Fractional-Modified Bessel Function of the First Kind of Integer Order
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...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/7/1630 |
| _version_ | 1797607511214260224 |
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| author | Andrés Martín Ernesto Estrada |
| author_facet | Andrés Martín Ernesto Estrada |
| author_sort | Andrés Martín |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T05:30:57Z |
| format | Article |
| id | doaj.art-a17233f8443c43f2bdaa661f89b8a9d9 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T05:30:57Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a17233f8443c43f2bdaa661f89b8a9d92023-11-17T17:08:23ZengMDPI AGMathematics2227-73902023-03-01117163010.3390/math11071630Fractional-Modified Bessel Function of the First Kind of Integer OrderAndrés Martín0Ernesto Estrada1Faculty of Science, University of Zaragoza, Pedro Cerbuna, 50009 Zaragoza, SpainInstitute of Interdisciplinary Physics and Complex Systems (IFISC), 07122 Palma de Mallorca, SpainThe 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.https://www.mdpi.com/2227-7390/11/7/1630modified Bessel functionscommunicability in graphsEstrada indexpower-seriesfractional calculusCaputo derivative |
| spellingShingle | Andrés Martín Ernesto Estrada Fractional-Modified Bessel Function of the First Kind of Integer Order Mathematics modified Bessel functions communicability in graphs Estrada index power-series fractional calculus Caputo derivative |
| title | Fractional-Modified Bessel Function of the First Kind of Integer Order |
| title_full | Fractional-Modified Bessel Function of the First Kind of Integer Order |
| title_fullStr | Fractional-Modified Bessel Function of the First Kind of Integer Order |
| title_full_unstemmed | Fractional-Modified Bessel Function of the First Kind of Integer Order |
| title_short | Fractional-Modified Bessel Function of the First Kind of Integer Order |
| title_sort | fractional modified bessel function of the first kind of integer order |
| topic | modified Bessel functions communicability in graphs Estrada index power-series fractional calculus Caputo derivative |
| url | https://www.mdpi.com/2227-7390/11/7/1630 |
| work_keys_str_mv | AT andresmartin fractionalmodifiedbesselfunctionofthefirstkindofintegerorder AT ernestoestrada fractionalmodifiedbesselfunctionofthefirstkindofintegerorder |