IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i22p4305-d975200.html
   My bibliography  Save this article

Taylor Series for the Mittag–Leffler Functions and Their Multi-Index Analogues

Author

Listed:
  • Jordanka Paneva-Konovska

    (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria)

Abstract

It has been obtained that the n -th derivative of the 2-parametric Mittag–Leffler function is a 3-parametric Mittag–Leffler function, with exactness to a constant. Following the analogy, the author later obtained the n -th derivative of the 2 m -parametric multi-index Mittag–Leffler function. It turns out that this is expressed via the 3 m -parametric Mittag–Leffler function. In this paper, upper estimates of the remainder terms of these derivatives are found, depending on n . Some asymptotics are also found for “large” values of the parameters. Further, the Taylor series of the 2 and 2 m -parametric Mittag–Leffler functions around a given point are obtained. Their coefficients are expressed through the values of the corresponding n -th order derivatives at this point. The convergence of the series to the represented Mittag–Leffler functions is justified. Finally, the Bessel-type functions are discussed as special cases of the multi-index ( 2 m -parametric) Mittag–Leffler functions. Their Taylor series are derived from the general case as corollaries, as well.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4305-:d:975200
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/22/4305/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/22/4305/
    Download Restriction: no
    ---><---

    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. Muhammad Aslam Noor & Khalida Inayat Noor, 2021. "Some New Classes of Higher Order Strongly Generalized Preinvex Functions," Springer Optimization and Its Applications, in: Ioannis N. Parasidis & Efthimios Providas & Themistocles M. Rassias (ed.), Mathematical Analysis in Interdisciplinary Research, pages 573-588, Springer.
    3. Sergei Rogosin, 2015. "The Role of the Mittag-Leffler Function in Fractional Modeling," Mathematics, MDPI, vol. 3(2), pages 1-14, May.
    4. Virginia Kiryakova, 2021. "A Guide to Special Functions in Fractional Calculus," Mathematics, MDPI, vol. 9(1), pages 1-40, January.
    5. F. Ghanim & Hiba F. Al-Janaby & Marwan Al-Momani & Belal Batiha, 2022. "Geometric Studies on Mittag-Leffler Type Function Involving a New Integrodifferential Operator," Mathematics, MDPI, vol. 10(18), pages 1-10, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Virginia Kiryakova, 2021. "A Guide to Special Functions in Fractional Calculus," Mathematics, MDPI, vol. 9(1), pages 1-40, January.
    3. Virginia Kiryakova, 2020. "Unified Approach to Fractional Calculus Images of Special Functions—A Survey," Mathematics, MDPI, vol. 8(12), pages 1-35, December.
    4. 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.
    5. Jordanka Paneva-Konovska, 2023. "Prabhakar Functions of Le Roy Type: Inequalities and Asymptotic Formulae," Mathematics, MDPI, vol. 11(17), pages 1-13, September.
    6. Faten Fakher Abdulnabi & Hiba F. Al-Janaby & Firas Ghanim & Alina Alb Lupaș, 2023. "Some Results on Third-Order Differential Subordination and Differential Superordination for Analytic Functions Using a Fractional Differential Operator," Mathematics, MDPI, vol. 11(18), pages 1-14, September.
    7. 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.
    8. El Anouz, K. & El Allati, A. & Metwally, N. & Obada, A.S., 2023. "The efficiency of fractional channels in the Heisenberg XYZ model," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    9. Alina Alb Lupaş, 2023. "Fuzzy Differential Subordination and Superordination Results for Fractional Integral Associated with Dziok-Srivastava Operator," Mathematics, MDPI, vol. 11(14), pages 1-20, July.
    10. Meysam Alvan & Rahmat Darzi & Amin Mahmoodi, 2016. "Existence Results for a New Class of Boundary Value Problems of Nonlinear Fractional Differential Equations," Mathematics, MDPI, vol. 4(1), pages 1-10, March.
    11. Khan, Tahir Ullah & Khan, Muhammad Adil, 2021. "New generalized mean square stochastic fractional operators with applications," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    12. Paolo Di Barba & Luisa Fattorusso & Mario Versaci, 2022. "Solution Properties of a New Dynamic Model for MEMS with Parallel Plates in the Presence of Fringing Field," Mathematics, MDPI, vol. 10(23), pages 1-20, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4305-:d:975200. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.