EXTENSION OF THE FRACTIONAL DERIVATIVE OPERATOR OF THE RIEMANN-LIOUVILLE D. BALEANU,2,3, P. AGARWAL 4, R. K. PARMAR 5, M. AL. QURASHI 6 AND S. SALAHSHOUR 7 Abstract. In this paper, by using the generalized beta function, we extend the definition of the fractional derivative operator of the Riemann-Liouville and discuses its properties. Moreover, we establish the some relations to extended special functions of two and three variables via generating functions.. Introduction, Definitions and Preliminaries In recent years, fractional derivative operators and their extensions have received considerable attention. There are many definitions of generalized fractional derivatives involving extended beta and hypergeometric functions 2, 3, 7, ]. In continuation, Özarslan and Özergin 5] was introduced and studied the extended fractional derivative operator. Definition. The extended fractional derivative operator defined by: ) z z t) η pz 2 ) z t)t e f t) dt Re η) < η,p Γ η) {f z)} : d m { dz m η m } m ) {f z)} Re η) < m m N).) Clearly, the special case of.), when p reduce immediately to Riemann Liouville fractional derivativesee,2, 3]). In recent years number of researchers has been systematically study the extended fractional derivative operators and discussed their applications in different fields see, 3, 4, 5, 7]). In view of the effectiveness of the above works, here by using the generalized beta function due to Choi et al. ], we extend the definition of the fractional derivative operator of the Riemann-Liouville and discuses its various properties. Moreover, we establish the some relations to extended special functions of two and three variables via generating functions. For our purpose we recall the some earlier works and definitions. 2 Mathematics Subject Classification. Primary 33C5, 33C5, 33C2; Secondary 33C65, 33C99. Key words and phrases. Beta function, Hypergeometric function of two and three variables, Fractional derivative operator, Generating functions, Mellin transform.
2D. BALEANU,2,3, P. AGARWAL 4, R. K. PARMAR 5, M. AL. QURASHI 6 AND S. SALAHSHOUR 7 Definition 2. The generalized beta function defined by see Choi et al. ]]: B p,q x, y ) : t x t) y e p t q t) dt.2) min{rx), Ry)} > ; min{rep), Req)} ) Definition 3. The generalized hypergeometric function defined as see Choi et al. ]]: B p,q b + n, c b ) z n ) 2F ;p,q a, b; c; z) : a) n p, q ; z < ; Rec) > Reb) > Bb, c b) n.3) 2. Extension of Hypergeometric Functions and Integral Representations By making use of.2), we consider another extensions of Appell s and the Lauricella functions of one, two and three variables. Definition 4. The extension of the hypergeometric functions of two and three variables are defined as: B p,q a + m + n, d a) x m y n F a, b, c; d; x, y; p, q) b) m c) n 2.) Ba, d a) m! m,n Rep) >, Req) > ; x <, y < ) B p,q b + m, d b)b p,q c + n, e c) F 2 a, b, c; d, e; x, y; p, q) a) m+n Bb, d b)bc, e c) m,n Rep) >, Req) > ; x + y < ) x m m! y n 2.2) F 3 Da, b, c, d; e; x, y, z; p, q) m,n,r B p,q a + m + n + r, e a)b) m c) n d) r Ba, e a) Rep) >, Req) > ; x <, y <, z < ) x m y n z r m! r! 2.3) Remark. For p q above definitions are similar to Özarslan and Özergin 5] and for p q similar to 2]. Theorem. The following integral holds true for 2.): F a, b, c; d; x, y; p, q) Ba, d a) t a t) d a xt) b yt) c e p t q t) dt 2.4)
EXTENDED FRACTIONAL DERIVATIVE OPERATOR 3 Proof. To prove the above Theorem, we start by assuming that I t a t) d a xt) b yt) c e p t q t) dt. Using the binomial series expansion for xt) b and yt) c and interchanging the order of summation and integration, we get { } I t a t) d a e p t t) q xt) n yt) m b) n c) m dt m! n m { b) n c) m n m t a+m+n t) d a e p t q t) dt } x n by applying.2) and 2.), we get the desired representation. y m m!, Theorem 2. The following integral holds true for 2.2): F 2 a, b, c; d, e; x, y; p, q) Bb, d b)bc, e c) t b t) d b s c s) e c xt ys) a e Proof. We start by expanding xt ys) a we have t b t) d b s c s) e c xt ys) a e p t q t p s q s) dtds t b t) d b e p t q t) s c s) e c e p s q Using the summation formula x + y)n fn) N! we get N t b t) d b s c s) e c xt ys) a e r l p t q t p s q s) fl + r) xr y l r! l! p t q t p s q N s) dtds t b t) d b e p t q t) s c s) e c e p s q s) r l s) dtds xt + ys) N a) N dtds N! a) r+l xt) r r! ys) l dtds l! 2.5)
