CHAPTER 3 MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS

CHAPTER 3 MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS [ThischpterisbsedonthelecturesofProfessorR.K.SxenofJiNrinVysUniversity, Jodhpur, R jsthn.] 3.. Introduction This section dels with Mittg-Leffler function nd its generliztions. Its importnce is relised during the lst one nd hlf decdes due to its direct involvement in the problems of physics, biology, engineering nd pplied sciences. Mittg-Leffler function nturlly occurs s the solution of frctionl order differentil equtions or frctionl order integrl equtions. Vrious properties of Mittg- Leffler functions re described. Among the vrious results presented by vrious reserchers, the importnt ones del with Lplce trnsform nd symptotic expnsions of these functions, which re directly pplicble in the solution of differentil equtions nd behvior of the solution for smll nd lrge vlues of the rgument. Hille nd Tmrkin [14, p.86] in 192 hve presented solution of Abel-Volterr type integrl eqution φx) λ φt)dt Γα) x t) 1 α=f x), < x<1 in terms of Mittg-Leffler function. Dzherbshyn [2] hs shown tht both the functions defined by 3.1.1) nd 3.1.2) re entire functions of order p= 1 nd type α σ=1. A detiled ccount of the bsic properties of these functions is given in the third volume of Btemn Mnuscript Project written by A. Erdélyi et l [3] nd published by McGrw-Hill in the yer 1955 under the heding Miscellneous Functions. 71

72 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.1. Mittg-Leffler Functions Nottion 3.1.1. E α x): Mittg-Leffler function Nottion 3.1.2. E α,β x): Generlized Mittg-Leffler function Note 3.1.1. : According to Erdélyi, A. et l, E α x) nd E α,β x) re clled Mittg-Leffler functions. Definition 3.1.1. E α z) := k= z k, α C,Rα)>). 3.1.1) Γαk+1) Definition 3.1.2. E α,β z) := k= z k, α,β C,Rα)>,Rβ)>). 3.1.2) Γαk+β) The function E α z) ws defined nd studied by Mittg-Leffler in the yer 193. It is direct generliztion of the exponentil function. The function defined by 3.1.2) gives generliztion of 3.1.1). This generliztion ws studied by Wimn in 195, Agrwl in 1953 nd Humbert nd Agrwl in 1953, nd others. Exmple 3.1.1. Prove tht E 1 z) = e z = E 1,1 z). It redily follows from 3.1.1) nd 3.1.2). Exmple 3.1.2. Prove tht E 1,2 z)= ez 1 z Solution: We hve E 1,2 z)= k= z k Γk+2) = z k k+1)! = 1 z k= k= z k+1 k+1)! = 1 z ez 1). Definition 3.1.3. Hyperbolic function of order n. z nk+r 1 h r z, n) := nk+r 1)! = zr 1 E n,r z n ), r N). 3.1.3) k=

3.1. MITTAG-LEFFLER FUNCTIONS 73 Definition 3.1.4. Trigonometric function of order n. 1) k z kn+r 1 k r z, n) : = kn+r 1)! = zr 1 E n,r z n ). 3.1.4) E 1 2,1z) = k= k= Γ k 2 where erfc is complementry to the error function efc. Definition 3.1.5. Error function. erfcz) := 2 π 1/2 To derive 3.1.5), we see tht [ [1], p,297, Eq.7.1.] reds s wheres [ [1],p. 297, Eq.7.1.8] is z z k ez2erfc z), 3.1.5) + 1)= e u2 du=1 erfz), z C, 3.1.6) wz)=e z2 erfc iz) 3.1.7) wz)= Γ n n= 2 iz) n + 1). 3.1.8) From 3.1.7) nd 3.1.8), we esily obtin 3.1.5). In pssing, we note thtwz) is lso n error function [1]. Definition 3.1.6. Mellin-Ross function. E t v, ) :=t v t) k Γv+k+1) = tv E 1,v+1 t). 3.1.9) k= Definition 3.1.7. Robotov s function. β, t) :=t α β k t kα+1) Γ1+α)k+ 1)) = tα E α+1,α+1 βt α+1 ). 3.1.1) F k= Exmple 3.1.3. Prove tht E 1,3 z)= ez z 1 z 2.

74 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS Solution: We hve E 1,3 z)= k= z k Γk+ 3) = 1 z 2 k= z k+2 k+2)! = 1 z 2 ez z 1). Exmple 3.1.4. Prove tht E 1,r z)= 1 { z r 1 e z The proof is similr to tht in Exmple 3.1.3 r 2 k= z k }, r. k! 3.1.1. Prove tht 3.1. Revision Exercises [ H 1,1 1,2 x,a) ]=A 1 1) k x k+)/a,a),,1) Γ[1+k+ )A]. k= 3.1.2. Prove tht d [ dx H1,1 1,2 x ],A),A),,1) [ = H 1,1 1,2 x ] A,A). A,A),,1) 3.1.3. Prove tht 1 2,1[ x H 1,1 1,A),1,1) 1,A) ]=A 1 k= 1) k 1 x ) k+1 A Γ[1 k+1 )/A]. 3.2. Bsic Properties of Mittg-Leffler Function As consequence of the definitions 3.1.1) nd 3.1.2) the following results hold: Theorem 3.2.1. There holds the following reltions:

3.2. BASIC PROPERTIES OF MITTAG-LEFFLER FUNCTION 75 i) E α,β z)=ze α,α+β z)+ 1 Γβ) 3.2.1) ii) E α,β z)=β E α,β+1 z)+αz d dz E α,β+1z) 3.2.2) d m [ ] iii) z dz) β 1 E α,β z α ) = z β m 1 E α,β m z α ), Rβ m)>, m ). 3.2.3) Solutions: i) We hve ii) We hve iii) since E α,β z)= k= z k Γαk+β) = k= 1 z k+1 Γα+β+αk) = z E α,α+β z)+ 1 Γβ),Rβ)>. R.H.S=βE α,β+1 z)+αz d dz =βe α,β+1 z)+ k= = E α,β z)=l.h.s. ) m d L.H.S = dz k= = ) m d ) z αk+β 1 = dz k= k= z k Γαk+β+1) αk+β β)z k Γαk+β+1) z αk+β 1 Γαk+β) z αk+β m 1 Γαk+β m),rβ m)>, Γαk+β) Γαk+β m) zαk+β m 1 = z β m 1 E α,β m z α ), m ) = R.H.S.

76 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS Following specil cses of 3.2.3) re worth mentioning. If we setα= m n, m, n ) then d m [ )] ) z dz) β 1 E m n,β z m n = z β m 1 E m n,β m z m n forrβ m)>. Replcing k by k+n) = z β m 1 = z β 1 E m n,β k= n z m n Γ = z β m 1 z mk n β+ mk n ) k= n )+z β 1 k=1γ d m ) [z dz) β 1 E m,β z )]=z m β 1 E m,β z m z m + Γβ m) Putting z=t n m in 3.2.4) it yields When m=1 3.2.6) reduces to z mk n ) mk Γ n +β m z mk n β mk n m n t1 m dt) n d m [ ] t β 1) m n E m n,βt) = t β 1) m n E m n,βt)+t β 1) m n n k=1 Γ t k β mk n ) m, n=1, 2, 3). 3.2.4) forrβ m)>. 3.2.5) ), Reβ m)>, m, n ). 3.2.6) t 1 n n d [ ] t β 1)n E 1 dt,βt) = t β 1)n E 1 n n,βt)+tβ 1)n n k=1 t k ), Γβ k n Rβ)>1, which cn be written s 1 n d n [t β 1)n E ]=t dt 1n,β t) βn 1 E 1n,β t)+tβn 1 k=1 t k Γβ k n ),Rβ)>1. 3.2.7)

3.2. BASIC PROPERTIES OF MITTAG-LEFFLER FUNCTION 77 3.2.1. Mittg-Leffler functions of rtionl order Now we consider the Mittg-Leffler functions of rtionl orderα= p q with p, q reltively prime. The following reltions redily follow from the definitions 3.1.1) nd 3.1.2). i) ii) pep d dz) z P ) = E p z P ) 3.2.8) d p dz p E p q z p q ) q 1 = E p z p q )+ q k=1 k=1 z kp q Γ1 kp 3.2.9) q ); q=1, 2, 3,. We now derive the reltion ) q 1 γ1 iii) E 1 z 1 q = e [1+ k z q z) ] q Γ1 k q ) ; 3.2.1) where q = 2, 3, nd γα, z) is the incomplete gmm function, defined by γα, z)= z e u u α 1 du. To prove 3.2.1), set p = 1 in 3.2.9) nd multiply both sides by e z nd use the definition ofγ, z). Thus we hve d [ dz e z E 1 q w 1 z 1 q )]=e z k=1 Γ Integrting 3.2.11) with respect to z, we obtin 3.2.1). z k q 1 k q 3.2.2. Euler trnsform of Mittg-Leffler function By virtue of bet function formul it is not difficult to show tht 1 z ρ 1 1 z) σ 1 E α,β xz γ )dz=γσ) 2 ψ 2 [ ρ,γ),1,1) ). 3.2.11) β,α),σ+ρ,γ) ] x 3.2.12) whererα)>,rβ)>,rρ)>,rσ)>,γ>. Here 2 ψ 2 is the generlized Wright function ndα,β,ρ,σ C. Specil cses of 3.2.12): i) Whenρ=β,γ=α, 3.2.12) yields.

