tdt hx, tdt x b b dt t 2 21 fx, t; 22 hx, tdt + hx, x. 23 x 2. Bsic properties Theorem 2.1 The series in 6 is bsolutely convergent for ll vlues of z provided tht q < p + Rα. Moreover if q p + Rα, then z converges for z < 1. Proof. Rewriting z in the form of power series Eγ,δ,q z γ qn where b n Γαn + βδ pn Γz + nd pplying Γz + b b + b 1 1 z b 1 + + O 2z z 2, we get b n

JFCA-212/3 A GENERALIZATION OF MITTAG-LEFFLER FUNCTION 5 c n+1 c n γ qn+q δ pn Γαn + β γ qn δ pn+p Γαn + β + α nq q q2q + γ 1 1 1 + + O 2qn nq 2 np p p2δ + p 1 1 1 + + O 2pn np 2 αn α α2β + α 1 1 1 + + O 2αn αn 2 then c n+1 s n nd q < p + Rα, c n +1 z qq p p α α n q n p+α, which mens tht the function z converges for ll z provided tht q < p + Rα. Moreover if q p + Rα, then z converges for z < 1. Theorem 2.2 If the condition 7 is stisfied, then nd Proof. z Eγ,δ 1,q z zp 1 δ z Eγ,δ 1,q z γ qn Γδ zp 1 δ d dz Eγ,δ,q z; δ 1 24 z βeγ,δ,q α,β+1,p z + z d dz Eγ,δ,q α,β+1,p z 25 Γαn + β Γδ + pn d dz Eγ,δ,q z hence 24 is proved. z γ qn Γαn + βδ pn βγ qn + αn + βγαn + βδ pn β α,β+1,p z + z d dz Eγ,δ,q α,β+1,p z which is 25. γ qn 1 1 Γαn + β δ pn δ 1 pn pn zp 1 δ 1 δ γ qn n 1 Γαn + βδ pn γ qn Γαn + βδ pn αn + β αn + βγαn + β αγ qn αn + βγαn + βδ pn Theorem 2.3 If the condition 7 is stisfied, then for m N m d dz z γ qn γ + qm qn n + 1 m δ pn δ + pm pn Γαn + αm + β zn ; 26 Proof. m d z β 1 dz wzα z β m 1 α,β m,p wzα 27 m d γ qn dz Γαn + βδ pn Γγ + qn + qm Γδn + 1 m ΓγΓδ + pn + pm Γαn + αm + β zn

6 TARIQ O. SALIM, AHMAD W. FARAJ JFCA-212/3 γ qn δ pn γ + qm qn δ + pm pn n + 1 m Γαn + αm + β zn ; m d z β 1 dz wzα z α β 1 γ qn w n z α n Γαn + β mδ pn γ qn w n Theorem 2.4 If the condition 7 is stisfied, then 1 Γδ x t d dz zα+β 1 Γαn + βδ pn d dz zα+β 1 z β m 1 α,β m,p wzα. x s δ 1 s t β 1 λs tα ds x t δ+β 1 α,β+δ,p λs tα Proof. Let u s t x t, then 1 x x s δ 1 s t β 1 Γδ λs tα ds t 1 x x t δ 1 1 u δ 1 x t β 1 u β 1 x t Γδ t x tδ+β 1 γ qn λx t α n Γαn + βγδ Γδ Γαn + βδ pn Γαn + β + δ x t δ+β 1 α,β+δ,p λs tα. In prticulr, setting t nd x 1 in 28, we get 1 Γδ 1 u β 1 1 u δ 1 zuα ds α,β+δ,p z. 3. z in Terms of Other Functions 28 γ qn λ n x t αn u αn du Γαn + βδ pn In this section we write z in terms of Wright generlized function, generlized hypergeometric function, Mellin-Brnes integrl nd Fox s H-function. z γ qn Γαn + βδ pn Γγ + qn Γγ Γδ Γn + 1 Γδ + pn Γαn + β hence, we cn write z in terms of the Wright generlized function s Γδ z Γγ Γγ + qn Γδ + pn Γn + 1 Γαn + β n! n! Γδ γ, q, 1, 1 Γγ 2 Ψ 2 δ, p, β, α ; z Theorem 3.1 Let 7 be stisfied with α k N, then z cn be written in terms of the generlized hypergeometric function s z 1 Γβ. q+1f p+k 1, q, γ k, β, p, δ ; zq q p p k k 29, 3

