Abstract
In this paper, our principle aim is to establish a new extension of the Caputo fractional derivative operator involving the generalized hypergeometric type function F p (a, b; c; z; k), introduced by Lee et al. Some extensions of the generalized hypergeometric functions and their integral representations are also presented. Furthermore, linear and bilinear generating relations for the extended hypergeometric functions are obtained. We also present some properties of the extended fractional derivative operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. A. Chaudhry, A. Qadir, M. Rafique, and S. M. Zubair, “Extension of Euler’s Beta Function,” J. Comput. Appl. Math. 78, 19–32 (1997).
M. A. Chaudhry, A. Qadir, H. M. Srivastava, and R. B. Paris, “Extended Hypergeometric and Confluent Hypergeometric Functions,” Appl. Math. Comput. 159 (2), 589–602 (2004).
M. A. Chaudhry and S. M. Zubair, “Generalized Incomplete Gamma Functions with Applications,” J. Comput. Appl. Math. 55, 99–124 (1994).
W. S. Chung, T. Kim and T. Mansour, “The q-Deformed Gamma Function and q-Deformed Polyfamma Function,” Bull. Korean Math. Soc. 51 (4), 1155–1161 (2014).
A. A. Kilbas and M. Saigo, H-Transforms: Theory and Applications (Chapman and Hall (CRC Press Company), Boca Raton, London, New York and Washington, D.C., 2004).
T. Kim, “q-Extension of the Euler Formula and Trigonometric Functions,” Russ. J. Math. Phys. 14 (3), 275–278 (2007).
V. Kiryakova, Generalized Fractional Calculus and Applications (Longman & J. Wiley, Harlow-N. York, 1994).
D. M. Lee, A. K. Rathie, R. K. Parmar, and Y. S. Kim, “Generalization of Extended Beta Function, Hypergeometric and Confluent Hypergeometric Functions,” Honam Math. J. 33 (2), 187–206 (2011).
A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Applications (Springer, New York, 2010).
R. K. Parmar, “Some Generalized Relations for Generalized Extended Hypergeometric Functions Involving Generalized Fractional Derivative Operator,” J. Concrete Appl.Math. 12 (3–4), 217–228 (2014).
H. M. Srivastava, P. Agarwal, and S. Jain, “Generating Functions for the Generalized Gauss Hypergeometric Functions,” Appl. Math. Comput. 247, 348–352 (2014).
H. M. Srivastava, A. C¸ etinkaya, and I. O. Kıymaz, “A Certain Generalized Pochhammer Symbol and Its Applications to Hypergeometric Functions,” Appl. Math. Comput. 226, 484–491 (2014).
H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions (Ellis Horwood Limited, Chichester, 1984).
H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals (Elsevier Science Publishers, Amsterdam, London and New York, 2012).
H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series (Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985).
M. A. Özarslan and E. Özergin, “Some Generating Relations for Extended Hypergeometric Functions via Generalized Fractional Derivative Operator,” Math. Comput. Modelling 52, 1825–1833 (2010).
E. Özergin, Some Properties of Hypergeometric Functions (Ph.D. Thesis, Eastern Mediterranean University, North Cyprus, Turkey, 2011).
E. Özergin, M. A. Özarslan, and A. Altın, “Extension of Gamma, Beta and Hypergeometric Functions,” J. Comput. Appl. Math. 235, 4601–4610 (2011).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Agarwal, P., Jain, S. & Mansour, T. Further extended Caputo fractional derivative operator and its applications. Russ. J. Math. Phys. 24, 415–425 (2017). https://doi.org/10.1134/S106192081704001X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S106192081704001X