Open Math. 2017; 15: 1667–1681

Open Mathematics

Research Article

P. Agarwal, Juan J. Nieto*, and M.-J. Luo Extended Riemann-Liouville type fractional derivative operator with applications https://doi.org/10.1515/math-2017-0137 Received November 16, 2017; accepted November 30, 2017. Abstract: The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modied Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by dening the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.

Keywords: Gamma function, Extended beta function, Riemann-Liouville fractional derivative, Hypergeomet- ric functions, Fox H-function, Generating functions, Mellin transform, Integral representations

MSC: 26A33, 33B15, 33B20, 33C05, 33C20, 33C65

1 Introduction

Extensions and generalizations of some known special functions are important both from the theoretical and applied point of view. Also many extensions of fractional derivative operators have been developed and applied by many authors (see [2–6, 11, 12, 19–21] and [17, 18]). These new extensions have proved to be very useful in various elds such as physics, engineering, statistics, actuarial sciences, economics, nance, survival analysis, life testing and telecommunications. The above-mentioned applications have largely motivated our present study. The extended incomplete gamma functions constructed by using the exponential function are dened by z p α, z; p tα−1 exp t dt R p 0; p 0, R α 0 (1) t 0 ( ) = ‹− −  ( ( ) > = ( ) > ) and S ∞ p Γ α, z; p tα−1 exp t dt R p 0 (2) t z ( ) = ‹− −  ( ( ) ≥ ) with arg z π, which have been studiedS in detail by Chaudhry and Zubair (see, for example, [2] and [4]). The extended incomplete gamma functions γ α, z; p and Γ α, z; p satisfy the following decomposition formulaS S < ( ) α~2 ( ) α, z; p Γ α, z; p Γp α 2p Kα 2 p R p 0 , (3) √ ( ) + ( ) = ( ) = ‰ Ž ( ( ) > ) P. Agarwal: Department of Mathematics, Anand International College of Engineering, Jaipur-303012, Republic of India, E-mail: [email protected] *Corresponding Author: Juan J. Nieto: Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticasd, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain, E-mail: [email protected] M.-J. Luo: Department of Mathematics, East China Normal University, Shanghai 200241, China, E-mail: [email protected]

Open Access. © 2017 Agarwal et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License. 1668 Ë P. Agarwal et al.

where Γp α is called extended gamma function, and Kα z is the modied Bessel function of the third kind, or the Macdonald function with its integral representation given by (see [7]) ( ) ( ) ∞ 1 dt Kα z exp κ z t , (4) 2 tα+1 0 ( ) = S [− ( S )] where R z 0 and z 1 κ z t t . (5) ( ) > 2 t 1 ( S ) = ‹ +  For α 2 , we have π −z = K 1 z e . (6) 2 ½2z Instead of using the exponential function, Chaudhry( ) = and Zubair extended (1) and (2) in the following form (see [3], see also [4]) x 2p α− 3 p 2 γµ α, x; p t exp t Kµ+ 1 dt (7) ¾ π 2 t 0 ( ) = (− ) ‹  and S ∞ 2p α− 3 p 2 Γµ α, x; p t exp t Kµ+ 1 dt, (8) ¾ π 2 t x ( ) = (− ) ‹  where R x 0, R p 0, α . S Inspired by their construction of (7) and (8), we aim to introduce a class of new special functions and ( ) > ( ) > −∞ < < ∞ fractional derivative operator by suitably using the modied Bessel function Kα z . The present paper is organized as follows: In Section 2, we rst dene the extended beta function and study some of its properties such as dierent integral representations and( its) Mellin transform. Then some extended hypergeometric functions are introduced by using the extended beta function. The extended Riemann-Liouville type fractional derivative operator and its properties are given in Section 3. In Section 4, the linear and bilinear generating relations for the extended hypergeometric functions are derived. Finally, the Mellin transforms of the extended fractional derivative operator are determined in Section 5.

2 Extended beta and hypergeometric functions

This section is divided into two subsections. In subsection-1, we dene the extended beta function Bµ x, y; p; m and study some of its properties. In subsection-2, we introduce the extended Gauss hyper- geometric function Fµ a, b; c; z; p; m , the Appell hypergeometric functions F1,µ, F2,µ and the Lauricella ( ) D hypergeometric function F3,µ and then obtain their integral representations. Throughout the present study, we shall assume that R( p 0 and m) 0.

( ) > > 2.1 Extended beta function

Denition 2.1. The extended beta function Bµ x, y; p; m with R p 0 is dened by

(1 ) ( ) > 2p x− 3 y− 3 p 2 2 Bµ x, y; p; m t 1 t Kµ+ 1 dt, (9) ¾ π 2 tm 1 t m 0 ( ) ∶= ( − ) Œ ‘ S ( − ) where x, y C, m 0 and R µ 0.

∈ > ( ) ≥ Extended Riemann-Liouville type fractional derivative operator with applications Ë 1669

Remark 2.2. Taking m 1, µ 0 and making use of (6), (9) reduces to the extended beta function Bµ x, y; p dened by Chaudhry et al. [5, Eq. (1.7)] = = ( ) 1 p B x, y; p B x, y; p; 1 tx−1 1 t y−1 exp dt, (10) 0 t 1 t 0 ( ) ∶= ( ) = ( − ) Œ− ‘ S ( − ) where R p 0 and x, y C.

