Classes of I- and bar-H- special functions related to Fractional Calculus and generalized fractional integrals


Friday, 17 May, 2024 - 14:30

Speaker : Virginia Kiryakova, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences

Time : 14:30 - 15:30 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

YouTube :

Organizers : Pavan Pranjivan Mehta* ( and Arran Fernandez** (

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Special Functions, Fractional Calculus, Integral Operators

Abstract: Recently, the interest of authors like Gerhold, Garra, Polito, Mainardi, Rogosin, Garrappa, Gorska and others has been attracted to the so-called Le Roy function that happened to appear almost same time as the famous Mittag-Leffler function and for almost same goals, initially not associated with Fractional Calculus (FC). They have introduced and studied its extension as a hybrid between the Mittag-Leffler and Le Roy functions, depending on fractional parameters both as arguments in the Gamma function and as its fractional power. Next came a series of multi-index generalizations as new classes of Special Functions (SF) related to FC (Kiryakova, Paneva-Konovska, Rogosin, Dubatovskaya: [2], [4], [5], [6], [7]).

The idea to relate the Le Roy type multi-index functions to the not so-popular I-functions of Rathie and bar-$H$-functions of Inayat-Hussain, as further extensions of the Fox H-functions, happened to be very fruitful. It happened that not only the Le Roy type functions belong to their class, but also functions appearing by needs in Physics and Statistics (as Feynman integrals) and some well known mathematical functions as polylogarithms, Riemann Zeta-functions, its extensions, etc., see [8]. 

First, this led us to introduce and study the analytical properties of a new entire function $\widetilde{{}_p\Psi_q}$ extending the Fox-Wright function ${}_p\Psi_q$, thus to encompass yet more of the SF related to Classical and Fractional Calculus, as previously listed in [4].

Then, it is the problem to identify the integral (and possibly differential) operators for which such SF as of Le Roy type play the role of eigen-functions, similarly like the exponential / trigonometric functions appear as eigen-functions of the operators of integer order Calculus, and the Mittag-Leffler type functions - for the R-L and Erdelyi-Kober operators of FC. The Gelfond-Leontiev theory for generalized integrations and differentiation (cf. [3]) helped to resolve the case, first in form of power series, and then also as integral operators with I-function kernels, [8]. 

The next step is to introduce integral operators with more general $I^{m,0}_{m,m}$ kernels, for which we prove to satisfy the semi-group property and other axioms of FC. These appear as further extensions of our operators of Generalized Fractional Calculus [1], with $H^{m,0}_{m,m}~$ kernels, and are also compositions of $m \geq 1$ generalized Erdelyi-Kober integrals.

The talk is based on recent results and publications by V. Kiryakova and J. Paneva-Konovska, partly collaborated also with S. Rogosin and M. Dubatovskaya.

Biography: Virginia Kiryakova is a Bulgarian mathematician, Professor Emeritus in Institute of Mathematics and Informatics at Bulgarian Academy of Sciences. Her research results in more than 140 publications for about 50 years are in fractional calculus, special functions and integral transforms, and on the history of fractional calculus, author of the monograph ``Generalized Fractional Calculus and Applications”, Longman-J. Wiley, 1994. These works are cited more than 7000 times and she is in the list of the top 2% of scientists worldwide according to the Standford University ranking. Aside from her role as Editor-in-Chief of the top-ranked specialized journal ``Fractional Calculus and Applied Analysis" (Springer), for long years she has a lot of organizational activities in scientific and social organizations, committees, etc. and holds some national and international prizes.


[1] V. Kiryakova. Generalized Fractional Calculus and Applications, Longman - J.Wiley, Harlow- New York (1994).

[2] V. Kiryakova. “The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus”. In: Comput. Math. Appl. 59 (2010), pp. 1885–1895;

[3] V. Kiryakova. “Gel’fond-Leont’ev integration operators of fractional (multi)order generated by some special functions”. In: AIP Conference Proc. 2048 (2018), # 050016, 10 pp.;

[4] V. Kiryakova. “A guide to special functions in fractional calculus”. In: Mathematics 9 (2021), # 106;

[5] V. Kiryakova, J. Paneva-Konovska. “Multi-index Le Roy functions of Mittag-Leffler-Prabhakar type”. In: Intern. J. Appl. Math. 35 (2022), pp. 743–766;

[6] J. Paneva-Konovska, V. Kiryakova, S. Rogosin, M. Dubatovskaya. “Laplace transform (Part 1) of the multi-index Mittag-Leffler-Prabhakar functions of Le Roy type”. In: Intern. J. Appl. Math. 36 (2023), pp. 455–474;

[7] V. Kiryakova, J. Paneva-Konovska, S. Rogosin, M. Dubatovskya. “Erd ́elyiKober fractional integrals (Part 2) of the multi-index Mittag-Leffler-Prabhakar functions of Le Roy type”. In: Intern. J. Appl. Math. 36 (2023), pp. 605–623;

[8] V. Kiryakova, J. Paneva-Konovska. “After “A Guide to Special Functions in Fractional Calculus”: Going Next. Discussion Survey”. In: Mathematics 12 (2024), # 319;