JFCA-212/3 A GENERALIZATION OF MITTAG-LEFFLER FUNCTION 7 where k, n is k tuple n k, n + 1 k,..., n + k 1. k Proof. Let α k N, then z γ qn 1 γ qn Γαn + βδ pn Γβ β αn δ pn 1 q qn q γ + i 1 1 n i1 q n Γβ p δ + j 1 p pn k j1 p kn k β + r 1 n! n r1 k n 1 Γβ. 1, q, γ q+1f p+k k, β, p, δ ; zq q p p k k. Now in order to write z s Mellin-Brnes type integrl z in terms of Fox s H-function, we first express Theorem 3.2 Let 7 be stisfied, then z is represented in the Mellin- Brnes type integrl s z 1 2πi L ΓsΓ1 sγγ qs z s ds, 31 Γβ αsγδ ps where rgz < π; the contour of integrtion begins t i nd ending t i, nd intended to seprte the poles of the integrnd t s n for ll n N to the left from those t s n + 1 nd t s γ + n for ll n N {} to the right. q Proof. Simply, by writing the Wright generlized function in 29 in terms of Mellin-Brnes integrl, we get Γδ z Γγ Γγ + qn Γδ + pn 1 Γδ 2πi Γγ Γδ Γγ H1,2 2,3 L z Γn + 1 Γαn + β n! Γδ γ, q, 1, 1 Γγ 2 Ψ 2 δ, p, β, α ; z ΓsΓ1 sγγ qs z s ds Γβ αsγδ ps, 1, 1 γ, q, 1, 1 β, α, 1 δ, p. 32 The lst eqution is just representtion of z in terms of Fox s H-function. 4. Integrl Trnsforms of z In this section, the imge of z under Bet, Lplce, Mellin nd Whittker trnsforms with some specil cses re proved in the following theorems Theorem 4.1 Bet Trnsform { } B xzσ ;, b ΓbΓδ. 3 Ψ 3 Γγ where 7 is stisfied nd R >, Rb >. Proof. γ, q,, σ, 1, 1 β, α, δ, p, + b, σ ; z, 33

8 TARIQ O. SALIM, AHMAD W. FARAJ JFCA-212/3 { } B xzσ ;, b ΓbΓδ Γγ 1 z 1 1 z b 1 xzσ dz γ qn x n B + σn, b Γαn + βδ pn. 3 Ψ 3 γ, q,, σ, 1, 1 β, α, δ, p, + b, σ ; x γ qn x n Γ + σnγb Γαn + βδ pn Γ + σn + b. Theorem 4.2 Lplce Trnsform { } L z 1 xzσ ; s Γδs γ, q,, σ, 1, 1. 3 Ψ 2 Γγ β, α, δ, p ; x s σ 34 Proof. { } L z 1 xzσ ; s z 1 e sz xzσ dz γ qn x n z +σn 1 e sz dz Γαn + βδ pn γ qn x n { } Γ + σn z +σn 1 L Γαn + βδ pn Γ + σn ; s Γδs γ, q,, σ, 1, 1. 3 Ψ 2 Γγ β, α, δ, p Theorem 4.3 Mellin Trnsform { } M wz; s Γδ ΓsΓ1 sγγ qs Γγ Γβ αsγδ ps w s 35 ; x s σ. Proof. According to Theorem 3.2 nd using 31, wz cn be written s wz 1 Γδ ΓsΓ1 sγγ qs 2πi Γγ Γβ αsγδ ps wz s ds 1 Γδ f sz s ds 2πi Γγ L where f ΓsΓ1 sγγ qs s Γβ αsγδ psw s begins t c i nd ends t c i ; c R. Hence Γδ wz Γγ M 1 {f s; z} Now pplying Mellin trnsform to both sides, we obtin M which proves 35. nd L is the contour of integrtion tht { } wz; s Γδ ΓsΓ1 sγγ qs Γγ Γβ αsγδ ps w s Theorem 4.4 Whittker Trnsform L e 1 2 ϕt t ζ 1 W λ,µ ϕt wtσ dt Γδϕ ζ Γγ γ, q, 1, 1, 1. 4 Ψ 2 + µ + ζ, σ, 1 2 µ + ζ, σ 3 β, α, δ, p, 1 λ + ζ, σ where 7 is stisfied nd Rζ >, Rϕ >. Proof. Setting v ϕt, then we get ; w ϕ σ 36