78 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 1 whereα>;β,σ C,Rβ)>,Rσ)> nd, ii) 1 z β 1 1 z) σ 1 E α,β xz α )dz=γσ)e α,σ+β x), 3.2.13) z) σ 1 1 z) β 1 E α,β [x1 z) α ]dz=γσ)e α,σ+β x), 3.2.14) whereα>;β,σ C,Rβ)>,Rσ)>. iii) Whenα=β=1, we hve 1 z ρ 1 1 z) σ 1 expxz γ )dz=γσ) 2 ψ 2 [ ρ,γ),1,1) whereγ>,ρ,σ C,Rρ)>,Rσ)>. =Γσ) 1 ψ 1 [ ρ,γ) 1,1),σ+ρ,γ) σ+ρ,γ) ] x ] x, 3.2.15) 3.2.3. Lplce trnsform of Mittg-Leffler function By the ppliction of Lplce integrl, it follows tht z ρ 1 e z E α,β xz γ )dz= 1 [ 1,1),ρ,γ) ρ 2 ψ 1 x ] β,α) γ, 3.2.16) whereρ,,α,β C,Rα)>,Rβ)>,Rγ)>,R)>,Rρ)>nd z <1. γ Specil cses of 3.2.16) re worth mentioning. i) Forρ=β,γ=α,Rα)>, 3.2.16) gives e z z β 1 E α,β xz α )dz= α β α x, 3.2.17) where,α,β C,Rα)>,Rβ)>,R)> x <1. α When =1, 3.2.17) yields known result. e z z β 1 E α,β xz α )dz= 1, x <1, 3.2.18) 1 x whererα)>,rβ)>. If we further tkeβ=1, 3.2.18) reduces to

3.2. BASIC PROPERTIES OF MITTAG-LEFFLER FUNCTION 79 ii) Whenβ=1, 3.2.17) gives e z E α xz α )dz= 1 1 x, x <1. wherer)>,rα)>, x α <1. 3.2.4. Appliction of Lplce trnsform From 3.2.17), we find tht e z E α xz α )dz= α 1 α x, 3.2.19) whererα)>,rβ)> L { x β 1 E α,β x α ) } = sα β s α 3.2.2) We lso hve L{f t); s}= e st f t)dt,rs)>. 3.2.21) Now L { x γ 1 E α,γ x α ) } = sα γ s α + 3.2.22) [ s α β ][ s α γ ]= s2α β+γ) s α s α + s 2α 2 forrs 2 )>R). 3.2.23) By virtue of the convolution theorem of the Lplce trnsform, it redily follows tht t u β 1 E α,β u α )t u) γ 1 E α,γ t u) α )du = t β+γ 1 E 2α,β+γ 2 t 2α ). 3.2.24) where Rβ) >, Rγ) >. Further, if we use the identity nd the reltion 1 sα β s2= s α 1 [s β 2 s β α 2 ] 3.2.25)

8 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS where Rρ) >, Rs) >, we obtin t where <β<2,rα)>. L { t ρ 1 ; s } =Γρ)s ρ, 3.2.26) [ t u) u β 1 E α,β u α 1 β ) Γ2 β) t u)α β+1 Γα β+2) ] du=t, 3.2.27) Next we note tht the following result 3.2.29) cn be derived by the ppliction of Lplce trnsform of the identity We hve [ s 2α β s 2α 1 1 Γα) ] [s α ]= s2α β s 2α 1 + sα β s α 1 forrsα )>1. 3.2.28) x t) α 1 E 2α,β t 2α )t β 1 dt = x β 1 E 2α,β x 2α )+ x β 1 E α,β x α ), 3.2.29) where Rα) >, Rβ) >. If we set β = 1 in 3.2.29), it reduces to where Rα) >. 1 Γα) x t) α 1 E 2α t 2α ) dt=e α x α ) E 2α x 2α ) 3.2.3) 3.2.5. Contour integrl representtion Lemm 3.2.1. Contour integrl representtion for E α z) is given by E α z)= 1 t α 1 e t dt 2πi H t α z, 3.2.31) where the pth of integrtion H is loop strting nd ending t, nd encircling the circulr disk t z 1/α in the positive sense : π<rg t πon H. To estblish the representtion 3.2.31) we expnd the integrnd in powers of z, integrte term by term nd pply the Hnkel integrl for the reciprocl of the gmm function, nmely e s s z ds= 2πi H Γz). 3.2.32) Similrly we cn estblish

3.2. BASIC PROPERTIES OF MITTAG-LEFFLER FUNCTION 81 Lemm 3.2.2. where α,β>. E α,β z)= 1 t α β e t dt 2πi H t α z. 3.2.33) Y X O = O = o E X H Y Figure 3.1: Showing Hnkel contour H 3.2.6. Reltion between Mittg-Leffler functions nd the H-function Both the Mittg-Leffler functions E α z) nd E α,β z) belong to H-function fmily. We derive their reltions with the H function. Lemm 3.2.3. integrl Letα =, ). Then E α z) is represented by the Mellin-Brnes E α z)= 1 2πi L Γs)Γ1 s) z) s ds, rg z <π), 3.2.34) Γ1 αs) where the contour of integrtion L, beginning t i nd ending t +i, seprtes ll poles s= k k ) to the left nd ll poles s=1+n n ) to the right. Proof. We now evlute the integrl 3.2.34) s the sum of the residues t the points s=, 1, 2,. We find tht

82 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 1 2πi L Γs)Γ1 s) z) s ds= Γ1 αs) = k= k= = E α z), [ Γs)Γ1 s) z) s] Res s= k Γ1 αs) 1) k Γ1+k) k!γ1+αk) z)k 3.2.35) which yields 3.2.34) in ccordnce with the definition 3.1.1). It redily follows from the definition of the H - function nd 3.2.34) tht E α z) cn be represented in the form where H 1,1 1,2 is the H-function, which is studied in Chpter 1. [ E α z)=h 1,1 1,2 z ],1), 3.2.36),1),,α) Lemm 3.2.4. Letα + =, ),β C, then E α,β z)= 1 2πi L Γs)Γ1 s) z) s ds. 3.2.37) Γβ αs) The proof of 3.2.37) is similr to tht of 3.2.34). Hence the proof is omitted. From 3.2.37) nd the definition of the H-function we obtin the reltion [ E α,β z)=h 1,1 1,2 z ],1). 3.2.38),1), β,α) In prticulr, E α z) cn be expressed in terms of generlized Wright function in the form Similrly, we hve E α z)= 1 ψ 1 [ 1,1) 1,α) ] z. 3.2.39) E α,βz)= 1 ψ 1 [ 1,1) β,α) ] z. 3.2.4) Next if we clculte the residues t the poles of the gmm functionγ1 s) t the points s=1+n n=, 1, 2, ) it gives

3.2. BASIC PROPERTIES OF MITTAG-LEFFLER FUNCTION 83 1 Γs)Γ1 s) 2πi L Γ1 αs) z)s ds= = = n= n= [ Γs)Γ1 s) z) s] Res s 1+n Γ1 αs) 1) n n1 n=1 Γ1+n) z) n 1 Γ1 α1+n)) z n Γ1 αn). 3.2.41) Similrly for E α,βz), if we clculte the residues t s=1+n n=, 1, 2, ), we obtin 1 Γs)Γ1 s) 2πi L Γβ αs) z) s ds= n=1 z n Γβ αn) 3.2.42) Exercises 3.2. 3.2.1. Let nd U 1 t)=t β 1 E m n,βt m n ) U 2 t)=t β 1 E m,β t m ) U 3 t)=t β 1) n m E m n,βt) U 4 t)=t β 1)n E 1 n,β t). Then show tht these functions respectively stisfy the following differentil equtions of Mittg-Leffler functions. i) ii) d m dt m U 1t) U 1 t)=t β 1 n k=1 Rβ)>m, m, n=1, 2, 3, ) t m n k Γβ mk n ) d m dt m U 2t) U 2 t)= t m+β 1,Rβ)>m, m=1, 2, 3, ) Γβ m)