Theorem( 2.3.) > The following∈ integral representations for the extended beta functions Bµ x, y; p; m with R p 0 are valid ( ) π ( ) > 2 2p 2(x−1) 2(y−1) 2m 2m Bµ x, y; p; m 2 cos θ sin θ Kµ+ 1 p sec θ csc θ dθ (11) ¾ π 2 0 ( ) = ∞S x− 3 Š2m  2p u 2 1 u + − Kµ+ 1 p du (12) ¾ π 1 u x y 1 2 um 0 ( + ) = Œ ‘ S 1 2m 2−x−y 2(p + ) x− 3 y− 3 2 p 2 2 2 1 u 1 u Kµ+ 1 du. (13) ¾ π 2 1 u2 m −1 = S ( + ) ( − ) Œ ‘ (2 − ) u 1+u Proof. These formulas can be obtained by using the transformations t cos θ, t 1+u and t 2 in (9), respectively. = = = Theorem 2.4. The following expression holds true

∞ 1 2p m m du Bµ x, y; p, m B x , y ; κ p u ; m , (14) 0 µ+ 3 2¾ π 2 2 2 0 u ( ) = S ‹ − − ( S )  x, y C, m 0, R p 0, R µ 0 where κ p u is given by (5). ( ∈ > ( ) > ( ) ≥ )

Proof. Expressing( S ) Bµ x, y; p, m in its integral form with the help of (9), and taking (4) into account, we obtain ( ) 1 2p x− 3 y− 3 p 2 2 Bµ x, y; p, m t 1 t Kµ+ 1 m dt ¾ π 2 tm 1 t 0 ( ) = S 1 ( − ) ∞Œ ‘ 1 2p x− 3 y− 3 ( − κ) p u du t 2 1 t 2 exp dt m m µ+ 3 2¾ π ⎧ t 1 t u 2 ⎫ 0 ⎪ 0 ( S ) ⎪ = ( − ) ⎨ − ⎬ S∞ 1 ⎪S ⎪ 1 2p x− 3 ⎪y− 3 κ( p−u ) d⎪u t 2 1 t⎩ 2 exp dt ⎭ , (15) m m µ+ 3 2¾ π ⎧ t 1 t ⎫ u 2 0 ⎪ 0 ( S ) ⎪ = S ⎨S ( − ) − ⎬ ⎪ ( − ) ⎪ where κ p u is given by (5). ⎩⎪ ⎭⎪ In order to write the inner integral as our extended beta function, we need the following variant of (6), that is, ( S )

−z 2 e z K 1 z . ¾ π 2 Then, we have = ( )

1 x− 3 y− 3 κ p u t 2 1 t 2 exp dt tm 1 t m 0 ( S ) S ( − ) − ( − ) 1670 Ë P. Agarwal et al.

1 x− 3 y− 3 2 κ p u κ p u 2 2 t 1 t m K 1 m dt ¿ π tm 1 t 2 tm 1 t 0 Á ÀÁ ( S ) ( S ) = S ( − )1 Œ ‘ 2κ p u x− m − 3 ( −y− m)− 3 (κ −p u) 2 2 2 2 t 1 t K 1 m dt ¾ π 2 tm 1 t ( S ) 0 ( S ) = m S m ( − ) Œ ‘ B x , y ; κ p u ; m . ( − ) (16) 0 2 2 Substituting (16) into (15)= we obtain‹ − the required− ( resultS ) (14).

Remark 2.5. It is interesting to note that using the denition of the extended beta function (see, [11, 12]) we can get the following expression for (9)

∞ 1 2p 1 1 du Bµ x, y; p, m B x , y , (17) κ(pSu);m,m µ+ 3 2¾ π 2 2 2 0 u ( ) = S ‹ − −  where the function Bb;ρ,λ x, y is given by [12, p. 631, Eq. (2)]

1 ( ) x−1 y−1 b Bb;ρ,λ x, y t 1 t exp dt x, y . ρ λ C 0 t 1 t ( ) = S ( − ) Œ− ‘ ( ∈ ) The following theorem establishes the relation between the Mellin( − transform) ∞ M f x ; x s xs−1f x dx 0 { ( ) → } = S ( ) and the extended beta function.

Theorem 2.6. Let x, y C, m 0, R µ 0 and 1 1 R x 1 1 R y ∈R s >max (R)µ≥ , , . 2 2m m 2 2m m ( ) ( ) Then we have the following( relation) > › ( ) − + − − + −

M Bµ x, y; p; m p s Γ s µ Γ s µ Γ x ms m−1 Γ y ms m−1 { ( 1 ) ∶ → 2} 2 2 2 2µ Γ s µ Γ x y 2ms m 1 ( + ) ‰ 2− 2Ž ‰ + + Ž ‰ + + Ž µ = 1 Γ s µ Γ s m 1 m 1 ‰2 +2 ŽB (x +ms+ + , y− ms) . (18) 2µ Γ s µ 2 2 ( + )2 ‰2 − Ž − − = ‹ + + + +  Proof. First, we have ‰ + Ž

M Bµ x, y; p; m p s ∞ 1 s−1 2p x− 3 y− 3 p { ( ) ∶ →2 } 2 p t 1 t Kµ+ 1 dtdp ¾ π 2 tm 1 t m 0 0 = S 1 S ( −∞) Œ ‘ 2 x− 3 y− 3 s+ 1 −1 ( −p) 2 2 2 t 1 t p Kµ+ 1 dp dt ¾ π ⎡ 2 tm 1 t m ⎤ 0 ⎢ 0 ⎥ ⎢ ⎥ = S1 ( − ) ⎢S Œ ∞ ‘ ⎥ 2 x+m‰s+ 1 Ž− 3 ⎢ y+m‰s+ 1 Ž− 3 ( s+−1 −1) ⎥ 2 2 ⎣ 2 2 2 ⎦ t 1 t dt u Kµ+ 1 u du ¾ π 2 0 0 = S m−1( − ) m−1 S∞ ( ) 2 Γ x ms Γ y ms s+ 1 −1 2 2 2 u Kµ+ 1 u du. (19) ¾ π Γ x y 2ms m 1 2 ‰ + + Ž ‰ + + Ž 0 = ( ) ( + + + − ) S Extended Riemann-Liouville type fractional derivative operator with applications Ë 1671

Since the Mellin transform of the Macdonald function Kv z is given by [9, p. 37, Eq.(1.7.41)]:

s−2 ( s) v s v M Kv z z s 2 Γ Γ , (20) 2 2 2 2 the last integral in (19) can be evaluated{ ( ) ∶ as → } = ‹ +  ‹ − 

