The main aim of the paper is to introduce an operator in the space of Lebesgue measurable real or complex functions L(a, b). Some properties of the Riemann–Liouville fractional integrals and differential operators associated with the function E γ,δ α,β,λ,μ,ρ,p (cz; s, r) are studied and the integral representations are obtained. Some properties of a special case of this function are also studied by the means of fractional calculus.
Similar content being viewed by others
References
A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Applications, Springer, New York (2010).
Applications of Fractional Calculus in Physics, Ed. R. Hilfer, World Scientific, Singapore, etc. (2000).
Fractional Time Evolution. Applications of Fractional Calculus in Physics, Ed. R. Hilfer, World Scientific, Singapore, etc. (2000).
A. A. Kilbas, M. Saigo, and R. K. Saxena, “Generalized Mittag-Leffler function and generalized fractional calculus operators,” Integral Transforms Spec. Funct., 15, 31–49 (2004).
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” North-Holland Math. Stud., Elsevier (North-Holland) Sci. Publ., Amsterdam, 204 (2006).
K. S. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993).
J. C. Prajapati, B. I. Dave, and B.V. Nathwani, “On a unification of generalized Mittag-Leffler function and family of Bessel functions,” Adv. Pure Math., 3, No. 1, 127–137 (2013).
E. D. Rainville, Special Functions, Macmillan Co., New York (1960).
T. O. Salim, “Some properties relating to the generalized Mittag-Leffler function,” Adv. Appl. Math. Anal., 4, No. 1, 21–30 (2009).
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach, Yverdon (Switzerland) (1993).
A. K. Shukla and J. C. Prajapati, “On a generalization of Mittag-Leffler functions and its properties,” J. Math. Anal. Appl., 337, 797–811 (2007).
H. M. Srivastava and R. K. Saxena, “Some Vottera type fractional integrodifferential equations with a multivariable confluent hypergeometric function as their kernel,” J. Integral Equat. Appl., 17, 199–217 (2005).
H. M. Srivastava and Z. Tomovski, “Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel,” Appl. Math. Comput., 211, No. 1, 198–210 (2009).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 8, pp. 1133–1145, August, 2014.
Rights and permissions
About this article
Cite this article
Prajapati, J.C., Nathwani, B.V. Fractional Calculus of a Unified Mittag-Leffler Function. Ukr Math J 66, 1267–1280 (2015). https://doi.org/10.1007/s11253-015-1007-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-015-1007-2