84 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.2.2. Prove tht 3.2.3. Prove tht 3.2.4. Prove tht 1 Γv) z 3.2.5. Prove tht m iii) n t1 m n d ) mu3 n t) U 3 t)=t β 1) m n t k dt k=1γβ mk n ) m, n=1, 2, 3, ) iv) 1 d n n[ dt U 4t)) ] t n 1 U 4 t)=t βn 1 t k Γβ k n ) n=1, 2, 3, ). k=1 λ x E α λt α )dt Γα) x t) 1 α= E αλx α ) 1, Rα)>. d [ ] x γ 1 E α,β x α ) = x γ 2 E α,β 1 x α )+γ β)x γ 2 E α,β x α ), β γ. dx t β 1 z t) v 1 E α,β λt α )dt=z β+v 1 E α,β+v λz α ), Rβ)>, Rv)>Rα)>. 1 Γα) 3.2.6. Prove tht 3.2.7. Prove tht 1 Γα) z 1 Γα) z z t) α 1 cosh λ t)dt=z α E 2,α+1 λz 2 ), Rα)>. z e λt z t) α 1 dt=z α E 1,α+1 λz),rα)>. λ ) sinh t z t) α 1 dt=z α+1 E 2,α+2 λz 2 ), Rα)>. λ

3.3. GENERALIZED MITTAG-LEFFLER FUNCTION 85 3.2.8. Prove tht e sρ ρ β 1 E α,β ρ α )dρ= sα β s α 1,Rs)>1. 3.2.9. Prove tht e st E α t α )dt= 1 s s 1 α,rs)>1. 3.2.1. Prove tht u γ 1 E α,γ yu α )x u) β 1 E α,β [zx u) α ]du = ye α,β+γyx α ) ze α,β+γ zx α ) y z wherey, z C;y z, γ>,β>. x β+γ 1 3.3. Generlized Mittg-Leffler Function Nottion 3.3.1. Eβ,γ δ z) : Generlized Mittg-Leffler function Definition 3.3.1. E δ β,γ z) := n= δ) n z n Γβn+γ)n!, 3.3.1) whereβ,γ,δ CwithRβ)>. Forδ=1, it reduces to Mittg-Leffler function 3.1.2). This function ws introduced by T. R. Prbhkr in 1971. It is n entire function of orderρ=[rβ)] 1. 3.3.1. Specil cses of E δ β,γ z) i) E β z)=eβ,1 1 z). 3.3.2) ii) E β,γz)=eβ,γ 1 z) 3.3.3) iii) φγ,δ; z)= 1 F 1 γ;δ; z)=γδ)e γ 1,δ z), 3.3.4) whereφγ,δ; z) is Kummer s confluent hypergeometric function.

86 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.3.2. Mellin-Brnes integrl representtion Lemm 3.3.1. Letβ + =, );γ,δ Cγ ). Then Eβ,γ δ z) is represented by the Mellin-Brnes integrl Eβ,γ δ z)= 1 Γs)Γδ s) 2πi Γδ) L Γγ βs) z) s ds, 3.3.5) where rg z) <π; the contour of integrtion beginning t i nd ending t+i, seprtes ll the poles t s= k k ) to the left nd ll the poles t s=n+δ n ) to the right. Proof. We will evlute the integrl on the R.H.S of 3.3.5) s the sum of the residues t the poles s=, 1, 2,. We hve which proves 3.3.5). 1 Γs)Γδ s) 2πi L Γγ βs) z) s ds= k= Res s= k [ Γs)Γδ s) Γγ βs) z) s ] 1) k Γδ+k) = k! Γγ+βk) z)k k= δ) k z k =Γδ) Γβk+γ) k! =Γδ)Eδ β,γ z) k= 3.3.3. Reltions with the H function nd Wright hypergeometric function It follows from 3.3.5) tht Eβ,γ δ z) cn be represented in the form Eβ,γ δ z)= 1 [ Γδ) H1,1 1,2 z ] 1 δ,1),1),1 γ,β) 3.3.6) where H 1,1 1,2 z) is the H function, the theory of which cn be found in Chpter 1. This function cn lso be represented by Eβ,γ δ z)= 1 [ δ,1) ] Γδ) 1 ψ 1 z, 3.3.7) γ,β) where 1 ψ 1 is the Wright hypergeometric function p ψ q z).

3.3.4. Cses of reducibility 3.3. GENERALIZED MITTAG-LEFFLER FUNCTION 87 In this subsection, we present some interesting cses of reducibility of the function Eβ,γ δ z). The results re given in the form of five theorems. The results re useful in the investigtion of the solutions of certin frctionl-order differentil nd integrl equtions. The proof of the following theorems cn be developed on similr lines to tht of eqution 3.2.1). Theorem 3.3.1. holds the reltion Ifβ,γ,δ C withrβ)>,rγ)>,rγ β)>, then there ze δ β,γ z)=eδ β,γ β z) Eδ 1 β,γ β z). 3.3.8) Corollry 3.3.1. Ifβ,γ,δ C,Rγ)>Rβ)>, then we hve z E δ β,γ z)=e β,γ βz) 1 Γγ β). 3.3.9) Theorem 3.3.2. Ifβ,γ,δ C,Rβ)>,Rγ)>1, then there holds the formul β E 2 β,γ z)=e β,γ 1z)+1+β γ)e β,γ z). 3.3.1) Theorem 3.3.3. If Rβ) >, Rγ) > 2 + Rβ), then there holds the formul zeβ,γ 3 z)= 1 [ E 2β 2 β,γ β 2 z) 2γ 3β 3)E β,γ β 1 z) ] + 2β 2 +γ 2 3βγ+3βγ 2γ+1) E β,γ β z). 3.3.11) Theorem 3.3.4. If Rβ) >, Rγ) > 2, then there holds the formul E 3 β,γ z)= 1 [ E 2β 2 β,γ 2 z)+3+3β 2γ)E β,γ 1 z) ] + 2β 2 +γ 2 + 3β 3βγ 2γ+1)E β,γ z). 3.3.12)

88 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.3.5. Differentition of generlized Mittg-Leffler function Theorem 3.3.5. Letβ,γ,δ,ρ,w. Then for ny n, there holds the formul, forrγ)>n, In prticulr for Rγ) > n, nd forrγ)>n, d n [ ] z dz) γ 1 E δ β,γ wzβ ) = z γ n 1 E δ β,γ n wzβ ). 3.3.13) d n [ ] z dz) γ 1 E β,γ wz β ) = z γ n 1 E β,γ n wz β ) 3.3.14) d n [ ] z dz) γ 1 φδ;γ;wz) = Γγ) Γγ n) zγ n 1 φδ,γ n;wz). 3.3.15) Proof. Using 3.3.1) nd tking term by term differentition under the summtion sign, which is possible in ccordnce with uniform convergence of the series in 3.3.1) in ny compct set of, we obtin d n [ ] z dz) γ 1 Eβ,γ δ wzβ ) = δ) k d ) n [ w Γβk+γ) k z βγ+γ 1 ] dz k! k= = z γ n 1 E δ β,γ n wzβ ) forrγ)>n, which estblishes 3.3.13). Note tht 3.3.14) follows from 3.3.13) when δ = 1 due to 3.3.3) nd 3.3.15) follows from 3.3.13) whenβ=1 on ccount of 3.3.4). 3.3.6. Integrl property of generlized Mittg-Leffler function Corollry 3.3.2. Letβ,γ,δ,w,Rγ)>,Rβ)>,Rδ)>. Then z nd 3.3.16) follows from 3.3.13). In prticulr, t γ 1 E δ β,γ wtβ )dt=z γ E δ β,γ+1 wzβ ) 3.3.16) nd z t γ 1 E β,γ wt β )dt=z γ E β,γ+1 wt β ) 3.3.17) z t γ 1 φγ,δ;wt)dt= 1 δ zδ φγ,δ+1;wx) 3.3.18)

