Abstract
Various extended beta functions have been investigated by many researchers (Mubeen et al. in FJMS 102:1545–1557, 2017; Özergin et al. in J Comput Appl Math 235:4601–4610, 2011; Parmar et al. in J Class Anal 11:91–106, 2017). Our aim in this paper is to develop \(\psi \)-generalized Riemann Liouville fractional derivative operator by using \(\psi \)-generalized beta function. We also present some new results on \(\psi \)-generalized beta function and introduce some integral representations of \(\psi \)-generalized hypergeometric functions. Moreover, we establish fractional derivative formulas of some known functions and discuss the properties of Mellin transform. Further we discuss some applications based on generating relation for the \(\psi \)-generalized hypergeometric functions with the help of new \(\psi \)-generalized Riemann–Liouville 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
Agarwal, P., Choi, J., Parisc, R.B.: Extended Riemann–Liouville fractional derivative operator and its applications. J. Nonlinear Sci. Appl. 8, 451–466 (2015)
Ata, E.: Generalized Beta Function Defined by Wright function. arXiv:1803.03121v1 [math.CA] (2018)
Baleanu, D., Agarwal, P., Parmar, R.K., Alqurashi, M.M., Salahshour, S.: Extended Riemann–Liouville fractional derivative operator and its applications. J. Nonlinear Sci. Appl. 10, 2914–2924 (2015)
Bansal, M.K., Kumar, D.: On the integral operators pertaining to a family of incomplete I-functions. AIMS Math. 5(2), 1247–1259 (2020)
Bansal, M.K., Kumar, D., Jain, R.: A study of Marichev Saigo Maeda fractional integral operators associated with S-generalized Gauss hypergeometric function. Kyungpook Math. J. 59, 433–443 (2019)
Bansal, M.K., Kumar, D., Nisar, K.S., Singh, J.: Certain fractional calculus and integral transform results of incomplete \(\aleph \)-functions with applications. Math Methods Appl. Sci. 43, 5602–5614 (2020)
Bohner, M., Rahman, G., Mubeen, S., Nisar, K.S.: A further extension of the extended Riemann–Liouville fractional derivative operator. Turk. J. Math. 42, 2631–2642 (2018)
Chaudhry, M.A., Qadir, A., Rafique, M., Zubair, S.M.: Extension of Euler’s beta function. J. Comput. Appl. Math. 78, 19–32 (1997)
Chaudhry, M.A., Qadir, A., Srivastava, H.M., Paris, R.B.: Extended hypergeometric and confiuent hypergeometric functions. Appl. Math. Comput. 159, 589–602 (2004)
Chaudhry, M.A., Zubair, S.M.: Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55, 99–124 (1994)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 3. McGraw-Hill, New York (1953)
Mubeen, S., Rahman, G., Nisar, K.S., Choi, J., Arshad, M.: An extended beta function and its properties. FJMS 102, 1545–1557 (2017)
Özergin, E., Özarslan, M.A., Altin, A.: Extension of gamma, beta and hypergeometric functions. J. Comput. Appl. Math 235, 4601–4610 (2011)
Özarslan, M.A., Özergin, E.: Some generating relations for extended hypergeometric functions via generalized fractional derivative operator. Math. Comput. Model. 52, 1825–1833 (2010)
Parmar, R.K., Chopra, P., Paris, R.B.: On an extension of extended beta and hypergeometric functions. J. Class. Anal. 11, 91–106 (2017)
Rahman, G., Nisar, K.S., Tomovski, Z.: On a certain extension of the Riemann–Liouville fractional derivative operator. Commun. Korean Math. Soc. 34, 507–522 (2019)
Shadab, M., Khan, M.F., Luis, J., Lopez-Bonilla, J.: A new Riemann–Liouville type fractional derivative operator and its application in generating functions. Adv. Differ. Equ. 2018, 167 (2018)
Srivastava, H.M., Agarwal, P.: Certain fractional integral operators and the generalized incomplete hypergeometric functions. Appl. Appl. Math. 8(2), 333–345 (2013)
Wang, J., Zhou, Y., O’Regan, D.: A note on asymptotic behaviour of Mittag-Leffler. Integral Transforms Spec. Funct. 29, 81–94 (2017)
Author information
Authors and Affiliations
Corresponding authors
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Singhal, M., Mittal, E. On a \(\psi \)-Generalized Fractional Derivative Operator of Riemann–Liouville with Some Applications. Int. J. Appl. Comput. Math 6, 143 (2020). https://doi.org/10.1007/s40819-020-00892-5
Accepted:
Published:
DOI: https://doi.org/10.1007/s40819-020-00892-5
Keywords
- \(\psi \)-Gamma function
- \(\psi \)-Beta function
- \(\psi \)-Hypergeometric function
- \(\psi \)-Generalized Riemann–Liouville fractional derivative operator
- Mellin transform
- Generating function