∞ s µ s+ 1 −1 s− 3 s µ 1 s µ − 1 −µ Γ s µ Γ 2 2 2 2 2 u Kµ+ 1 u du 2 Γ Γ 2 π µ , (21) 2 2 2 2 2 2 Γ s 0 √ ( + )2 ‰2 − Ž S ( ) = ‹ + +  ‹ −  = where we have used ‰ + Ž 22z−1 1 Γ 2z Γ z Γ z . π 2 Finally, we get ( ) = √ ( ) ‹ + 

s µ m−1 m−1 1 Γ s µ Γ 2 2 Γ x ms 2 Γ y ms 2 M Bµ x, y; p; m p s . 2µ Γ s µ Γ x y 2ms m 1 ( + ) ‰ 2− 2Ž ‰ + + Ž ‰ + + Ž { ( ) ∶ → } = Now we can derive the Fox H-function representation of‰ the+ extendedŽ ( + beta+ function+ − dened) in (9). Let m, n, p, q be integers such that 0 m q, 0 n p, and for parameters ai , bi C and for parameters + αi , βj R i 1, ... , p; j 1, ... , q , the H-function is dened in terms of a Mellin-Barnes integral in the following manner ([8, pp. 1–2]; see also [10,≤ p.≤ 343, Denition≤ ≤ E.1.] and [13, p. 2, Denition∈ 1.1.]): ∈ ( = = )

ai , αi 1,p a1, α1 , , ap , αp m,n m,n 1 −s Hp,q ⎡z R ⎤ Hp,q ⎡z ⎤ Θ s z ds, (22) ⎢ R( ) ⎥ ⎢ R( ) ⋯ ( )⎥ 2πi ⎢ R bj , βj ⎥ ⎢ R b1, β1 , , bq , βq ⎥ L ⎢ R 1,q ⎥ ⎢ R ⎥ ⎢ R ⎥ = ⎢ R ⎥ = S ( ) ⎢ R ⎥ ⎢ R ⎥ where ⎢ R( ) ⎥ ⎢ R( ) ⋯ ( ) ⎥ ⎣ R ⎦ m ⎣ R n ⎦ j=1 Γ bj βj s i=1 Γ 1 ai αi s Θ s p q , (23) Γ ai αi s Γ 1 bj βj s i∏=n+1 ( + ) ∏j=m+1( − − ) ( ) = with the contour L suitably chosen.∏ As convention,( + the) ∏ empty product( − is− equal) to one. The theory of the H- function is well explained in the books of Mathai [14], Mathai and Saxena ([15], Ch.2), Srivastava, Gupta and m,n Goyal ([22], Ch.1) and Kilbas and Saigo ([8], Ch.1 and Ch.2). Note that (18) and (20) mean Hp,q z in (22) is the inverse Mellin transform of Θ s in (23). ( ) Theorem 2.7. Let R p 0, x(, y) C, m 0 and R µ 0, then

( ) > ∈ > ( )µ≥, 1 , x y m 1, 2m 1 2 2 B x, y; p; m H4,0 p , µ µ 2,4 ⎡ R ⎤ 2 ⎢ R ‰ µ 1Ž ( + m+−1 − ) m−1 ⎥ ⎢ R µ, 1 , , , x , m , y , m ⎥ ⎢ R 2 2 2 2 ⎥ ( ) = ⎢ R ⎥ ⎢ R ⎥ ⎢ R ⎥ where Bµ x, y; p; m is as dened in (9). ⎣ R( ) ‰− Ž ‰ + Ž ‰ + Ž⎦

Proof. The( result is obtained) by taking the inverse Mellin transform of (18) in Theorem 2.6 and using (22) and (23).

2.2 Extended hypergeometric functions

Denition 2.8. The extended Gauss hypergeometric function Fµ a, b; c; z; p; m is dened by

∞ n Bµ b n(, c b; p; m z) Fµ a, b; c; z; p; m a n , (24) n=0 B b, c b n! ( + − ) ( ) ∶= Q( ) where R p 0, R µ 0, 0 R b R c and z 1. ( − )

( ) > ( ) ≥ < ( ) < ( ) S S < 1672 Ë P. Agarwal et al.

Denition 2.9. The extended Appell hypergeometric function F1,µ is dened by

∞ n k Bµ a n k, d a; p; m x y F1,µ a, b, c; d; x, y; p; m b n c k , (25) = B a, d a n! k! n,k 0 ( + + − ) ( ) ∶= Q ( ) ( ) where R p 0, R µ 0, 0 R a R d and x 1, y 1. ( − )

Denition( ) 2.10.> The( ) extended≥ < Appell( ) < hypergeometric( ) S S < functionS S < F2,µ is dened by

∞ n k Bµ b n, d b; p; m Bµ c k, e c; p; m x y F2,µ a, b, c; d, e; x, y; p; m a n+k , (26) = B b, d b B c, e c n! k! n,k 0 ( + − ) ( + − ) ( ) ∶= Q ( ) where R p 0, R µ 0, 0 R b R d , 0 R c ( R e− and) x y 1(. − )

( ) > ( ) ≥ < ( ) < ( ) < ( ) < ( ) 3S S + S S < Denition 2.11. The extended Lauricella hypergeometric function FD,µ is

∞ n k r 3 Bµ a n k r, e a; p; m x y z FD,µ a, b, c, d; e; x, y, z; p; m b n c k d r , (27) = B a, e a n! k! r! n,k,r 0 ( + + + − ) ( ) ∶= Q ( ) ( ) ( ) where R p 0, R µ 0, 0 R a R e and x 1, y 1, z 1. ( − )

Here, it is( important) > ( to) ≥ mention< that( ) when< ( we) takeS mS < 1,S µS < 0 andS S < then letting p 0, function (24) reduces to the ordinary Gauss hypergeometric function dened by = = → ∞ n a, b a n b n z 2F1 a, b; c; z 2F1 ; z , (28) c = c n! n 0 ( ) (n ) ( ) ≡  = Q where x n denotes the Pochhammer symbol dened, in terms of the( familiar) gamma function, by

( ) Γ x n 1 n 0; x C 0 x n Γ x ⎧x x 1 x n 1 n N; x C . ( + ) ⎪ ( = ∈ ∖ { }) ( ) = = ⎨ For conditions of convergence and( ) other⎪ related( + details) ⋯ ( of+ this− function,)( ∈ see [1],∈ [9]) and [16]. Similarly, we can ⎩ 3 reduce the functions (25), (26) and (27) to the well-known Appell functions F1, F2 and Lauricella function FD, respectively (see [16] and [23]). Now, we establish the integral representations of the extended hypergeometric functions given by (24), (25), (26) and (27) as follows.

