tailieunhanh - A further extension of the extended Riemann–Liouville fractional derivative operator
The main objective of this paper is to establish the extension of an extended fractional derivative operator by using an extended beta function recently defined by Parmar et al. by considering the Bessel functions in its kernel. | Turk J Math (2018) 42: 2631 – 2642 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article A further extension of the extended Riemann–Liouville fractional derivative operator Martin BOHNER1 ,, Gauhar RAHMAN2 ,, Shahid MUBEEN3 ,, Kottakkaran Sooppy NISAR4,∗, 1 Department of Mathematics, Missouri University of Science and Technology, Rolla, MO, USA 2 Department of Mathematics, International Islamic University, Islamabad, Pakistan 3 Department of Mathematics, University of Sargodha, Sargodha, Pakistan 4 Department of Mathematics, College of Arts and Science-Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia Received: • Accepted/Published Online: • Final Version: Abstract: The main objective of this paper is to establish the extension of an extended fractional derivative operator by using an extended beta function recently defined by Parmar et al. by considering the Bessel functions in its kernel. We also give some results related to the newly defined fractional operator, such as Mellin transform and relations to extended hypergeometric and Appell’s function via generating functions. Key words: Hypergeometric function, extended hypergeometric function, Mellin transform, fractional derivative, Appell’s function 1. Introduction Recently, the applications and importance of fractional calculus have received more attention. In the field of mathematical analysis, fractional calculus is a powerful tool. Various extensions and generalizations of fractional derivative operators were recently investigated in [7, 8, 10, 16]. Euler’s beta function is defined by ∫1 B(σ1 , σ2 ) = tσ1 −1 (1 − t)σ2 −1 dt, Re(σ1 ) > 0, Re(σ2 ) > 0, () 0 and its relation with the gamma function is given by B(σ1 , σ2 ) = Γ(σ1 )Γ(σ2 ) . Γ(σ1 + σ2 ) The Gauss hypergeometric and the confluent hypergeometric functions are defined (see [15]) by 2 F1 (σ1 , σ2 ; σ3 ;
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