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

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1 Journl of Frctionl Clculus nd Applictions, Vol. 3. July 212, No. 5, pp ISSN: 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 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,

2 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 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

3 JFCA-212/3 A GENERALIZATION OF MITTAG-LEFFLER FUNCTION 3 nd D λ φ n d x I n λ φx n Rλ 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 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 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 O 2z z 2, we get b n

5 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 O 2qn nq 2 np p p2δ + p O 2pn np 2 αn α α2β + α 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 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

7 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 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

9 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 µ + ζ + σnγ 1 2 µ + ζ + σn Γαn + βγpn + δn! ϕ σ Γ1 λ + ζ + σn which directly yields 36. Γqn + γγn + 1Γ 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

10 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 u Rαn+Rβ 1 du B

11 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 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 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; 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; 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 G.M. Mittg-Leffler, Sur l nouvelle fonction. C.R. Acd. Sci. Pris, Vol. 137, 193, pp T.R. Prbhkr, A Singulr integrl eqution with generlized Mittg-Leffler function in the kernel. Yokohm Mth. J., Vol. 19,1971, pp E.D. Rinville, Specil Functions. New York : Chelse Publ. Co.; T.O. Slim, Some properties relting to the generlized Mittge-Leffler function, Adv. Appl. Mth. Anl., Vol. 4, 29, pp S.G. Smko, A.A. Kilbs, O.I. Mrichev, Frctionl Integrls nd Derivtives: Theory nd Applictions, Yverdon Switzerlnd: Gordon nd Brech Science Publishers; A.K. Shukl, J.C. Prjpti, On generliztion of Mittg Leffler function nd its properties, J. Mth. Anl. Appl., Vol. 336, 27, pp I.N. Sneddon, The Use of Integrl Trnsforms. New Delhi: Tt McGrw Hill; H.M. Srivstv, H.L. Mnoch, A Tretise on Generting Functions. New York: John Wiley nd Sons; 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 A. Wimn, Uber den fundmentl stz in der theori der functionen, Act Mth., Vol. 29, 195, pp E.T. Wittker, G.N. Wtson, A Course of Modern Anlysis. Cmbridge: Cmbridge Univ. Press; 1962.

13 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