Theorem 2.12. The following integral representation for the extended Gauss hypergeometric function Fµ a, b; c; z; p; m is valid

1 ( ) 2p 1 b− 3 c−b− 3 −a p 2 2 Fµ a, b; c; z; p; m t 1 t 1 zt Kµ+ 1 dt, (29) ¾ π B b, c b 2 tm 1 t m 0 ( ) = ( − ) ( − ) Œ ‘ where arg 1 z π, R p 0, m( 0 and− )RS µ 0. ( − )

Proof. SBy using( − (9))S < and employing( ) > the> binomial( ) expansion≥ ∞ n −a zt 1 zt a n zt 1 , (30) n=0 n! ( ) we get the above integral representation.( − ) = Q( ) (S S < )

Theorem 2.13. The following integral representation for the extended hypergeometric function F1,µ is valid

2p 1 F1,µ a, b, c; d; x, y; p; m ¾ π B a, d a ( 1 ) = a− 3 d−a− 3 −b −c p 2 2 ( − ) t 1 t 1 xt 1 yt Kµ+ 1 dt, (31) 2 tm 1 t m 0 × ( − ) ( − ) ( − ) Œ ‘ S ( − ) Extended Riemann-Liouville type fractional derivative operator with applications Ë 1673

Proof. For simplicity, let I denote the left-hand side of (31). Then, using (25) yields

∞ n k Bµ a n k, d a; p; m x y I b n c k , (32) = B a, d a n! k! n,k 0 ( + + − ) = Q ( ) ( ) By applying (9) to the integrand of (31), after a little simplication,( − ) we have

∞ 1 n k 2p a+n+k− 3 d−a− 3 p b n c x y 2 2 k I t 1 t Kµ+ 1 dt . (33) 2 m m n,k=0 ⎧¾ π t 1 t ⎫ B a, d a n! k! ⎪ 0 ⎪ ( ) ( ) = Q ⎨ S ( − ) Œ ‘ ⎬ ⎪ ( − ) ⎪ ( − ) By interchanging the⎩⎪ order of summation and integration in (33), we get ⎭⎪

1 2p 1 a− 3 d−a− 3 p 2 2 I t 1 t Kµ+ 1 ¾ π B a, d a 2 tm 1 t m 0 = ∞ b S ∞ ( c− ) Œ ‘ ( −n xt) n k yt k dt ( − ) n=0 n! = k! ( ) k 0 ( ) × œQ ( )1 ¡ œQ ( ) ¡ 2p 1 a− 3 d−a− 3 t 2 1 t 2 ¾ π B a, d a 0 = S ( − ) ( −−b) −c p 1 xt 1 yt Kµ+ 1 dt, (34) 2 tm 1 t m × ( − ) ( − ) Œ ‘ which proves the integral representation (31). ( − ) To establish Theorem 2.13, we need to recall the following elementary series identity involving the bounded ∞ sequence of f N N=0 stated in the following result.

{ ( )} ∞ Lemma 2.14. For a bounded sequence f N N=0 of essentially arbitrary complex numbers, we have

∞ { (x )}y N ∞ ∞ xn yk f N f n k . (35) N=0 N! n=0 = n! k! ( + ) k 0 Q ( ) = Q Q ( + ) Theorem 2.15. The following integral representation for the extended hypergeometric function F2,µ is valid 2p 1 F2,µ a, b, c; d, e; x, y; p; m π B b, d b B c, e c ( 1 1 ) = b− 3 d−b− 3 b− 3 e−c− 3 −a t 2 1 t 2 w ( 2 1− w) ( 2 −1 )xt yw 0 0 × S S (p − ) ( −p ) ( − − ) Kµ+ 1 Kµ+ 1 dtdw. (36) 2 tm 1 t m 2 wm 1 w m × Œ ‘ Œ ‘ Proof. Let denote the left-hand side of( (36).− ) Then, using (26)( yields− )

∞ n k L Bµ b n, d b; p; m Bµ c k, e c; p; m x y a n+k . (37) = B b, d b B c, e c n! k! n,k 0 ( + − ) ( + − ) L = Q ( ) By applying (9) to the integrand of (32), we( have − ) ( − )

∞ 1 2p b+n− 3 d−b− 3 p 2 2 t 1 t Kµ+ 1 dt 2 m m π n,k=0 ⎧ t 1 t ⎫ ⎪ 0 ⎪ L = Q ⎨ ( − ) Œ ‘ ⎬ ⎪S 1 ( − ) ⎪ ⎪ b+n− 3 e−c− 3 p ⎪ ⎩ 2 2 ⎭ w 1 w Kµ+ 1 dw ⎧ 2 wm 1 w m ⎫ ⎪ 0 ⎪ × ⎨S ( − ) Œ ‘ ⎬ ⎪ ( − ) ⎪ ⎩⎪ ⎭⎪ 1674 Ë P. Agarwal et al.

a xn yk n+k . (38) B b, d b B c, e c n! k! ( ) × Next, interchanging the order of summation( − and) integration( − ) in (38), which is guaranteed, yields 1 1 2p 1 b− 3 d−b− 3 b− 3 e−c− 3 t 2 1 t 2 w 2 1 w 2 π B b, d b B c, e c 0 0 L = ( − ) ( − ) ( − ) ( p− ) S S p Kµ+ 1 Kµ+ 1 2 tm 1 t m 2 wm 1 w m × ∞ Œ xt n ‘ yw k Œ ‘ a n(+k− ) dtd(w. − ) (39) = n! k! ⎛n,k 0 ( ) ( ) ⎞ × Q ( ) Finally, applying (35) to the double⎝ series in (39), we obtain⎠ the right-hand side of (36).

