Fractional Hypergeometric Functions
Abstract
:1. Introduction
2. Preliminaries
3. Main Results
3.1. Marichev–Saigo–Maeda Fractional Integral of the Function and
3.2. Marichev–Saigo–Maeda Fractional Derivative of the Function and
3.3. Examples of the Marichev–Saigo–Maeda Fractional Integral of the Function and
3.4. Examples of the Marichev–Saigo–Maeda Fractional Derivative of the Function and
4. Concluding Remarks
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Marichev, O.I. Volterra equation of Mellin convolution type with a Horn function in the kernel. Izv. Akad. Nauk BSSR Seriya Fiz.-Mat. Nauk 1974, 1, 128–129. [Google Scholar]
- Saigo, M. On generalized fractional calculus operators. In Proceedings of the Recent Advances in Applied Mathematics, Kuwait City, Kuwait, 4–7 May 1996; Kuwait University Press: Kuwait City, Kuwait, 1996; pp. 441–450. [Google Scholar]
- Saigo, M.; Maeda, N. More generalization of fractional calculus. In Transform Methods and Special Functions (Varna, Bulgaria); Bulgaria Academy of Sciences: Sofia, Bulgaria, 1998; pp. 386–400. [Google Scholar]
- Saigo, M. A remark on integral operators involving the Gauss hypergeometric functions. Math. Rep. Kyushu Univ. 1978, 11, 135–143. [Google Scholar]
- Srivastava, H.M.; Saigo, M. Multiplication of fractional calculus operators and boundary value problems involving the Euler- Darboux equation. J. Math. Anal. Appl. 1987, 121, 325–369. [Google Scholar] [CrossRef] [Green Version]
- Srivastava, H.M.; Agarwal, P. Certain Fractional Integral Operators and the Generalized Incomplete Hypergeometric Functions. Appl. Appl. Math. 2013, 8, 333–345. [Google Scholar]
- Agarwal, P.; Choi, J. Fractional calculus operators and their image formulas. J. Korean Math. Soc. 2016, 53, 1183–1210. [Google Scholar] [CrossRef] [Green Version]
- Virchenko, N.; Lisetska, O.; Kalla, S.L. On some fractional integral operators involving generalized Gauss hypergeometric functions. Appl. Appl. Math. 2010, 5, 1418–1427. [Google Scholar]
- Choi, J.; Agarwal, P. A Note on Fractional Integral Operator Associated with Multiindex Mittag-Leffler Functions. Filomat 2016, 30, 1931–1939. [Google Scholar] [CrossRef]
- Agarwal, P.; Nieto, J.J. Some fractional integral formulas for the Mittag-Leffler type function with four parameters. Open Math. 2015, 13, 537–546. [Google Scholar] [CrossRef]
- Rao, A.; Garg, M.; Kalla, S.L. Caputo-type fractional derivative of a hypergeometric integral operator. Kuwait J. Sci. Eng. 2010, 37, 15–29. [Google Scholar]
- Agarwal, P.; Choi, J.; Paris, R.B. Extended Riemann–Liouville fractional derivative operator and its applications. J. Nonlinear Sci. Appl. 2015, 8, 451–466. [Google Scholar] [CrossRef]
- Agarwal, P.; Jain, S.; Mansour, T. Further extended Caputo fractional derivative operator and its applications. Russ. J. Math. Phys. 2017, 24, 415–425. [Google Scholar] [CrossRef]
- Kiryakova, V. On two Saigo’s fractional integral operators in the class of univalent functions. Fract. Calc. Appl. Anal. 2006, 9, 159–176. [Google Scholar]
- Kiryakova, V. A brief story about the operators of the generalized fractional calculus. Fract. Calc. Appl. Anal. 2008, 11, 203–220. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations. In North-Holland Mathematical Studies; Elsevier (North-Holland) Science Publishers: Amsterdam, The Netherlands; London, UK; New York, NY, USA, 2006; Volume 204. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach: New York, NY, USA, 1993. [Google Scholar]
- Sneddon, I.N. The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations. In Proceedings of the Fractional Calculus and Its Applications, West Haven, CT, USA, 15–16 June 1974; Ross, B., Ed.; Lecture Notes in Mathematics. Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1975; Volume 457, pp. 37–79. [Google Scholar]
- Rainville, E.D. Special Functions; Macmillan: New York, NY, USA, 1960. [Google Scholar]
- Olver, W.J.F.; Lozier, W.D.; Boisvert, F.R.; Clark, W.C. NIST Handbook of Mathematical Functions; Cambridge University Press: New York, NY, USA, 2010. [Google Scholar]
- Goyal, R.; Agarwal, P.; Momami, S.; Rassias, M.T. An Extension of Beta Function by Using wiman’s function. Axioms 2021, 10, 187. [Google Scholar] [CrossRef]
- Wiman, A. U ber der Fundamentalsatz in der Theorie der Funktionen Eα(x). Acta Math. 1905, 29, 191–201. [Google Scholar] [CrossRef]
- Jain, S.; Goyal, R.; Agarwal, P.; Lupica, A.; Cesarano, C. Some results of extended beta function and hypergeometric functions by using Wiman’s function. Mathemtics 2021, 9, 2944. [Google Scholar] [CrossRef]
- Pohlen, T. The Hadamard Product and Universal Power Series; Universitát Trier: Trier, Germany, 2009. [Google Scholar]
- Appell, P. Sur les Fonctions Hypergéométriques de Plusieurs Variables; Mémorial des Sciences Mathématiques, No. 3; Gauthier-Villars: Paris, France, 1925. [Google Scholar]
- Appell, P.; De Fériet, J.K. Fonctions Hypergéométriques et Hypersphériques: Polynomes d’Hermite; Gauthier-Villars: Paris, France, 1926; Volume 140. [Google Scholar]
- Srivastava, H.M.; Karlsson, P.W. Multiple Gaussian Hypergeometric Series; Halsted Press (Ellis Horwood Limited): Chichester, UK; John Wiley and Sons: New York, NY, USA; Chichester, UK; Brisbane, Australia; Toronto, ON, Canada, 1985. [Google Scholar]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Jain, S.; Cattani, C.; Agarwal, P. Fractional Hypergeometric Functions. Symmetry 2022, 14, 714. https://doi.org/10.3390/sym14040714
Jain S, Cattani C, Agarwal P. Fractional Hypergeometric Functions. Symmetry. 2022; 14(4):714. https://doi.org/10.3390/sym14040714
Chicago/Turabian StyleJain, Shilpi, Carlo Cattani, and Praveen Agarwal. 2022. "Fractional Hypergeometric Functions" Symmetry 14, no. 4: 714. https://doi.org/10.3390/sym14040714
APA StyleJain, S., Cattani, C., & Agarwal, P. (2022). Fractional Hypergeometric Functions. Symmetry, 14(4), 714. https://doi.org/10.3390/sym14040714