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A Guide to Special Functions in Fractional Calculus

Author

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  • Virginia Kiryakova

    (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria; virginia@diogenes.bg)

Abstract

Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project . Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions , Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “ Guide to the Functions ”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G -function and, specially, of the generalized hypergeometric functions p F q . These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC) . Generally, under this notion, we have in mind the Fox H -functions, their most widely used cases of the Wright generalized hypergeometric functions p Ψ q and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

Suggested Citation

  • Virginia Kiryakova, 2021. "A Guide to Special Functions in Fractional Calculus," Mathematics, MDPI, vol. 9(1), pages 1-40, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:1:p:106-:d:475128
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    References listed on IDEAS

    as
    1. Kiryakova, Virginia, 2017. "Fractional calculus operators of special functions? The result is well predictable!," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 2-15.
    2. Francesco Mainardi & Armando Consiglio, 2020. "The Wright Functions of the Second Kind in Mathematical Physics," Mathematics, MDPI, vol. 8(6), pages 1-26, June.
    3. Sergei Rogosin, 2015. "The Role of the Mittag-Leffler Function in Fractional Modeling," Mathematics, MDPI, vol. 3(2), pages 1-14, May.
    4. Roberto Garrappa & Eva Kaslik & Marina Popolizio, 2019. "Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial," Mathematics, MDPI, vol. 7(5), pages 1-21, May.
    5. H. J. Haubold & A. M. Mathai & R. K. Saxena, 2011. "Mittag-Leffler Functions and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-51, May.
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    Cited by:

    1. Asifa Tassaddiq & Rekha Srivastava, 2023. "New Results Involving the Generalized Krätzel Function with Application to the Fractional Kinetic Equations," Mathematics, MDPI, vol. 11(4), pages 1-17, February.
    2. Jordanka Paneva-Konovska & Virginia Kiryakova, 2024. "The Generalized Fox–Wright Function: The Laplace Transform, the Erdélyi–Kober Fractional Integral and Its Role in Fractional Calculus," Mathematics, MDPI, vol. 12(12), pages 1-25, June.
    3. Jordanka Paneva-Konovska, 2021. "Series in Le Roy Type Functions: A Set of Results in the Complex Plane—A Survey," Mathematics, MDPI, vol. 9(12), pages 1-15, June.
    4. Jordanka Paneva-Konovska, 2023. "Prabhakar Functions of Le Roy Type: Inequalities and Asymptotic Formulae," Mathematics, MDPI, vol. 11(17), pages 1-13, September.
    5. Virginia Kiryakova & Jordanka Paneva-Konovska, 2024. "Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey," Mathematics, MDPI, vol. 12(2), pages 1-39, January.
    6. Jordanka Paneva-Konovska, 2022. "Taylor Series for the Mittag–Leffler Functions and Their Multi-Index Analogues," Mathematics, MDPI, vol. 10(22), pages 1-15, November.

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