3.3. GENERALIZED MITTAG-LEFFLER FUNCTION 89 Remrk 3.3.1. The reltions 3.3.15) nd 3.3.18) re well known. 3.3.7. Integrl trnsform of E δ β,γ z) By ppeling to the Mellin inversion formul 3.3.5) yields the Mellin-trnsform of the generlized Mittg-Leffler function. t s 1 Eβ,γ δ wt)dt= Γs)Γδ s) Γδ)w s Γγ sβ). 3.3.19) If we mke use of the integrl t v 1 e t 2 Wλ,µ t)dt= Γ 1 2 +µ+v)γ 1 2 µ+v) Γ1 λ+v) 3.3.2) whererv±µ)> 1 2, we obtin the Whittker trnsform of the Mittg-Leffler function t ρ 1 e 1 2 pt W λ,µ pt)e δ β,γ wtα )dt= p ρ Γδ) 3 ψ 2 [ δ,1), 1 2 ±µ+ρ,α) γ,β),1 λ+ρ,α) w ] p α 3.3.21) where 3 ψ 2.) is the generlized Wright function, ndrρ)> Rµ) 1 2,Rp)>o, w p α <1. Whenλ=ndµ= 1 2, then by virtue of the identity W ± 1 2, t)=exp t ), 3.3.22) 2 the Lplce trnsform of the generlized Mittg-Leffler function is obtined: [ t ρ 1 e pt Eβ,γ δ wtα )dt= p ρ δ,1),ρ,α) Γδ) 2 ψ 1 w ] p α γ,β) 3.3.23) whererβ)>,rγ)>,rρ)>,rp)>, p> w Rα). In prticulr, forρ=γ nd α = β we obtin result given by Prbhkr [ p.8, Eq. 2.5]. whererβ)>,rγ)>,rp)> nd p> w t γ 1 e pt E δ β,γ wtβ )dt= p γ 1 wp β ) δ 3.3.24) 1 Rβ). The Euler trnsform of the generlized Mittg-Leffler function follows from the bet function: 1 t 1 1 t) b 1 Eβ,γ δ xtα )dt= Γb) [ δ,1),,α) Γδ) 2 ψ 2 wherer)>,rb)>,rδ)>,rβ)>,rγ)>,α>. 1 γ,β),+b,α) ] x, 3.3.25)

9 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS Theorem 3.3.6. We hve e pt t αk+β 1 E k) α,β ±tα )dt,= k!pα β p α ) k+1, 3.3.26) whererp)> 1/α,Rα)>,Rβ)>, nd E k) dk α,β y)= E dy k α,β y). Solution: We will use the following result: The given integrl e t t β 1 E α,β zt α )dt= 1 1 z z <1). 3.3.27) = dk d k = dk d k e pt t β 1 E α,β ±t α )dt p α β p α ) = k!p α β p α ) k+1, Rβ)>. Corollry 3.3.3. where Rp)> 2. e pt t k 1 2 E k) 1 ± t)dt= 2, 1 2 k! p ) k+1 3.3.28) Exercises 3.3. 3.3.1. prove tht 1 Γα) 1 3.3.2. Prove tht where 1 Γα) t u γ 1 1 u) α 1 E δ β,γ zuβ )du=e δ β,γ+α z),rα)>,rβ)>,rγ)>. x s) α 1 s t) γ 1 E δ β,γ [λs t)β ]ds=x t) γ+α 1 E δ β,γ+α [λx t)β ] Rα)>,Rγ)>.

3.3. GENERALIZED MITTAG-LEFFLER FUNCTION 91 3.3.3. Prove tht where 1 Γα) t s t) α 1 x s) γ 1 E δ β,γ [λx s)β ]ds=x t) γ+α 1 E δ β,γ+α [λx t)β ] Rα)>,Rγ)>,Rβ)>. 3.3.4. Prove tht for n=1, 2, E δ n,γ z)=π n 1 2 n 1 2 γ Γγ) 1F n δ; n;γ); n n z), where n;γ) represents the sequence of n prmeters γ n, r+1 n,γ+n 1 n. 3.3.5. Show tht for Rβ) >, Rγ) >, d dz ) m E δ β,γ z)=δ) me δ+m β,γ+mβ z). 3.3.6. Prove tht for Rβ) >, Rγ) >, z d dz +δ ) E δ β,γ z)=δeδ+1 β,γ z). 3.3.7. Prove tht forrγ)>1, 3.3.8. Prove tht γ βδ 1) E δ β,γ z)= Eδ β,γ 1 z) βδeδ+1 β,γ z). t v 1 x t) µ 1 E γ ρ,µw[x t] ρ ) E σ ρ,v wtρ )dt=x µ+v 1 E γ+σ ρ,µ+v wxρ ), whereρ,µ,γ,v,σ,w ;Rρ),Rµ),Rv)>. 3.3.9. Find nd give the conditions of vlidity. L 1 { s λ 1 z s ρ ) α }

92 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.3.1. Prove tht L 1 [ s λ 1 z 1 s ) α 2 ]= tλ 1 Γλ) Φ 2[α 1,α 2 ;λ; z 1 t, z 2 t], s ) α 1 1 z 2 whererλ)>,rs)>mx[,rz 1 ),Rz 2 )] ndφ 2.) is the confluent hypergeometric function of two vribles defined by Φ 2 b, b ; c; u, z)= k,l= 3.3.11. From the bove result deduce the formul b) k b ) l u k z l. c) k+l k!l! L 1{ s λ 1 z/s) α} = tλ 1 Γλ) whererλ)>,rs)>mx[, z ]. φα,λ; zt), 3.4. Frctionl Integrls This section dels with the definition nd properties of vrious opertors of frctionl integrtion nd frctionl differentition of rbitrry order. Among the vrious opertors studied, it involves the Riemnn-Liouville frctionl integrl opertors, Riemnn-Liouville frctionl differentition opertors, Weyl opertors nd Kober opertors etc. Besides the bsic properties of these opertors, their behviours under Lplce, Fourier nd Mellin trnsforms re lso presented. Appliction of Riemnn-Liouville opertors in the solution of frctionl order differentil nd frctionl order integrl equtions is demonstrted. 3.4.1. Riemnn-Liouville frctionl integrls of rbitrry order Nottion 3.4.1. Definition 3.4.1. Ix n, D n x, n {} : Frctionl integrl of integer order n Ix n f x)= D n x f x)= 1 Γn) where n {}. x t) n 1 f t)dt 3.4.1) We begin our study of frctionl clculus by introducing frctionl integrl of integer order n in the form Cuchy formul):

3.4. FRACTIONAL INTEGRALS 93 is, D n x f x)= 1 x t) n 1 f t)dt. 3.4.2) Γn) It will be shown tht the bove integrl cn be expressed in terms of n-fold integrl, tht D n x f x)= 1 2 dx 1 dx 2 n 1 dx 3 Proof. When n = 2, then using the well-known Dirichlet formul, nmely f t)dt. 3.4.3) 3.4.3) becomes b dx f x,y)dy= b b dy f x, y)dx 3.4.4) y dx 1 1 f t)dt= = dt f t) t dx 1 x t) f t)dt. 3.4.5) This shows tht the two-fold integrl cn be reduced to single integrl with the help of Dirichlet formul. For n=3, the integrl in 3.4.3) gives D 3 x f x)= = 1 2 dx 1 dx 2 f t)dt [1 2 dx 1 dx 2 Using the result 3.4.5) the integrls within big brckets simplify to yield D 3 x f x)= dx 1 [1 If we use 3.4.4), then the bove expression reduces to D 3 x f x)= dt f t) t Continuing this process, we finlly obtin D n x f x)= x 1 n 1)! x 1 t)dx 1 = ] f t)dt. 3.4.6) ] x 1 t) f t)dt. 3.4.7) x t) 2 f t)dt. 3.4.8) 2! x t) n 1 f t)dt. 3.4.9)

94 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS It is evident tht the integrl in 3.4.9) is meningful for ny number n provided its rel prt is greter thn zero. 3.4.2. Riemnn-Liouville frctionl integrls of orderα Nottion 3.4.2. orderα. Ix α, D α x, Iα + ; Riemnn-Liouville left-sided frctionl integrl of Nottion 3.4.3. orderα. Definition 3.4.2. xib α, xd α b, Iα b ; Riemnn-Liouville right-sided frctionl integrl of Let f x) L, b),α C Rα)>), then Ix α f x)= D α x f x)=i+ α f x)= 1 f t)dt Γα) x t) 1 α, x> 3.4.1) is clled Riemnn-Liouville left-sided frctionl integrl of orderα. Definition 3.4.3. Let f x) L, b),α C Rα)>), then xi α b f x)= xd α b f x)= I α b f x)= 1 b f t)dt Γα) x t x) 1 α, x<b 3.4.11) is clled Riemnn-Liouville right-sided frctionl integrl of orderα. Exmple 3.4.1. If f x)=x ) β 1, then find the vlue of I α x f x). Solution: We hve Ix α f x)= 1 x t) α 1 t ) β 1 dt Γα) If we substitute t=+yx ) in the bove integrl, it reduces to where Rβ) >. Thus Γβ) Γα+β) x )α+β 1 I α x f x)= Γβ) Γα+β) x )α+β 1. 3.4.12)

