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Fractional-Modified Bessel Function of the First Kind of Integer Order

Author

Listed:
  • Andrés Martín

    (Faculty of Science, University of Zaragoza, Pedro Cerbuna, 50009 Zaragoza, Spain)

  • Ernesto Estrada

    (Institute of Interdisciplinary Physics and Complex Systems (IFISC), 07122 Palma de Mallorca, Spain)

Abstract

The modified Bessel function (MBF) of the first kind is a fundamental special function in mathematics with applications in a large number of areas. When the order of this function is integer, it has an integral representation which includes the exponential of the cosine function. Here, we generalize this MBF to include a fractional parameter, such that the exponential in the previously mentioned integral is replaced by a Mittag–Leffler function. The necessity for this generalization arises from a problem of communication in networks. We find the power series representation of the fractional MBF of the first kind as well as some differential properties. We give some examples of its utility in graph/networks analysis and mention some fundamental open problems for further investigation.

Suggested Citation

  • Andrés Martín & Ernesto Estrada, 2023. "Fractional-Modified Bessel Function of the First Kind of Integer Order," Mathematics, MDPI, vol. 11(7), pages 1-13, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1630-:d:1109354
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    References listed on IDEAS

    as
    1. Robert, Christian, 1990. "Modified Bessel functions and their applications in probability and statistics," Statistics & Probability Letters, Elsevier, vol. 9(2), pages 155-161, February.
    2. Zhen-Hang Yang & Jing-Feng Tian & Ya-Ru Zhu, 2020. "New Sharp Bounds for the Modified Bessel Function of the First Kind and Toader-Qi Mean," Mathematics, MDPI, vol. 8(6), pages 1-13, June.
    Full references (including those not matched with items on IDEAS)

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