3 Theorem 2.16. The following integral representation for the extended hypergeometric function FD,µ is valid

1 2p F3 a, b, c, d; e; x, y, z; p; m D,µ B a, e a ¾ π 1( a− 3 e−a− 3 ) = t 2 1 t 2 ( − ) p Kµ+ 1 dt (40) 1 xt b 1 yt c 1 zt d 2 tm 1 t m 0 ( − ) × S Œ ‘ Proof. A similar argument in the( proof− of) Theorem( − ) ( 2.15− will) be able to establish( − ) the integral representation in (40). Therefore, details of the proof are omitted.

3 Extended Riemann-Liouville fractional derivative operator

We rst recall that the classical Riemann-Liouville fractional derivative is dened by (see [23, p. 286])

z 1 Dν f z z t −ν−1f t dt, z Γ ν 0 ( ) ∶= S ( − ) ( ) where R ν 0 and the integration path is a( line− ) from 0 to z in the complex t-plane. It coincides with the fractional integral of order ν. In case m 1 R ν m, m N, it is customary to write ( ) < m m z ν − d ν−m − < d( ) < 1 ∈ m−ν−1 Dz f z Dz f z z t f t dt . dzm dzm ⎧ Γ m ν ⎫ ⎪ 0 ⎪ ( ) ∶= ( ) = ⎨ ( − ) ( ) ⎬ ⎪ ( − ) S ⎪ We present the following new extended Riemann-Liouville-type⎩⎪ fractional derivative operator.⎭⎪

Denition 3.1. The extended Riemann-Liouville fractional derivative is dened as

z 2m ν,µ;p;m 1 2p −ν−1 pz Dz f z z t f t Kµ+ 1 dt, (41) Γ ν ¾ π 2 tm z t m 0 ( ) ∶= S ( − ) ( ) Œ ‘ where R ν 0, R p 0, R m 0(−and) R µ 0. ( − )

For n( )1< R (ν ) >n, n (N,) we> write ( ) ≥

n n z 2m ν,µ;p;m − < d( ) <ν−n,µ;p∈;m d 1 2p n−ν−1 pz Dz f z Dz f z z t f t Kµ+ 1 dt . (42) dzn dzn ⎧ Γ n ν ¾ π 2 tm z t m ⎫ ⎪ 0 ⎪ ( ) ∶= ( ) = ⎨ ( − ) ( ) Œ ‘ ⎬ ⎪ ( − ) S ( − ) ⎪ Remark 3.2. If we take m 0, µ 0, and⎩⎪ p 0, then the above extended Riemann-Liouville fractional⎭⎪ derivative operator reduces to the classical Riemann-Liouville fractional derivative operator. = = → Extended Riemann-Liouville type fractional derivative operator with applications Ë 1675

Now, we begin our investigation by calculating the extended fractional derivatives of some elementary functions. For our purpose, we rst establish two results involving the extended Riemann-Liouville fractional derivative operator.

Lemma 3.3. Let R ν 0, then we have

λ−ν ( ) < ν,µ;p;m λ z 3 1 D z Bµ λ , ν ; p; m . (43) z Γ ν 2 2 š Ÿ = ‹ + − +  Proof. Using Denition 3.1 and 1, we have (− )

z 2m ν,µ;p;m λ 1 2p −ν−1 λ pz Dz z z t t Kµ+ 1 dt Γ ν ¾ π 2 tm z t m 0 š Ÿ = λ−ν S 1( − ) Œ ‘ z(− ) 2p ‰−ν+ 1 Ž− 3 ‰λ+ 3 Ž(− 3− ) p 2 2 2 2 1 u u Kµ+ 1 du Γ ν ¾ π 2 um 1 u m 0 = λ−ν ( − ) Œ ‘ z(− ) S3 1 ( − ) Bµ λ , ν ; p; m . Γ ν 2 2 = ‹ + − +  Next, we apply the extended Riemann-Liouville(− ) fractional derivative to a function f z analytic at the origin.

Lemma 3.4. Let R ν 0 and suppose that a function f z is analytic at the origin( ) with its Maclaurin ∞ n expansion given by f z n=0 an z z ρ for some ρ R+. Then we have ( ) < ( ) ∞ ( ) = ∑ ν(S,µS;p<;m ) ∈ ν,µ;p;m n Dz f z an Dz z . n=0 { ( )} = Q ™ ž Proof. Using Denition 3.1 to the function f z with its series expansion, we have

( ) z 2m ∞ ν,µ;p;m 1 2p −ν−1 pz n 1 Dz f z z t Kµ+ m m an t dt. Γ ν ¾ π 2 t z t = 0 n 0 { ( )} = S ( − ) Œ ‘ Q 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 0 to a xed z for z ρ. This fact guarantees term-by-term integration as follows: S S < ∞ z 2m ∞ ν,µ;p;m 1 2p −ν−1 pz n ν,µ;p;m n 1 Dz f z an z t Kµ+ m m t dt an Dz z . n=0 ⎧ Γ ν ¾ π 2 t z t ⎫ n=0 ⎪ 0 ⎪ { ( )} = Q ⎨ ( − ) Œ ‘ ⎬ = Q ™ ž ⎪ (− ) S ( − ) ⎪ As a consequence we have⎩⎪ the following result. ⎭⎪

Theorem 3.5. Let R ν 0 and suppose that a function f z is analytic at the origin with its Maclaurin ∞ n expansion given by f z n=0 an z z ρ for some ρ R+. Then we have ( ) < ( ) ∞ λ−ν−1 ∞ ν,µ;p;m λ−1( ) = ∑ (Sν,µS;p<;m ) λ+n−1 z∈ 1 1 n Dz z f z an Dz z an Bµ λ n , ν ; p; m z . n=0 Γ ν n=0 2 2 š ( )Ÿ = Q š Ÿ = Q ‹ + + − +  We present two subsequent theorems which may be useful(− to) nd certain generating function.