JFCA-212/3 A GENERALIZATION OF MITTAG-LEFFLER FUNCTION 9 e 1 2 ϕt t ζ 1 W λ,µ ϕt wtσ dt e v 2 Γδϕ ζ Γγ Γδϕ ζ Γγ Γδϕ ζ Γγ σn v ζ 1 v W γ qn w n ϕ 1 λ,µv ϕ Γαn + βδ pn ϕ dv n Γqn + γ w Γαn + βγpn + δ ϕ σ e v 2 v ζ+σn 1 W λ,µ vdv n Γqn + γγn + 1 w Γ 1 2 + µ + ζ + σnγ 1 2 µ + ζ + σn Γαn + βγpn + δn! ϕ σ Γ1 λ + ζ + σn which directly yields 36. Γqn + γγn + 1Γ 1 2 + µ + ζ + σnγ 1 2 µ + ζ + σn Γαn + βγpn + δγ1 λ + ζ + σn w ϕ σ 5. Integrl Opertors with Generlized Mittg-Leffler Function in the Kernel In this section, we consider composition of the Riemnn-Liouville frctionl integrl nd derivtive nd Hilfer s frctionl derivtive 9-11 with Mittg-Leffler function defined by 7. Theorem 5.1 Let R +, α, β, γ, δ, λ, w C, min {Rα, Rβ, Rγ, Rδ, Rλ} > nd p, q >, then for x > we hve D+ λ t β 1 wt α x x β λ 1 α,β λ,p wx α 37 Proof. Beginning with I + λ t β 1 x Γβ I λ + t β 1 wt α x I+ λ n! n Γβ + λ x β+λ 1, then γ qn w n t αn+β 1 x Γαn + βδ pn γ qn w n t αn+β 1 Γαn + βδ pn Γαn + β Γαn + β + λ x αn+β+λ 1 38 x β+λ 1 α,β+λ,p wx α Now mking use of 9, 27 nd 38 yields D+ λ t β 1 wt α x d m I+ m λ t β 1 wt α x m d x β+m λ 1 wx α x β λ 1 α,β λ,p wx α. Now, mking use of the formuls in 27 nd 38, we cn get the following result contined in

1 TARIQ O. SALIM, AHMAD W. FARAJ JFCA-212/3 Theorem 5.2 Let R +, α, β, γ, δ, w C, min {Rα, Rβ, Rγ, Rδ} >, < u < 1, v, Rβ > u + v uv nd p, q >, then for x > we hve D u,v + t β 1 wt α x x β u 1 α,β u,p wx α. 39 Consider the integrl opertor defined in 8 contining the Mittg-Leffler function γ,δ,q z in the kernel. First of ll we will prove tht the opertor E,w, + is bounded on L, b. Theorem 5.3 Let α, β, γ, δ, w C, min {Rα, Rβ, Rγ, Rδ} >, b > nd p, q >, then the opertor,w, is bounded on L, b nd +,w,+ φ 1 B φ 1 4 where B b Rβ γ qn wb Rα n Γαn + β δ pn Rαn + Rβ Proof. First of ll, let C n denote the n th term of 41, then c n+1 c n γ qn+q Γαn + β δ pn Rαn + Rβ γ qn Γαn + β + α δ pn+p Rαn + Rα + Rβ wb Rα wb Rα qn q s n, provided tht q < p + Rα. Hence α n Rα pn p c n+1 c n s n, which mens tht the right hnd side of 41 is convergent nd finite under the given condition. Now ccording to 8 nd 22,w,+ φ 1 b b x t β 1 wx tα φtdt b b x t β 1 wx b b t tα φt dt u Rβ 1 wuα du φt dt b b u Rβ 1 wuα du φt dt. But we hve b u Rβ 1 γ qn w n wuα du Γαn + β δ pn so tht B b Rβ Hence,w,+ φ 1 b b γ qn wb Rα n Γαn + β δ pn Rαn + Rβ B φt dt B φ 1. 41 u Rαn+Rβ 1 du B

JFCA-212/3 A GENERALIZATION OF MITTAG-LEFFLER FUNCTION 11 Equlity 28 cn simply be written by mens of the opertor,w, s + Corollry 5.4 Let α, β, γ, δ, ζ, w C, min {Rα, Rβ, Rγ, Rδ, Rλ} > nd p, q >, then,w,+ t ζ 1 x Γζx β+ζ 1 α,β+ζ,p wx α. 42 6. Composition of Frctionl Clculus Opertors nd Integrl Opertor with Generlized Mittg-Leffler Function in the Kernel We consider now composition of the Riemnn-Liouville frctionl integrtion opertor I λ with the opertor +,w,+ Theorem 6.1 Let α, β, γ, δ, λ, w C, min {Rα, Rβ, Rγ, Rδ, Rλ} > nd p, q >, then I λ γ,δ,q +E,w,+ φ α,β+λ,p,w,+ φ,w,+ Iλ +φ 43 holds for ny summble function φ L, b. Proof. I λ +,w,+ φ x 1 x u x u λ 1 u t β 1 Γλ wu tα φtdt du x 1 x x u λ 1 u t β 1 Γλ wu tα du φtdt letting τ u t implies t I λ +,w,+ φ x x I λ x τ β 1 wτ α x 1 x t Γλ x tφtdt x t β+λ 1 α,β+λ,p wx tα Similrly, we cn prove the other side. x t τ λ 1 τ β 1 wτ α dτ φtdt x τ β+λ 1 α,β+λ,p wτ α φtdt φtdt α,β+λ,p,w,+ φ x Theorem 6.2 If the conditions of Theorem 6.1 is stisfied, then D λ γ,δ,q +E,w,+ φ x α,β λ,p,w,+ φ x. 44 Proof. Let n Rλ + 1 nd using 9, we get n D λ d +,w,+ φ x I n λ +,w,+ φ x n d α,β+n λ,p,w,+ φ x n d x x t β+n λ 1 α,β+n λ,p wx tα φtdt Since the integrl is continuous, 23 yields