Exmple 3.4.2. It cn be similrly shown tht 3.4. FRACTIONAL INTEGRALS 95 xib α Γβ) gx)= Γα+β) b x)α+β 1, x<b, 3.4.13) whererβ)>ndgx)=b x) β 1. Note 3.4.1. It my be noted tht 3.4.12) nd 3.4.13) give the Riemnn-Liouville integrls of the power functions f x)=x ) β 1 ndgx)=b x) β 1,Rβ)>. 3.4.3. Bsic properties of frctionl integrls Property 3.4.1. Frctionl integrls obey the following property: Proof. I α x I x β φ= I x α+β φ= I x β I x α φ, xi b α x I b β φ= x I b α+β φ= x I b β x I b α φ. 3.4.14) By virtue of the definition 3.4.1), it follows tht Ix α I β x φ= 1 dt 1 Γα) x t) 1 α Γβ) 1 x = du φu) Γα)Γβ) If we use the substitutiony= t u x u, the vlue of the second integrl is 1 Γα)Γβ)x u) 1 α β 1 u t φu)du t u) 1 β dt x t) 1 α t u) 1 β 3.4.15) y β 1 1 y) α 1 dy= x u)α+β 1, Γα+β) which, when substituted in 3.4.15) yields the first prt of 3.4.14). The second prt cn be similrly estblished. In prticulr, I n+α x f= I x n I x α f, n,rα)>) 3.4.16) which shows tht the n-fold differentition d n dx n I x n+α f x)= I x α f x), n,rα)>) 3.4.17) for ll x. Whenα=, we obtin I x f x)= f x); I n x f x)= dn dx n f x)= f n) x). 3.4.18) Note 3.4.2. The property given in 3.4.14) is clled semigroup property of frctionl integrtion.

96 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS Nottion 3.4.4. L, b), spce of Lebesgue mesurble rel or complex vlued functions. Definition 3.4.4. L, b), consists of Lebesgue mesurble rel or complex vlued functions f x) on [, b]: L, b)={ f : f 1 b f t) dt<+ }. 3.4.19) Note 3.4.3. The opertors I α x nd xi α b re defined on the spce L, b). Property 3.4.2. The following results holds. b f x) I α xg)dx= b gx) x Ib α f )dx. 3.4.2) 3.4.2) cn be estblished by interchnging the order of integrtion in the integrl on the left-hnd side of 3.4.2) nd then using the Dirichlet formul 3.4.4). The bove property is clled the property of integrtion by prts for frctionl integrls. 3.4.4. A useful integrl We now evlute the following integrl given by Sxen nd Nishimoto [Journl of Frctionl clculus, Vol.6, 1994), 65-75]. b t ) α 1 b t) β 1 ct+ d) γ dt=c+d) γ b ) α+β 1 [ Bα,β) 2 F 1 α, γ;α+β; b)c ], 3.4.21) c+d whererα)>,rβ)>, rg d+bc d+c <π,, c nd d re constnts. Solution: Let I= b t ) α 1 b t) β 1 ct+ d) γ dt = c+d) γ k= 1) k γ) k c k c+d) k b t ) α+k 1 b t) β 1 dt = c+d) γ b ) α+β 1 Bα,β) 2 F 1 γ, α;α+β; b)c c+d ).

3.4. FRACTIONAL INTEGRALS 97 In evluting the inner integrl, the modified form of the bet function, nmely b whererα)>,rβ)>, is used. t ) α 1 b t) β 1 dt=b ) α+β 1 Bα,β), 3.4.22) Exmple 3.4.3. As consequence of 3.4.21), it follows tht Ix α [x ) β 1 cx+d) γ ]=c+d) γ x ) α+β 1 Γβ) Γα+β) x)c 2 F 1 β, γ;α+β; ), 3.4.23) c+d whererα)>,rβ)>, rg x)c c+d <π;, c nd d being constnts. In similr mnner we obtin the following result. Exmple 3.4.4. We lso hve xib α [b x)β 1 cx+d) γ ]=cx+d) γ b x) α+β 1 Γβ) Γα+β) whererα)>,rβ)>, rg x b)c cx+d <π. 2 F 1 α, γ;α+β; x b)c ), 3.4.24) cx+d Exmple 3.4.5. On the otherhnd if we set γ = α β in 3.4.21) it is found tht whererα)>,rβ)>. D α x [x ) β 1 cx+d) α β ] = Γβ) Γα+β) c+d) α x ) α+β 1 d+ cx) β, 3.4.25) Exmple 3.4.6. Similrly, we hve whererα)>,rβ)>. xi α b [b x)β 1 cx+d) α β ] = Γβ) Γα+β) cx+d) β bc+d) α b x) α+β 1 3.4.26)

98 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.4.5. The Weyl integrl Nottion 3.4.5. xw α, x I α, Weyl integrl of orderα. Definition 3.4.5., is defined by xw α The Weyl frctionl integrl of f x) of order α, denoted by xw α f x)= 1 t x) α 1 f t)dt, < x< ) 3.4.27) Γα) x whereα C,Rα)>. 3.4.27) is lso denoted by I α f x). Exmple 3.4.7. Prove tht Solution: We hve xw e α λx = e λx, whererα)>. 3.4.28) λα xw α e λx = 1 Γα) = e λx Γα)λ α x = e λx λ α,rα)>. t x) α 1 e λt dt,λ> u α 1 e u du Nottion 3.4.6. xd α, Dα : Weyl frctionl derivtive. Definition 3.4.6. : is defined by The Weyl frctionl derivtive of orderα, denoted by x D α d ) m xd α f x)= D α f x)= 1) m dx d ) m 1 = 1) m dx Γm α) where m 1 α<m, m N,α c. Exmple 3.4.8. Find x D α e λx,λ>. xw m α x ) f x) f t)dt t x) 1+α m, < x< ) 3.4.29)

3.4. FRACTIONAL INTEGRALS 99 Solution: We hve xd α e λx = 1) m d dx ) m xw m α e λx = 1) m d dx) m λ m α) e λx 3.4.3) =λ α e λx. 3.4.6. Bsic properties of Weyl integrl Property 3.4.3 : The following reltion holds. ) φx) Ixψx) α dx= ) xw φx) α ψx)dx. 3.4.31) 3.4.31) is clled the formul for frctionl integrtion by prts. It is lso cled Prsevl equlity. 3.4.31) cn be estblished by interchnging the order of integrtion. Property 3.4.4 : Weyl frctionl integrl obeys the semigroup property. Tht is, ) ) ) xw α xw β f = xw α+β f = xw β xw α f. 3.4.32) Proof. We hve xw α xw β f x)= 1 Γα) 1 Γβ) x t dtt x) α 1 u t) α 1 f u)du. Using the modified form of Dirichlet formul 3.4.4), nmely x dtt x) α 1 u t) β 1 f u)du = Bα,β) t nd letting, 3.4.33) yields the desired result t u t) α+β 1 f u)du, 3.4.33) ) ) xw α xw β f = xw α+β f. 3.4.34)

1 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS Nottion 3.4.7. Definition 3.4.7. W α x,i α +: Weyl integrl with lower limit. Another compnion to the opertor 3.4.27) is the following: Wx α f x)=i+ α f x)= 1 x t) α 1 f t)dt, < x< ) 3.4.35) Γα) whereα c, Rα)>). Note 3.4.4 : The opertor defined by 3.4.34) is useful in frctionl diffusion problems of physics nd relted res. Exmple 3.4.9. Prove tht Solution: We hve, by setting x t = u. Wx α e x = ex α. 3.4.36) Note 3.4.5 : where An lterntive form of 3.4.34) in terms of convolution is given by Wx α f x)= 1 t+ α 1 f x t)dt 3.4.37) Γα) t α 1 + = t α 1, t>, t< 3.4.38) Exmple 3.4.1. Prove tht where >, <Rv)<1. xw v cos x)= v cosx+ 1 πv) 3.4.39) 2 Solution: The result follows from the known integrl x u) v 1 cos x dx= Γv) cosu+ vπ u v 2 ) 3.4.4) where >, <Rv)<1.