1 Theorem 3.6. For R ν R λ 2 , we have

ν−1 λ−ν,µ;p;m λ−1 ( )−>α ( ) >z − 1 1 1 Dz z 1 z B λ , ν λ Fµ α, λ ; ν 1; z; p; m z 1; α C . Γ ν λ 2 2 2 œ ( − ) ¡ = ‹ + − +  ‹ + +  (S S < ∈ (44)) ( − ) 1676 Ë P. Agarwal et al.

Proof. Using (30) and applying Lemmas 3.3 and 3.4, we obtain ∞ l λ−ν,µ;p;m λ−1 −α λ−ν,µ;p;m λ−1 z Dz z 1 z Dz z α l l=0 l! ∞ α{ l λ(−ν,−µ;p);m }λ=+l−1 œ Q( ) ¡ Dz z = l! l 0 ( ) ∞ 1 1 = Q α Bµ λ l š , ν λŸ ; p; m l 2 2 zν+l−1. = l! Γ ν λ l 0 ( ) ( + + − + ) = Q By using (24), we can get ( − ) ν−1 λ−ν,µ;p;m λ−1 −α z 1 1 1 D z 1 z B λ , ν λ Fµ α, λ ; ν 1; z; p; m . z Γ ν λ 2 2 2 { ( − ) } = ‹ + − +  ‹ + +  1 ( − ) Theorem 3.7. Let R ν R λ 2 , R α 0, R β 0; az 1 and bz 1. Then we have

λ(−ν),µ>;p;m( )λ−>1− ( )−α> ( )−>β S S < S S < Dz z 1 az 1 bz

zν−1 œ ( 1− ) ( 1− ) ¡ 1 B λ , ν λ F1,µ λ , α, β; ν 1; az, bz; p; m . (45) Γ ν λ 2 2 2 = ‹ + − +  ‹ + +  Proof. Use the binomial( theorem− ) for 1 az −α and 1 bz −β . Apply Lemmas 3.3 and 3.4 to obtain

λ−ν,µ;p;m λ−1 −α −β Dz z 1( az− ) 1 bz( − )

∞ ∞ l k λ−νœ,µ;p;m( −λ−1 ) ( − ) ¡az bz Dz z α l β k = = l! k! l 0 k 0 ( ) ( ) ∞ l k = œ λ−Qν,µ;Qp;m( )λ(+l+)k−1 a b ¡ α l β k Dz z = l! k! l,k 0 ( ) ( ) = Q ( ∞) ( ) š Ÿ1 1 n k ν−1 Bµ λ l k 2 , ν λ 2 ; p; m az bz z α l β k . = Γ ν λ l! k! l,k 0 ‰ + + + − + Ž ( ) ( ) = Q ( ) ( ) By using (25), we get ( − )

λ−ν,µ;p;m λ−1 −α −β Dz z 1 az 1 bz

zνœ−1 ( − 1) ( − 1) ¡ 1 B λ , ν λ F1,µ λ , α, β; ν 1; az, bz; p; m . Γ ν λ 2 2 2 = ‹ + − +  ‹ + +  ( − ) 1 Theorem 3.8. Let R ν R λ 2 , R α 0, R β 0, R γ 0, az 1, bz 1 and cz 1. Then we have ( ) > ( ) > − ( ) > ( ) > ( ) > S S < S S < S S < zν−1 Dλ−ν,µ;p;m zλ−1 1 az −α 1 bz −β 1 cz −γ z Γ ν λ

œ 1 ( − 1) (3 − ) 1( − ) ¡ = B λ , ν λ F λ , α, β, γ; ν 1; az, bz, cz; p; m . (46) 2 2 D,µ 2 ( − ) × ‹ + − +  ‹ + +  Proof. As in the proof of Theorem 3.7, taking the binomial theorem for 1 az −α, 1 bz −β and 1 cz −γ and applying Lemmas 3.3 and 3.4 and taking Denition 5 into account, one can easily prove Theorem 3.8. ( − ) ( − ) ( − ) 1 x Theorem 3.9. Let R ν R λ 2 , R α 0, R γ R β 0; 1−z 1 and x z 1. Then we have

( )λ>−ν,µ(;p;)m > −λ−1 ( ) >−α ( ) > ( )x> S S < S S + S S < D z 1 z Fµ α, β; γ; ; p; m z 1 z œ 1 ( − ) 1 ‹  ¡ ν−1 B λ 2 , ν λ 2 1 z F2,µ α, β, λ − ; γ, ν 1; x, z; p; m (47) Γ ν λ 2 ‰ + − + Ž = ‹ + +  ( − ) Extended Riemann-Liouville type fractional derivative operator with applications Ë 1677

Proof. By using (30) and applying the Denition 2.8 for Fµ, we get

λ−ν,µ;p;m λ−1 −α x D z 1 z Fµ α, β; γ; ; p; m z 1 z ∞ n λ−ν,µ;p›;m λ(−1 − ) −α ‹ α n Bµ β n, 㐠β; p; m x Dz z 1 z − n=0 n! B β, γ β 1 z ∞ ( ) ( + − ) n = Bµœβ n(, γ− β); p;Qm λ−ν,µ;p;m λ−1 −α−n‹ x  ¡ α n Dz (z −1 )z −. n=0 B β, γ β n! ( + − ) = Q(λ−)ν,µ;p;m λ−1 −α−n š ( − ) Ÿ Using Theorem 3.6 for Dz z( 1− z) and interpreting the extended hypergeometric function Fµ as its series representation, we get ™ ( − ) ž λ−ν,µ;p;m λ−1 −α x D z 1 z Fµ α, β; γ; ; p; m z 1 z ν−1 ∞ z › ( 1− ) ‹1 Bµ β n, γ β; p; m B λ , ν λ −α n+k Γ ν λ 2 2 n,k=0 ⎧ B β, γ β ⎪ ( + − ) = ‹ + − B+ λ Qk ⎨(1 , ν) λ 1 ; p; m n k ( − ) µ ⎪2 2 ( x− z ) ⎩ 1 1 B λ , ν λ n! k! ⎫ ‰ + + 2 − + 2 Ž ⎪ ν−1 × ⎬ z 1 1 1 ⎪ B λ , ν λ F‰2,µ+ α, β,−λ + ;Žγ, ν 1; x, z⎪; p; m . Γ ν λ 2 2 2 ⎭ This completes the= proof. ‹ + − +  ‹ + +  ( − )

4 Generating functions involving the extended Gauss hypergeometric function

Here, we establish some linear and bilinear generating relations for the extended hypergeometric function Fµ by using Theorems 3.6, 3.7 and 3.9.

