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

Andrés Martín and Ernesto Estrada ()
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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

Mathematics, 2023, vol. 11, issue 7, 1-13

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.

Keywords: modified Bessel functions; communicability in graphs; Estrada index; power-series; fractional calculus; Caputo derivative; Riemann–Liouville integral; paths; cycles; Mittag–Leffler function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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