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) √ ( ) + ( ) = ( ) = ( ( ) > ) 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 ( ) ∶= ( − ) 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 ( ) ∶= ( ) = ( − ) − 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 2m 2p u 2 1 u + − Kµ+ 1 p du (12) ¾ π 1 u x y 1 2 um 0 ( + ) = 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 ( + ) ( − ) (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 ( − ) ∞ 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 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 ( − ) 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 ( − ) − ( ∈ ) 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− 2 + + + + µ = 1 Γ s µ Γ s m 1 m 1 2 +2 B (x +ms+ + , y− ms) .