1 Theorem 4.1. Let R λ 0 and R β R α 2 . Then we have ∞ λ n 1 n −λ 1 z Fµ( )λ> n, α (; β) >1; z(; p); m> −t 1 t Fµ λ, α ; β 1; ; p; m . (48) n=0 n! 2 2 1 t ( ) Q ‹ + + +  = ( − ) ‹ + +  z min 1, 1 t −

Proof. We start by recalling the elementary(S identityS < { S − S}) z −λ 1 z t −λ 1 t −λ 1 1 t and expand its left-hand side to obtain[( − ) − ] = ( − ) ‹ −  − ∞ λ t n z −λ 1 z −λ n 1 t −λ 1 t 1 z . n=0 n! 1 z 1 t ( ) ( − ) Q ‹  =α−(1 − ) ‹ −  (S S < S − S) Multiplying both sides of the above equality− by z and applying the− extended Riemann-Liouville fractional α−β;µ;p;m derivative operator Dz on both sides, we nd ∞ n −λ α−β;µ;p;m λ n t α−1 −λ−n α−β;µ;p;m −λ α−1 z Dz z 1 z Dz 1 t z 1 . n=0 n! 1 t ( ) œQ ( − ) ¡ = œ( − ) ‹ −  ¡ Uniform convergence of the involved series makes it possible to exchange the summation− and the fractional operator to give ∞ −λ λ n α−β;µ;p;m α−1 −λ−n n −λ α−β;µ;p;m α−1 z Dz z 1 z t 1 t Dz z 1 . n=0 n! 1 t ( ) Q š ( − ) Ÿ = ( − ) œ ‹ −  ¡ − 1678 Ë P. Agarwal et al.

The result then follows by applying Theorem 3.6 to both sides of the last identity.

1 Theorem 4.2. Let R λ 0, R γ 0 and R β R α 2 . Then we have

∞ ( ) > λ (n ) > ( 1) > ( ) > − n Fµ γ n, α ; β 1; z; p; m t n=0 n! 2 ( ) Q ‹ −−λ + +1  zt 1 t F µ α , γ, λ; β 1; z; ; p; m . 1, 2 1 t − = ( − ) ‹ + +  z 1; t 1 z ; z t 1 t −

Proof. Considering the following identity(S S < S S < S − S S SS S < S − S)

zt −λ 1 1 z t −λ 1 t −λ 1 1 t [ − ( − ) ] = ( − ) ‹ +  and expanding its left-hand side as a power series, we get − ∞ λ zt −λ n 1 z n tn 1 t −λ 1 t 1 z . n=0 n! 1 t ( ) − Q ( − ) = ( − ) ‹ −  (S S < S − S) Multiplying both sides by zα−1 1 z −γ and applying the− denition of the extended Riemann-Liouville α−β;µ;p;m fractional derivative operator Dz on both sides, we nd ( − ) ∞ α−β;µ;p;m λ n α−1 −γ n n Dz z 1 z 1 z t n=0 n! ( ) œQ ( − ) ( − ) ¡ zt −λ Dα−β;µ;p;m 1 t −λzα−1 1 z −γ 1 . z 1 t − = œ( − ) ( − ) ‹ −  ¡ The given conditions are found to allow us to exchange the order of the− summation and the fractional derivative to yield

∞ λ n α−β;µ;p;m α−1 −γ+n n Dz z 1 z t n=0 n! ( ) Q š ( − ) Ÿ zt −λ 1 t −λDα−β;µ;p;m zα−1 1 z −γ 1 . z 1 t − = ( − ) œ ( − ) ‹ −  ¡ Finally the result follows by using Theorems 3.6 and 3.7. −

1 1 Theorem 4.3. Let R ξ R δ 2 , R β R α 2 and R λ 0. Then we have

∞ ( λ) >n ( ) > − 1( ) > ( ) > − ( ) >1 n Fµ λ n, α ; β 1; z; p; m Fµ n, δ ; ξ 1; u; p; m t n=0 n! 2 2 ( ) Q ‹ +−λ + + 1 1 ‹− + z + ut  1 t F µ λ, α , δ ; β 1, ξ 1; , ; p; m . 2, 2 2 1 t 1 t − = ( − ) ‹1 u+ + z + +ut  z 1; t 1; −1 − 1 z 1 t 1 t − ‹S S < V V < V V + V V <  Proof. Replacing t by 1 u t in (48) and multiplying− both− sides of− the resulting identity by uδ−1 gives

∞ ( − λ)n 1 δ−1 n n Fµ λ n, α ; β 1; z; p; m u 1 u t n=0 n! 2 ( ) Q δ−1 ‹ + +−λ + 1  ( −z ) u 1 1 u t Fµ λ, α ; β 1; ; p; m . 2 1 1 u t

= [ − ( − ) ] Œ + + 1 ‘ R λ 0, R β R α − ( − ) 2 ‹ ( ) > ( ) > ( ) > −  Extended Riemann-Liouville type fractional derivative operator with applications Ë 1679

δ−ξ,µ;p;m Applying the fractional derivative Du to both sides of the resulting identity and changing the order of the summation and the fractional derivative yields

∞ λ n 1 δ−ξ,µ;p;m δ−1 n n Fµ λ n, α ; β 1; z; p; m Du u 1 u t n=0 n! 2 ( ) Q δ−ξ,µ;p;m‹ +δ−1 + + −λ  1 š ( −z ) Ÿ D u 1 1 u t Fµ λ, α ; β 1; ; p; m . u 2 1 1 u t = œ [ − ( − ) ] Œ + + ‘¡ 1 u t 1; ut 1 t − ( − )

The last identity can be written as follows:(S( − ) S < S S < S − S) ∞ λ n 1 δ−ξ,µ;p;m δ−1 n n Fµ λ n, α ; β 1; z; p; m Du u 1 u t n=0 n! 2 ( ) −λ z Q −λ ‹ δ+−ξ,µ;p;m+ δ−1+ ut  š 1 ( − ) 1Ÿ−t 1 t Du u 1 Fµ λ, α ; β 1; ; p; m . 1 t 2 1 −ut − 1−t = ( − ) œ  −  Œ + + ‘¡ Finally the use of Theorems 3.6 and 3.9 in the resulting− identity is seen to give the− desired result.

