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Therefore, the machine learning community has directed its attention to the development of computationally efficient activation functions. In this paper we introduce a new family of activation functions based on the hypergeometric functions. These functions have trainable parameters, and therefore after the training process, the NN will end up with different activation functions. To the best of our knowledge, this work is the first attempt to consider hypergeometric functions as activation functions in NNs. Special attention is given to the Bessel functions of the first kind J\u03bd, which is a sub-family of the general family of hypergeometric functions. The new (Bessel-type) activation functions are implemented on different benchmark data sets and compared to the widely adopted ReLU activation function. The results demonstrate that the Bessel activation functions outperform the ReLU activation functions in both accuracy aspects and computational time.<\/jats:p>","DOI":"10.3390\/math13142232","type":"journal-article","created":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T07:38:27Z","timestamp":1752133107000},"page":"2232","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Hypergeometric Functions as Activation Functions: The Particular Case of Bessel-Type Functions"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8756-4893","authenticated-orcid":false,"given":"Nelson","family":"Vieira","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193 Aveiro, Portugal"},{"name":"Department of Mathematics, University of Aveiro, Campus Universit\u00e1rio de Santiago, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3270-3395","authenticated-orcid":false,"given":"Felipe","family":"Freitas","sequence":"additional","affiliation":[{"name":"EDF Research and Development, 329 Portland Rd, Brighton and Hove, Hove BN3 5SU, UK"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-5980-8880","authenticated-orcid":false,"given":"Roberto","family":"Figueiredo","sequence":"additional","affiliation":[{"name":"Department of Electronics, Telecommunications and Informatics, University of Aveiro, Campus Universit\u00e1rio de Santiago, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6424-6590","authenticated-orcid":false,"given":"Petia","family":"Georgieva","sequence":"additional","affiliation":[{"name":"Department of Electronics, Telecommunications and Informatics, University of Aveiro, Campus Universit\u00e1rio de Santiago, 3810-193 Aveiro, Portugal"},{"name":"Instituto de Telecomunica\u00e7\u00f5es, 3810-193 Aveiro, Portugal"},{"name":"Institute of Electronics and Informatics Engineering of Aveiro (IEETA), 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"114048","DOI":"10.1016\/j.eswa.2020.114048","article-title":"Ensemble of convolutional neural networks trained with different activation functions","volume":"166","author":"Maguolo","year":"2021","journal-title":"Expert Syst. 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