4D. BALEANU,2,3, P. AGARWAL 4, R. K. PARMAR 5, M. AL. QURASHI 6 AND S. SALAHSHOUR 7 Here, involving series and the integrals are convergent, then by interchanging the order of summation and integration, we get t b t) d b s c s) e c xt ys) a e x r y l a) l+r r! l! r l p t q t p s q s) dtds t b+r t) d b e p t q t) dt s c+l s) e c e p s q s) ds, by applying.2) and 2.2), we get the desired representation. Theorem 3. The following integral holds true for 2.3): F 3 Da, b, c, d; e; x, y, z; p, q) Γe) Γa)Γe a) t a t) e a xt) b yt) c yt) d e p t q t) dt Proof. The proof of Theorem 3 is as similar to proof of Theorem. Therefore, we omit its detail here. 3. Extended Riemann-Liouville Fractional Derivative Operator Here, we introduce new extended Riemann-Liouville type fractional derivative operator as follows: Definition 5. The extended Riemann-Liouville type fractional derivative operator defined by z z t) η e pz qz ) t z t) f t) dt Re η) < Γ η) η {f z) ; p, q} : d m { } dz m η m ) {f z) ; p, q} m Re η) < m m N) 3.) where Rep) >, Req) > and the path of integration is a line from to z in the complex t-plane. Clearly for p q, 3.) reduces to.) and for p q, we obtain its classical form see, for details 2, 2, 3]). Now, we establishing some theorems involving the extended fractional derivatives. Theorem 4. The following representation for 3.) holds true: η z λ ; p, q] B p,qλ +, η) z λ η Reη) < ) 3.2) Γ η)
EXTENDED FRACTIONAL DERIVATIVE OPERATOR 5 Proof. Using 3.) and.2), we get η z λ ; p, q] z t λ z t) η pz e t z t) qz dt Γ η) replacing t uz, we have D η z z λ ; p, q] Γ η) zλ η Γ η) uz) λ z uz) η pz e u λ u) η e p u By applying Definition.2) to yield 5.4) directly. uz u) q du. qz z uz ) zdu 3.3) Theorem 5. Let Rη) < and suppose that a function fz) is analytic at the origin with its Maclaurin expansion given by fz) a n z n z < ρ) for some ρ R +. Then we have D η z fz); p, q] n a n η z n ; p, q] 3.4) n Proof. We begin from Definition 5 to the function fz) with its series expansion, we get η fz); p, q] z Γ η) n a n t n z t) η pz e t z t) qz dt Since the power series converges uniformly on any closed disk centered at the origin with its radius smaller than ρ, so does the series on the line segment from to a fixed z for z < ρ. This fact guarantees term-by-term integration as follows: D η z fz); p, q] This completes the proof. n a n Γ η) z t n z t) η pz e t z t) qz dt a n η z n ; p, q] 3.5) n Theorem 6. The following representation holds true: λ η z λ z) α ; p, q] Γλ) Γη) zη 2F ;p,q α, λ; η; z) 3.6) Reη) > Reλ) > and z < )
6D. BALEANU,2,3, P. AGARWAL 4, R. K. PARMAR 5, M. AL. QURASHI 6 AND S. SALAHSHOUR 7 Proof. Direct calculations yield λ η z λ z) α ; p, q] Γη λ) zη λ Γη λ) zη λ z λ Γη λ) z z t λ t) α pz e t z t) qz z t) η λ dt t λ t) α t z ) η λ pz e t z t) qz dt u λ uz) α u) η λ e p u u) q du Using.3) and after little simplification, we have the 3.6). This completes the proof. Theorem 7. The following representation for holds true: λ η z λ az) α bz) β ; p, q] Γλ) Γη) zη F λ, α, β; η; az, bz; p, q) 3.7) Reη) > Reλ) >, Reα) >, Reβ) > ; az < and bz < ) More generally, we have λ η z λ az) α bz) β cz) γ ; p, q] Γλ) Γη) zη FDλ, 3 α, β, γ; η; az, bz, cz; p, q) 3.8) Reη) > Rλ) >, Reα) >, Reβ) >, Reγ) >, az <, bz < and cz < ) Proof. To prove 3.7), using the following power series expansion for az) α and bz) β az) α bz) β az) l bz) k α) l β) k, l! k! then applying Theorem 4, we obtain D λ η l k z z λ az) α bz) β ; p, q] a) l b) k α) l β) k l! k! Dλ η z z λ+l+k ; p, q] l k 3.9) a) l b) k B p,q λ + l + k, η λ) α) l β) k z l+k+η. l! k! Γη λ) 3.) l k Now, applying 2.), we get D λ η z z λ az) α bz) β ; p, q] Γλ) Γη) zη F λ, α, β; η; az, bz; p, q). 3.) Similarly, as in the proof of 3.7), taking the binomial theorem for az) α, bz) β and cz) γ, then applying Theorem 4 and 2.3) into account, one can easily prove 3.8). Therefore, we omit the details of its proof. This completes the proof.
