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

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  • 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
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    References listed on IDEAS

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    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.
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