3.4. FRACTIONAL INTEGRALS 11 Exmple 3.4.11. Prove tht Hint : Use the integrl xw sin v x)= v sinx+ 1 πv) 3.4.41) 2 where >, <Rv)<1. u x u) v 1 sin x dx= Γv) v sinu+ vπ 2 ) 3.4.42) Exercises 3.4. 3.4.1. Prove tht I α x x )β 1 ) = Γβ) Γα+β) x )α+β 1, Rβ)>. 3.4.2. Prove tht I α x x±c) γ 1 )= ±c)γ 1 Γα+1) x )α 2F 1 1, 1 γ;α+1; whererβ)>,γ, < x<b. 3.4.3. Prove tht ]) Ix [x ) α β 1 b x) γ 1 = Γβ) x ) α+β 1 Γα+β) b ) 1 γ 2 F 1 β, 1 γ;α+β; x ) b whererβ)>,γ, < x<b. x ) ±c 3.4.4. Prove tht Rβ)>, < x<b. [ x ) Ix α β 1]) b x) α+β = Γβ) Γα+β) x ) α+β 1 b ) α b x) β

12 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.4.5. Prove tht ]) Ix [x ) α β 1 x±c) γ 1 = Γβ) x ) α+β 1 Γα+β) ±c) 1 r 2 F 1 β, 1 γ;α+β; x ), ±c whererβ)>,γ, ±c>. 3.4.6. Prove tht 3.4.7. Prove tht where x>. [ x ) Ix α β 1]) x±c) α+β = Γβ) Γα+β) x ) α+β 1 ±c) α x±c) β ) Ix α [eλx ] = e λ x ) α E 1,α+1 λx λ). 3.4.8. Prove tht where Rβ) >. 3.4.9. Prove tht ) Ix α [e λx x ) β 1 ] = Γβ) ) Γα+β) eλ x ) α+β 1 1F 1 β;α+β;λx λ, I α x [ ]) x ) β 1 lnx ) = x ) α+β 1 Γβ) Γα+β) [lnx )+ψβ) ψα+β)]. whererβ)>, whereψ.) is the logrithmic derivtive of the gmm function. 3.4.1. Prove tht where Rν) > 1. [ Ix ν x ) ν 2 J ν λ ]) 2 ) ν +ν x ) = x ) 2 J +ν λ x ), λ

3.4. FRACTIONAL INTEGRALS 13 3.4.11. Prove tht where Rβ) >. I ν x [ ]) x ) β 1 2F 1 µ,ν;β;λx )) = Γβ) Γα+β) x )ν+β 1 2F 1 µ,ν;ν+β;λx λ), 3.4.12. Prove tht I ν x [x ) β 1 E µ,β x ) µ )])=x ) ν+β 1 E µ,ν+β [x ) µ ]. 3.4.13. Show tht [ Ix ν ]) x µ 1 sin x where >,Rν)>,Rµ)>. 3.4.14. Estblish the formul [ Ix ν = xµ+ν 1 Γµ) 2i Γµ+ν) [ 1F 1 µ;µ+ν; ix) 1 F 1 µ;µ+ν; ix) ]) x µ 1 cos x where >,Rν)>,Rµ)>. 3.4.15. Prove the following results: I ν x = xµ+ν 1 Γµ) 2Γµ+ν) [ 1F 1 µ;µ+ν; ix) 1 F 1 µ;µ+ν; ix) { sinλx ) cos λx ) })= i 1±1)/2 2Γν+1) x )ν [ ] 1F 1 1;ν+1; iλx )) 1 F 1 1;ν+1; iλx )) ], ],

14 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.4.16. Prove the following results: [ { Ix ν x ) 1 }]) 2 cosλ x sinλ x = λ ) 1 2 ν π x ) 2ν 1)/4 2 J ν 1 2 λ x ) I ν 1 2 λ x ) 3.4.17. Prove tht [ ]) Ix ν x α 1 2F 1, b; c; wx) = Bα,ν) x α+ν 1 3F 2, b,α; c,α+ν; wx), Γν) where x,rα),rν)>, rg 1+wy) <π. 3.4.18. Prove tht [ Ix ν x α 1 2F 1, b; c; 1 wx) ])= xα,ν 1) Γν) 3 F 2, b,α; +b c+1,α+ν;wx) [ + wc b c, +b c, c b+α ] Γ Γν) xc b+α+ν 1, b, c b+α+ν [ c, c b,α,ν ] Γ c, c b, α+ν 3 F 2 c, c b, c b+α; c b+1; c b+α+ν;wx) [ b c ] where x,rα),rν),rc b+α)>, rgw <π ndγ stnds for the rtio of product of gmm functions Γ)Γb)Γc) Γd)Γe)Γ f ). 3.4.19. Prove tht d e f. [ ]) Ix ν x c 1 1 xz) ρ 2F 1, b; c;wx) = Γc) Γν+c) xc+ν 1 1 xz) ρ F 3 ρ,,ν, b, c+ν; where x,rc),rν)>; rg 1 wx) <π nd rg 1 z) <π; F 3 hypergeometric function of two vribles, defined by F 3, ; b, b ; c;y, z)= ) k ) l b) k b ) l y k z l. c) k+l k!l! k,l= xz xz 1,wx), is the Appell s

3.4.2. Prove tht 3.4. FRACTIONAL INTEGRALS 15 [ x ) Ix ν α 1] = x )α+ν 1 Γα) x y x y)γα+ν) 2 F 1 1,ν;α+ν; x ) x y whererα),rβ)>;y<< x. 3.4.21. Prove tht ]) Ix [x ν α 1 1 ux) ρ 1 νx) λ = xα+ν 1 Γα) F 1 α,ρ,λ,α+ν; ux,νx), Γα+ν) x u <1, rg u <π; rg ν <π, x ν <1; where x,rα),rν)>. F 1 defined by 3.4.22. Prove tht [ ]) Ix ν x α 1 ln n cx+d) is the Appell s hypergeometric function of the two vribles F 1, b, b ; c;y, z)= = xα+ν 1 Γα) Γα+ν) ) k+l b) l b ) l y k z l. c) k+l k!l! k,l= n [ ρ n where x,rα),rβ)>; rg cx+d) <π nd t x. 3.4.23. Prove tht d ρ 2F 1 ρ,α;α+ν; xc )] d ρ=, where Rν) >. [ ]) Ix ν lncx+d) = 1 Γν) t ν 1 lncx+d ct)dx 3.4.24. Prove tht [ ]) Ix ν x α 1 lncx+d) = xα+ν 1 Γα) ln d Γα+ν) + cd 1 x α+ν Γα+1) Γα+ν+1) 3 F 2 α+1, 1, 1; 2,α+ν+1; cx d ), where x,rα),rν)>; rg cx+d) <π nd t x.

16 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.4.25. Prove tht ]) Ix [e ν λx = x ν E 1,ν+1 λx), Rα)>). 3.4.26. Prove tht [ sinh λx Ix ν ])= x ν+1 E 2,ν+2 λx 2 ), Rα)>). λx 3.4.27. Prove tht [ ]) Ix ν λ, E ν λx ν ) = E ν λx ν ) 1,Rν)>. 3.4.28. Prove tht ]) Ix [x ν ρ 1 x+) σ = σ x ρ+ν 1 Γρ) Γρ+ν) 2 F 1 σ,ρ;ν+ρ; x ), whererν)>;rρ)>, rg x ) <π. 3.4.29. Prove tht ]) Ix [x ν ρ 1 x k + k ) σ = kσ x ν+ρ 1 Γρ) Γρ+ν) ) k+1 F k σ, k;λ); k;ρ+ν); xk k, where k,rν)>;rρ)>, rg x ) < π k, nd k; ) represents the sequence of prmeters k, +1 k, +k 1 k. 3.4.3. Prove tht [ ]) Ix ν x ρ 1 expx k ) = Γρ) Γρ+ν) xρ+ν 1 kf k k;ρ); k;ρ+ν); x ), k whererν)>;rρ)>, k=2, 3, 4,.