EXTENDED FRACTIONAL DERIVATIVE OPERATOR 7 Theorem 8. The following representation holds true: ) ] z λ η λ z) α x F p,q α, β; γ; ; p, q z Bβ, γ β)γη λ) zη F 2 α, β, λ; γ, η; x, z; p, q) Reµ) > Reλ) >, Reα) >, Reβ) >, Reγ) > ; Proof. Applying 2.2) on the LHS of 3.2), we get ) ] z λ η λ z) α x F p,q α, β; γ; ; p, q z { λ η z λ z) α Bβ, γ β) Dλ η z n z λ n n x z α) n B p,q β + n, γ β) Bβ, γ β) 3.2) < and x + z < ) ) } ] x n ; p, q z α) n B p,q β + n, γ β) xn { z) α n } ] ; p, q. Using power series expansion for z) α n, applying Theorem 4 and 2.2), we get ) ] z λ η λ z) α x F p,q α, β; γ; ; p, q z { ) } ] λ η z λ z) α α) n B p,q β + n, γ β) x n ; p, q Bβ, γ β) z n Bβ, γ β) Dλ η z z λ α) n B p,q β + n, γ β) xn { z) α n } ] ; p, q. This completes the proof. 4. Mellin Transform Representations The double Mellin transforms 8, p. 293, Eq. 7..6)] of a suitable classes of integrable function fx, y) with index r and s is defined by M {fx, y) : x r, y s} : provided that the improper integral in 4.) exists. Theorem 9. The following Mellin transform formula holds true: { } M z λ ) : p r, q s D µ,p,q z x r y s fx, y) dx dy, 4.) : Γr)Γs) Γ µ) Bλ + r +, s µ)zλ µ 4.2) Rλ) >, Rµ) <, Rs) > Rr) > )
8D. BALEANU,2,3, P. AGARWAL 4, R. K. PARMAR 5, M. AL. QURASHI 6 AND S. SALAHSHOUR 7 Proof. Applying definition 4.) on to 3.), we get { M D µ,p,q z } z λ ) : p r, q s : Γ µ) z µ Γ µ) z µ Γ µ) zλ µ Γ µ) zλ µ Γ µ) p r q s p r q s z z p r q s p r q s D z µ,p,q z λ )dpdq t λ z t) µ exp pz t t λ t ) µ exp pz z t u λ z λ u) µ exp p u p r q s u λ u) µ exp u λ u) µ p r exp p u qz ) z t q u qz ) z t dt dpdq dt dpdq ) ] dt dpdq q u ) ] dt dpdq ) ) p ) ) q dp q s exp dq du u u where we have changed the order of integration by absolutely convergent under the stated conditions. Using the definition of gamma function, we have { M D µ,p,q z } z λ ) : p r, q s : zλ µ Γ µ) zλ µ Γr)Γs) Γ µ) u λ+r u) s µ du zλ µ Γr)Γs) Bλ + r +, s µ) Γ µ) u λ u) µ u r Γr) u) s Γs)du Which completes the proof. Theorem. The following formula for 4.3) holds true: M { D µ,p,q z z) α ) : p r, q s } : Γr)Γs)z µ F α, r+; r+s µ+; z) Γ µ)br +, s µ) 4.3) Rµ) <, Rs) >, Rr) >, Rα) > and z < )
EXTENDED FRACTIONAL DERIVATIVE OPERATOR 9 Proof. Applying Theorem 9 with λ n, we can write that M { D z µ,p,q z) α ) : p r, q s } α) n : M { D z µ,p,q z) α ) : p r, q s } Γr)Γs) Γ µ) n Γr)Γs)z µ Γ µ) Which completes the proof. n α) n Bn + r +, s µ)z n µ Bn + r +, s µ) α) nz n n Γr)Γs)z µ F α, r + ; r + s µ + ; z) Γ µ)br +, s µ) 5. Generating Relations and Further Results Here, we obtain some generating relations of linear and bilinear type for the extended hypergeometric functions. Theorem. The following generating relation hold true: λ) n n 2F ;p,q λ + n, α; β; x)t n t) λ 2 F ;p,q λ, α; β; x < min, t )andrλ) >, Rβ) > Rα) > ) Proof. Let us consider the elementary identity x) t] λ t) λ x ] λ, t Using power series expension, we have ) λ) n t n x) λ t) λ x ] λ x t n n ) x t Now, multiplying both sides of the above equality by x α and applying the operator Dx α β,p,q on both sides, we can get ) ] Dx α β,p,q λ) n t n x) λ x α t) λ Dx x α β,p,q α x ) ] λ x t Interchanging the order, which is valid under the stated conditions, we get λ) n Dx α β,p,q x α x) λ n] t n t) λ Dx x α β,p,q α x ) ] λ t n Using Theorem 5, we get the desired result.