5 Mellin transforms and further results

In this section, we rst obtain the Mellin transform of the extended Riemann-Liouville fractional derivative operator.

Theorem 5.1. Let R ν 0, m 0, R µ 0 and

1 1 R λ 1 R ν ( ) < R s> max( )R≥ µ , , . 2 m m 2 m ( ) ( ) Then we have the following relation( ) > › ( ) − − − − +

λ−ν s µ m m ν,µ;p;m λ z Γ s µ Γ 2 2 Γ λ ms 2 1 Γ ν ms 2 M Dz z s 2µΓ ν Γ s µ Γ λ ν 2ms m 1 ( + ) ‰ 2− Ž2 ‰ + + + Ž ‰− + + Ž µ  š Ÿ ∶  = zλ−ν Γ s µ Γ s m m (− ) 2‰ +2 ŽB λ( −ms+ +1, +ν )ms . (49) 2µΓ ν Γ s µ 2 2 ( + )2 ‰2 − Ž = ‹ + + + − + +  Proof. Taking the Mellin transform and(− using) Lemma‰ + 3.3,Ž we have

∞ λ−ν ∞ ν,µ;p;m λ s−1 ν,µ;p;m λ z s−1 3 1 M D z s p D z dp p Bµ λ , ν ; p; m dp. z z Γ ν 2 2 0 0  š Ÿ ∶ = S š Ÿ = S ‹ + − +  Applying Theorem 2.6 to the last integral yields the desired( result.− )

Theorem 5.2. Let R ν 0, m 0, R µ 0, z 1 and

1 1 1 R ν ( ) < R> s (max) ≥ RS Sµ< , , . 2 m 2 m ( ) Then we have the following relation( ) > › ( ) − − − +

−ν s µ ν,µ;p;m −α z Γ s µ Γ 2 2 m m M Dz 1 z s B ms 1, ν ms 2µΓ ν Γ s µ 2 2 ( + )2 ‰2 − Ž
™( − ) ž ∶  = m ‹ + + − + +  2F1 (α−, ms) ‰1;+ ν Ž 2ms m 1; z , (50) 2 × ‹ + + − + + +  where 2F1 is a well known Gauss hypergeometric function given by (28). 1680 Ë P. Agarwal et al.

Proof. Using the binomial series for 1 z −λ and Theorem 5.1 with λ n yields

∞ ν,µ;p;m −α ( − ) ν,µ;p;m α n n = M Dz 1 z s M Dz z s n=0 n! ∞ ( )  α ™n( − ν),µ;pž;m∶ n=  œQ ¡ ∶ M Dz z s n=0 n! ( ) = ∞
n−ν ™ ž ∶  s µ m m Q α n z Γ s µ Γ 2 2 Γ n ms 2 1 Γ ν ms 2 µ s µ . n=0 n! 2 Γ ν Γ Γ n ν 2ms m 1 ( ) ( + ) ‰ 2− Ž2 ‰ + + + Ž ‰− + + Ž = Q (− ) ‰ + Ž ( − + + + ) Then the last expression is easily seen to be equal to the desired one.

Now we present the extended Riemann-Liouville fractional derivative of zλ in terms of the Fox H-function.

Theorem 5.3. Let R p 0, R µ 0, R ν 0 and m 0. Then we have

( ) > ( ) ≥ ( ) < >µ , 1 , λ ν m 1, 2m zλ−ν 2 2 Dν,µ;p;m zλ H4,0 p . z µ 2,4 ⎡ R ⎤ 2 Γ ν ⎢ R ‰ µ 1 Ž ( − m+ + ) m ⎥ ⎢ R µ, 1 , , , λ 1, m , ν , m ⎥ ⎢ R 2 2 2 2 ⎥ š Ÿ = ⎢ R ⎥ (− ) ⎢ R ⎥ ⎢ R ⎥ Proof. The result can be obtained by taking⎣ theR( inverse) ‰− MellinŽ transform‰ + + of theŽ result‰− + in LemmaŽ⎦ 3.3 with the aid of Theorem 2.7.

Applying the result in Theorem 3.3 to the Maclaurin series of ez and the series expression of the Gauss z hypergeometric function 2F1 gives the extended Riemann-Liouville fractional derivatives of e and 2F1 asserted by the following theorems.

Theorem 5.4. If R ν 0, then we have

−ν ∞ n ( ) < ν,µ;p;m z z 3 1 z Dz e Bµ n , ν ; p; m . Γ ν n=0 2 2 n! { } = Q ‹ + − +  Theorem 5.5. If R ν 0, then we have (− )

−ν ∞ n ν,µ;p;m ( ) < z a n b n 3 1 z Dz 2F1 a, b; c; z Bµ n , ν ; p; m z 1 . Γ ν n=0 c n 2 2 n! ( ) ( ) { ( )} = Q ‹ + − +  (S S < ) Acknowledgement: The research of J.J.(− Nieto) has( been) partially supported by the Ministerio de Economía y Competitividad of Spain under grants MTM2016–75140–P, MTM2013–43014–P, Xunta de Galicia, Grants GRC2015-004 and R2016-022, and co-nanced by the European Community fund FEDER.

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