12 TARIQ O. SALIM, AHMAD W. FARAJ JFCA-212/3 n 1 D λ d x +,w,+ φ x +limx t β+n λ 1 t x α,β+n λ,p wx tα n 1 d x x t β+n λ 2 γ qn wx t α n Γαn + β + n λ 1δ pn n 1 d x x t β+n λ 2 α,β+n λ 1,p wx tα φtdt Repeting this process n 1 times, then we get D λ +,w,+ φ x x x t β+n λ 1 α,β+n λ,p x wx tα φtdt φtdt x t β λ 1 α,β λ,p wx tα φtdt α,β λ,p,w,+ φ x. Theorem 6.3 Let α, β, γ, δ, w C, min {Rα, Rβ, Rγ, Rδ} > < u < 1, v 1, Rβ > u + v uv nd p, q >, then D u,v +,w,+ φ x α,β u,p,w,+ φ x. 45 Acknowledgement: The uthors wish to thnk the nonymous referee for vluble comments nd suggestions. References 1 R. Finney, D. Ostberg, R. Kuller, Elementry Differentil Equtions with Liner Algebr, Addison-Weley Publishing Compny; 1976.. 2 R. HilferEd., Applictions of Frctionl Clculus in Physics, Singpore, New Jersey, London nd Hong Kong : World Scientific Publishing Compny; 2. 3 A.A. Kilbs, M. Sigo, H Trnsforms: Theory nd Applictions., London, NewYork: Chpmn nd Hll/CRC; 24. 4 A.A. Kilbs, M. Sigo, R.K. Sxen, Generlized Mittg Leffler function nd generlized frctionl clculus opertors, Integrl Trnsforms Spec. Funct., Vol. 15,24,pp. 31 49. 5 G.M. Mittg-Leffler, Sur l nouvelle fonction. C.R. Acd. Sci. Pris, Vol. 137, 193, pp. 554 558. 6 T.R. Prbhkr, A Singulr integrl eqution with generlized Mittg-Leffler function in the kernel. Yokohm Mth. J., Vol. 19,1971, pp. 7 15. 7 E.D. Rinville, Specil Functions. New York : Chelse Publ. Co.; 196. 8 T.O. Slim, Some properties relting to the generlized Mittge-Leffler function, Adv. Appl. Mth. Anl., Vol. 4, 29, pp. 21-3. 9 S.G. Smko, A.A. Kilbs, O.I. Mrichev, Frctionl Integrls nd Derivtives: Theory nd Applictions, Yverdon Switzerlnd: Gordon nd Brech Science Publishers; 1993.. 1 A.K. Shukl, J.C. Prjpti, On generliztion of Mittg Leffler function nd its properties, J. Mth. Anl. Appl., Vol. 336, 27, pp. 797 811.. 11 I.N. Sneddon, The Use of Integrl Trnsforms. New Delhi: Tt McGrw Hill; 1979. 12 H.M. Srivstv, H.L. Mnoch, A Tretise on Generting Functions. New York: John Wiley nd Sons; 1984. 13 H.M. Srivstv, Z. Tomovski, Frctionl clculus with n integrl opertor contining generlized Mittg-Leffler function in the kernel, Appl. Mth. Comput.,Vol. 211,29, pp.198-21. 14 A. Wimn, Uber den fundmentl stz in der theori der functionen, Act Mth., Vol. 29, 195, pp. 191 21. 15 E.T. Wittker, G.N. Wtson, A Course of Modern Anlysis. Cmbridge: Cmbridge Univ. Press; 1962.

JFCA-212/3 A GENERALIZATION OF MITTAG-LEFFLER FUNCTION 13 Triq O. Slim, Deprtment of Mthemtics, Al-Azhr University-Gz, P.O.Box 1277, Gz, Plestine E-mil ddress: trslim@yhoo.com, t.slim@lzhr.edu.ps Ahmd W. Frj, Deprtment of Mthemtics, Al-Azhr University-Gz, P.O.Box 1277, Gz, Plestine