D. BALEANU,2,3, P. AGARWAL 4, R. K. PARMAR 5, M. AL. QURASHI 6 AND S. SALAHSHOUR 7 Theorem 2. The following generating relation holds true: λ) n 2F ;p,q ρ n, α; β; x)t n t) λ F α, ρ, λ; β; x, xt ) t ; p, q n Rβ) > Rα) >, Rρ) >, Rλ) > ; t < Proof. To prove above theorem we use the elementary identity x)t] λ t) λ + xt ] λ t Expanding the left hand side, we have λ) n x) n t n t) λ xt t n n + x Now, multiplying both sides of the above equality by x α x) ρ and applying the operator Dx α β,p,q on both sides, we get ] Dx α β,p,q λ) n x α x) ρ+n t n t) λ Dx x α β,p,q α x) ρ xt ) ] λ t Interchanging the order, which is valid for Reα) > and xt < t, we get λ) n D α β,p,q x x α x) ρ+n] t n t) λ Dx x α β,p,q α x) ρ xt ) ] λ t n Using Theorem 5, we get the desired result. ] λ Theorem 3. The following bilinear generating relation holds true: λ) n 2F ;p,q γ, n; δ; y) 2 F ;p,q λ + n, α; β; x)t n t) λ x F 2 λ, α, γ; β, δ; t, yt ) t ; p, q n Rδ) > Rγ) >, Rα) >, Rλ) >, Rβ) > ; t < x and x < ) + y Proof. Replacing t y)t in Theorem, multiplying the resulting equality by y γ and then applying the operator Dy γ δ,p,q, we get ] Dy γ δ,p,q λ) n y γ 2 F ;p,q λ + n, α; β; x) y) n t n n D γ δ,p,q y y)t) λ y γ 2 F ;p,q λ, α; β; )] x y)t
n EXTENDED FRACTIONAL DERIVATIVE OPERATOR Interchanging the order, which is valid under stated conditions, we can write that λ) n D γ δ,p,q y y γ y) n] 2 F ;p,qλ + n, α; β; x)t n t) λ D γ δ,p,q y Using Theorems 5 and 6, we get the result. y γ yt ) λ F ;p,q λ, α; β; t 2 )] x t yt t Remark 2. For p q, the results presented here would reduce to the corresponding well-known results see, for details, 2,, 2, 3]). Theorem 4. Let Rp) >, Rq) >, Rµ) > Rλ) > ; γ, δ C and the extended Riemann-Liouville fractional derivative 3.). Then there holds the formula: ] z λ µ,p,q λ E µ γ,δ z) zµ µ) n Γµ λ) Γγ n + δ) B p,qλ + n, µ λ) zn, 5.) n where E µ γ,δ z) is a well known generalized Mittag-Leffler function due to Prabhakar 9] defined as: E µ γ,δ z) n µ) n z n Γγ n + δ) Proof. Applying 5.2) to 5.) and using Theorem 8 and 4, we get { λ µ,p,q z λ E µ γ,δ z)] Dλ µ,p,q z z λ n n γ, δ, µ C; Rγ) > ). 5.2) n µ) { n λ µ,p,q Γγ n + δ) µ) n Γγ n + δ) µ) n Γγ n + δ) ]} λ+n z { Bp,q λ + n, µ λ) Γµ λ) } ] z n z µ+n }. 5.3) Remark 3. If we set p q in 5.), we get the interesting known result given by Özarslan and Yilmaz 6, Theorem 9]. Theorem 5. Let Rp) >, Rq) >, Rµ) > Rλ) > ; γ, δ C and the extended Riemann-Liouville fractional derivative 3.). Then there holds the formula: ]] z λ µ,p,q λ mψ n z a i, α i ) m,m zµ i Γ a i + α i k) b j, β j ),n Γµ λ) n j Γ b j + β j k) B p,qλ+k, µ λ) zk k!, k 5.4)
2 D. BALEANU,2,3, P. AGARWAL 4, R. K. PARMAR 5, M. AL. QURASHI 6 AND S. SALAHSHOUR 7 where p Ψ q z) is the Fox-Wright function defined by see 2, pp. 56 58]) ] mψ n z) m Ψ n z a i, α i ) m,m i : Γ a i + α i k) z k b j, β j ) n,n j Γ b j + β j k) k!. 5.5) Proof. Applying the result in Theorem 4 to the 5.5) and using same process as similar to Theorem 4, we get desired result. Remark 4. If we set p q in 5.4), we get the interesting known result given by Shrma and Devi, p. 49, Theorem 8]. k 6. Conclusion The fractional derivative operator D z µ,p,q {f z)} in 3.) is defined for {Rp), Rq)}. The extended fractional derivatives for the some elementary functions are given by Theorem 4-8. The Mellin transform of the 3.) and generating relations of linear and bilinear type for the extended hypergeometric functions are given by Theorem to 3, respectively. All of this show that this paper has the distinctive advantage in the field of applied mathematics. References ] Choi J., Rathie A.K. and Parmar R.K., Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J. 362) 24) 339367. 2] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies; Elsevier North-Holland) Science Publishers: Amsterdam, The Netherlands, 26; Volume 24. 3] Luo M. J., Milovanovic G. V. and Agarwal P., Some results on the extended beta and extended hypergeometric functions, Appl. Math. Comput. 248 24), 63 65. 4] Olver F. W. J., Lozier D. W., Boisvert R. F. and Clark C. W. eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2. 5] Özarslan M.A., Özergin E., Some generating relations for extended hypergeometric function via generalized fractional derivative operator, Math. Comput. Modelling 52 2), 825 833. 6] Özarslan M. A. and Yilmaz B., The extended Mittag-Leffer function and its properties, J. Inequal Appl.24). doi:.86/29-242x-24-85. 7] Parmar R. K., Some generating relations for generalized extended hypergeometric functions involving generalized fractional derivative operator, J. Concr. Appl. Math. 2 24), 27 228. 8] Paris R.B. and Kaminski D., Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, Cambridge, 2. 9] Prabhakar T. R., A Singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math J 9 97), 7 5. ] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives: Theory and Applications, Translated from the Russian: Integrals and Derivatives of Fractional Order and Some of Their Applications Nauka i Tekhnika, Minsk, 987); Gordon and Breach Science Publishers: Reading, UK, 993. ] Sharma S. C. and Devi M.,Certain Properties of Extended Wright Generalized Hypergeometric Function, Annals of Pure and Applied Mathematics 9) 24), 45 5. 2] Srivastava H.M. and Karlsson P. W., Multiple Gaussian Hypergeometric Series, Halsted Press Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 985.
EXTENDED FRACTIONAL DERIVATIVE OPERATOR 3 3] Srivastava H.M. and Manocha H. L., A Treatise on Generating Functions, Halsted Press Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 984.,2,3 D. Baleanu Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 824, Jeddah 2589, Saudi Arabia. Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, Ankara, Turkey. Institute of Space Sciences, P.O. Box MG-23, 769 Magurele Bucharest, Romania E-mail address: dumitru@cankaya.edu.tr 4 P. Agarwal Department of Mathematics, Anand International College of Engineering, Jaipur-332, Republic of India E-mail address: goyal.praveen2@gmail.com 5 R. K. Parmar Department of Mathematics, Government College of Engineering and Technology, Bikaner- 3344, Rajasthan State, India E-mail address: rakeshparmar27@gmail.com 6 M. Al. Qurashi College of Science, Department of Mathematics, King Saud University,Saudi Arabia E-mail address: e-mail:maysaa@ksu.edu.sa 7 S. Salahshour Department of Computer Engineering, Mashhad Branch, IAU, Iran. E-mail address: soheilsalahshour@yahoo.com