3.4.31. 3.4. FRACTIONAL INTEGRALS 17 [ ]) Ix ν x ρ 1 pf q 1,, p ;ρ, b 2,, b q ; x) = xρ+ν 1 Γρ) Γν+ρ) p F q 1,, p ;ρ+ν, b 2,, b q ; x) where p q+1,rν)>;rρ)>, x <1 if p=q+1. 3.4.32. [ Ix ν x ρ 1 G m,n p,q x )]) 1,, p = x ρ+ν 1 G m,n+1 b 1,,b q p+1,q+1 x ) 1 ρ, 1,, p b 1,,b q ;1 ρ ν where G m,n p,q.) is Meiger s G-Function; m+n> 1 2 p+q), rg < m+n p 2 q 2 Rρ+b j )>, j=1,, m,rν)>. ) π 3.4.33. Prove tht [ Ix α x ρ 1 H m,n p,q x p,a p ) b q,b q ) )]) [ = x ρ+α 1 H m,n+1 p+1,q+1 x ] 1 ρ,1), p,a p ) b q,b q ), 1 ρ α,1) [ b whererρ+min j 1 j m B j )]>,Rα)>, rg < 1 2 πc ; c = n A j j=1 p j=n+1 A j + m B j j=1 q j=m+1 ndµ = p j=1 A j q j=1 B j< orµ = nd < x <β 1 ; p β= A j ) A q j B j ) B j. 3.4.34. Prove tht [ Ix α x ρ 1 H m,n p,q j=1 x λ p,a p ) b q,b q ) )]) j=1 B j > [ = x ρ+α 1 H m,n+1 p+1,q+1 x λ ] 1 ρ,λ), p,a p ) b q,b q ),1 ρ α,λ) where R [ ρ+λmin 1 j m b j B j )]>,Rα)>,λ> rg < 1 2 πc ; c is defined in Exercise 3.4.33 bove, c > ;µ < orµ = nd < x λ <β 1 ;

18 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.4.35. Prove tht β= p A j ) A j j=1 q B j ) B j ;µ is given in Exercise 3.4.33. j=1 ]) xi [x ν α 1 x u) ρ x v) λ = xα+ν λ ρ 1 Γ1+λ+ρ α ν) Γ1+λ+ρ α) F 1 1+λ+ρ α ν,ρ,λ, 1+λ+ρ α; u/x,v/x), where x>; rg u <π nd u < x; rgv <π nd v < x; <Rν)<Rλ+ρ α)+1. 3.4.36. Prove tht [ ]) xi ν x α 1 lncx+d) = x α+ν 1Γ1 α ν) Γ1 α) [ ] lncx)+ψ1 α) ψ1 α ν) + dc 1 x α+ν 2Γ2 α ν) 3F 2 2 α ν, 1, 1; 2, 2 α; d Γ2 α) cx ) where x,rβ)>,rα+ν)<1; rg cx+d) <π nd x t<. 3.4.37. Prove tht ]) xw [x ν λ x+) µ = xµ+ν λ Γλ µ ν) 2 F 1 µ,λ µ ν;λ µ; /x) Γλ µ) where <Rν)<Rλ µ); rg /x) <πor /x) <1,Rν)>. 3.4.38. Prove tht whererν)>,rx)>. 3.4.39. Prove tht xw [x ν ν 1 e ])=π x 2 1 x/) ν 1 2 e 1 1 ) 2 x K ν 12 2 x ]) xw [x ν λ e x = B 2 1 x B 1 2 e 1 2 x W A,B x) where 2A=1 λ ν, 2B=λ ν;rx)>,rν)>.

3.4. FRACTIONAL INTEGRALS 19 3.4.4. Prove tht where Rν) >. [ ]) xw ν x 2ν e π ) 1 2 1 ) ) x = ν 2 exp I x 2x ν 1, 2 2x 3.4.41. Prove tht where <Rν)<Rλ). [ xw ν x λ exp x)]) = Γλ ν) x ν λ 1F 1 λ ν;λ; Γλ) x ), 3.4.42. Prove tht [ ) xw ν exp x 1 2 )])=2 ν+ 2 1 π 1 1 2 ν 2 x 1 2 ν+ 4 1 Kν+ x 1 2, 12 whererx 1 2 )>,Rν)>. 3.4.43. Prove tht xw +ν whererx 1 2 )>,Rν)>. 3.4.44. Prove tht [x 2 1 exp x 1 2 [ ]) xw ν x λ log x = Γλ ν) Γλ) where <Rν)<Rλ). 3.4.45. Prove tht, x>; <Rν)< 1 2 Rµ)+ 3 4. 3.4.46. Prove tht ) )])=2 ν+ 1 2 π 1 1 2 x ν 2 x 1 2 ν 1 4 Kν x 1 2, 12 x ν λ [log x+ψλ) ψλ ν)], ) xw [x ν µ 2 Jµ x 1 2 )])=2 ν ν x 1 2 ν 1 2 µ J µ ν x 1 2, )]) xw [x ν λ J µ x 1 2 where >, x>, <Rν)< 1 4 Rλ). = 2 2λ 2λ x ν G 2, 1,3 2 x 4 ) ν,λ+ 1 2 µ,λ 1 2 µ,

11 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.4.47. Prove tht [ xw α x λ 1 2F 1, b; c; 1 wx) ])= w Γα) xλ+α 1 Γ [ c, b,α, λ α+1 b, c, λ+1 3 F 2, c b, λ α+1; λ+1, b+1; + w b Γα) xλ+α b 1 Γ [ c, b,α, b α+1, c b, b λ+1 3 F 2 b, c, b α+1; b +1, b λ+1; where x,rα)>,rλ+α )>1,Rλ+α b)<1; rgw <π. 3.4.48. Prove tht ] 1 wx ) 1 wx ) [ ]) xw ν x λ 1 2F 1, b; c; wx) = 1 Γν) xα+ν 1 Bν, 1 λ ν) [ 3 F 2, b,λ; c,λ+ν; wx)+ w1 λ ν c, λ ν+1, b λ ν+1,λ+ν 1 ] Γ Γν), b, c λ ν+1 3 F 2 1 ν, λ ν+1, b λ ν+1; 2 λ ν, c λ ν+1; wx), ] where x,rν)>;rλ+ν ),Rν b)<1; rgw <π. 3.4.49. Prove tht [ xw ν x α 1 G m,n p,q x 1,, p b 1,,bq )]) [ = x α+ν 1 G m+1,n p+1,q+1 x ] 1,, p,1 α 1 α ν,b 1,,b q whererν)>,r[α+ν+mx 1 j n j )]<2 Exercise 3.4.33 rg < 1 2 πc, c >. c is defined in 3.4.5. Prove tht [ xi ν x α 1 H m,n p,q x λ p,a p ) b q,b q ) )]) [ = x α+ν 1 H m+1,n p+1,q+1 x λ ] p,a p ),1 α,λ) 1 α ν,λ),b q,b q ) whereλ>, Rν)>, rg < 1 2 πc, c >, c is defined in Exercise 3.4.33; R[α+ν+mx 1 j n j 1) A j ]<1.

3.5. DERIVATIVES OF FRACTIONAL ORDER 111 3.5. Derivtives of Frctionl Order In this section, we study vrious frctionl order derivtives which occur in certin rection relxtion) nd diffusion problems. 3.5.1. Riemnn - Liouville frctionl derivtives of rbitrry order Nottion 3.5.1. {α} mens the frctionl prt of number α, {α} < 1. Nottion 3.5.2. [α] mens the integrl prt of number α. Note 3.5.1 We note tht α={α}+[α]. 3.5.1) Nottion 3.5.3. D α +, D α xφx), Riemnn - Liouville derivtive of the function φx) of order α, left - hnd). Nottion 3.5.4. D α b, bd α xφx), Riemnn - Liouville derivtive of the function φx) of order α, right-hnd). Definition 3.5.1. The left-hnd Riemnn - Liouville derivtive of order α > is defined by D α +φx)= D α xφx)= 1 d ) n Γn α) dx φt)dt n=[α]+1). 3.5.2) x t) α n+1 Exmple 3.5.1. Prove tht D α x x γ = Γγ+1) Γγ+1 α) xγ α,α,γ> 1, x>. 3.5.3) Solution: We hve

112 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS D α 1 d n x xγ )= t γ x t) n α 1 dt Γn α) dx n Γγ+ 1) = Γγ+1+n α) γ α+1) nx γ α forγ+1>,γ+1+n α> Γγ+ 1) = Γγ+1 α) xγ α, forγ+1>, γ α+1) n. Note 3.5.2 It is interesting to observe tht forγ=, We obtin D α x 1)= x α, α 1, 2,. 3.5.4) Γ1 α) which is remrkble result nd indictes tht frctionl derivtive of constnt is not zero. Definition 3.5.2. The right - hnd Riemnn - Liouville frctionl derivtive of order α, of the function φx) is defined by D α b φx)= xd α 1)n d ) n b bφx)= Γn α) dx x In short, we cn express 3.5.2) in the form φt)dt n=[α]+1). 3.5.5) t x) α n+1 nd 3.5.3) s D α xφx)= dn dx n I n α x φx) 3.5.6) xd α dn bφx)= dx n x Ib n α φx). 3.5.7) We shll lso employ the nottions Similrly, we hve 1 D α xφ= Ix α φ= Ix) α φ,α. 3.5.8) xd α b φ= bi α x φ= bi α x) 1φ,α. 3.5.9)

3.5. DERIVATIVES OF FRACTIONAL ORDER 113 Exmple 3.5.2. Prove tht I ν x ln x= x ν [ln x γ ψν+1)]. Γν+1) Solution: We hve Ix ν ln x= 1 x t) ν 1 ln t dt Γν) If we mke the chnge of vrible t= xu, then We know tht Ix ν ln x= 1 1 x ν 1 u) ν 1 ln x+ln u)du Γν) = xν lnx) 1 Γν+1) + xν 1 u) ν 1 ln u du. Γν) 1 t α 1 1 t) β 1 ln t dt=bα,β)[ψα) ψα+β)], whererα)>,rβ)>. Applying the bove formul forα=1, nd noting tht ψ1)= γ, we see tht I ν x ln x= x ν [ln x γ ψν+1)]. Γν+1) Similrly we cn estblish the result in the next exercise. Exmple 3.5.3. Prove tht D α x ln x)= x ν [ln x γ ψ ν+1)] Γ1 ν) Exmple 3.5.4. Prove tht D α x ex )= x α Γ1 α) 1 F 1 1; 1 α; x) Solution: We hve

114 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS D α x ex )= r= = = r r! D α x xr ) r Γr+ 1) r! Γr α+1) xr α r= x α Γ1 α) 1 F 1 1; 1 α; x). Exmple 3.5.5. Prove tht D α x x+)p = p x α Γ1 α) 2 F 1 1, p; 1 α; x ). Solution: We hve D α x x+)p = p 1) r r= = p r= r! p) r D α x xr ) r 1) r p) r Γr+ 1) x r α r! r Γr α+1) = p x α Γ1 α) 2 F 1 1, p; 1 α; x ). In bove expression, the following result hs been used: D α x xp )= Γp+1)xp α Γp α+1). 3.5.1. Prove tht ) D α x [sin x] = Exercises 3.5. x α [ ] 1F 1 1; 1 α; ix) 1 F 1 1; 1 α; ix). 2iΓ1 α)

3.5. DERIVATIVES OF FRACTIONAL ORDER 115 3.5.2. Prove tht ) D α x [cos x] = x α [ ] 1F 1 1; 1 α; ix)+ 1 F 1 1; 1 α; ix). 2Γ1 α) 3.5.3. Prove tht ) D α x [xp ln x] = Γp+1) Γp+1 α) xp α [ln x+ψp+1) ψp α+1)] whereψ.) is the digmm function. 3.5.4. Prove tht D α x [xp + x) q ] )= Γp+1)xp α q 2F 1 q, p+1; p α+1; x Γp α+1) ), for p α+1) n, p 1, 2, 3,, x <1. 3.5.5. Prove tht D α x [xp e x ] )= Γp+1)xp α Γp α+1) 1 F 1 p+1; p α+1; x) for p α+1+k) n, k=, 1,, p 1, 2, 3,, Specil functions cn be expressed s frctionl derivtives. This cn be seen from the following exercises. 3.5.6. Prove tht forrc b)>,rb)>. 3.5.7. Prove tht forr c)>,rc)>. 2F 1, b; c; x)= Γc) Γb) x1 c D b c x [x b 1 1 x) ] 1F 1 ; c; x)=φ, c; x)= Γc) Γ) x1 c D c x e x x 1 )

116 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.5.8. Prove tht forrc d)>,rd)>. 3.5.9. Prove tht [ 1,, p ] p+1f q+1 ; x = Γd) { [ 1 b 1,,b q Γc) x1 d D c d x x c 1,, p ]} PF q ; x b 1,,b q 3.5.1. Prove tht 3.5.11. Prove tht J ν x)=π 1 2 2 1 ν x ν D ν+ 2 1 x sin x). ψx)= γ+ln z Γx)z 1 x Dz 1 x ln z). γ, z)=γ)e z D z e z ) whereγ, z) is incomplete gmm function. 3.5.12. Prove tht ]) D ν x [x µ 2 Jµ x 1 2 ) = 2 ν x 1 2 µ ν) J µ ν x 1 2 ), where Rµ) > 1. 3.5.13. Prove tht [ ]) D ν x sin x 1 2 = 1 2 π 1 1 1 2 2x 2 ) ν 2 J 1 2 ν x 1 2 ). 3.5.14. Prove tht ]) D ν x [x 1 1 2 cos x 2 = π 2x 1 2 ) ν 2 1 J ν 1 x 1 2 ). 2 3.5.15. Prove tht ]) D ν x [x µ 2 Iµ x 1 2 ) = 2 ν x 1 2 µ ν) I µ ν x 1 2 ).

3.5. DERIVATIVES OF FRACTIONAL ORDER 117 3.5.16. Show tht nd [ ]) D ν x sinh x 1 2 = 2 1 1 1 π 2x 2 ) ν 2 I 1 1 νx 2 ) 2 [ ]) D ν x cosh x 1 2 = π 2x 1 2 ) 1 2 ν I 1 2 ν x 1 2 ). 3.5.17. Prove tht [ ]) D ν x x λ 1F α; ; x) = Γλ+1) Γλ ν+1) xλ ν 2 F 1 λ+1,α,λ ν+1; x). 3.5.18. Estblish the result D ν x [ ] x λ 2F 1, b; c; x) = Γλ+1) Γλ ν+1) xλ ν 3 F 2 λ+1,, b, c;λ ν+1; x) whererλ)> 1,Rλ+1 ν)>, nd c, 1, 2, nd x <1. 3.5.19. Prove tht D ν x whererλ)> 1,Rλ ν+1)>. 3.5.2. Prove tht [ ] x λ pf q 1, p ; b 1, b q ; x) = Γλ+1) Γλ ν+1) xλ ν p+1 F q+1 λ+1, 1, p ; b 1, b q,λ ν+1; x) [ ]) D ν x x λ pf q 1, p ; b 1, b q ; x 2 ) = Γλ+1) Γλ ν+1) xλ ν p+2 F q+2 [ 1 2 λ+1), 1 2 λ+2), 1,, p ; b 1,, b q, 1 2 λ ν+1), 1 2 λ ν+2); x2 ] whererλ)> 1,Rλ ν+1)> nd x 2 <1; b j, 1, 2, j=1,, p).

118 3. MITTAG-LEFFLER FUNCTIONS AND FRACTIONAL CALCULUS 3.5.21. Show tht 3.5.22. Show tht 3.5.23. Prove tht ) D ν x J µ [x] = D ν x [ex2 ] )= x ν Γ2n+1)x 2 ) n Γ2n ν+1)n!. n= x µ ν 2µΓµ ν+1) 2 F 3 [ 1 2 µ+1), 1 2 µ+2);µ+1, 1 2 µ ν+1), 1 1 ) 2 ] 2 µ ν+2); 2 x. [ n,α1,,α p ;x] p+1f q = Γβ 1) Γβ q ) β 1,,β q Γα 1 ) Γα p ) x1 β q ) x α p β q 1 D α p β q x D α p 1 β q 1 x x α 3 β 2 D α 2 β 2 x x α 2 β 1 D α 1 β 1 x ) x α p 1 β q 2 ] [x α1 1 1 x) n. 3.5.24. Prove tht Ixe ν λ x)= x ν E 1,ν+1 λx) whererα)>. ) 3.5.25. Prove tht Ix ν[cosh λ x] = x ν E 2,ν+1 λx 2 ), whererα)>. 3.5.26. Prove tht [ Ix ν sinh λ x ])= x ν+1 E λ x 2,ν+2 λx 2 ). References Abrmowitz, M nd Stegun, I.A. 1965). Hndbook of Mthemticl Functions, Dover, New York. Agrwl, R.P. 1953). A propos d une note de M.Pierre Humbert, C.R.Acd.Sci.Pris 236, 231-232 Anh, V.V nd Leonenko, N.N. 21). Spectrl nlysis of frctionl kinetic equtions with rndom dt, J. Sttist. Phys. 14, 1349-1387.

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