A Generalization of the Fox H-function
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908782140850176 |
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| author | Vaz, Jayme |
| author_facet | Vaz, Jayme |
| contents | In this paper we present a generalization of the Fox H-function called Fox-Barnes J-function. Like the Fox H-function, it is defined as a contour integral in the complex plane, but instead of an integrand given by a ratio of products of gamma functions involving several parameters, we use a ratio of products of double gamma functions. We study the conditions for its existence and how to choose a contour of integration based on the involved parameters. We discuss how the Fox H-function appears as a particular case and prove some properties of the Fox-Barnes J-function. As an application, we show how the Laplace transform of the Kilbas-Saigo function can be conveniently written in terms of the Fox-Barnes J-function, even in cases where the usual series representation is not convergent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15920 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Generalization of the Fox H-function Vaz, Jayme General Mathematics 33E20 (Primary) 44A10, 33B99 (Secondary) In this paper we present a generalization of the Fox H-function called Fox-Barnes J-function. Like the Fox H-function, it is defined as a contour integral in the complex plane, but instead of an integrand given by a ratio of products of gamma functions involving several parameters, we use a ratio of products of double gamma functions. We study the conditions for its existence and how to choose a contour of integration based on the involved parameters. We discuss how the Fox H-function appears as a particular case and prove some properties of the Fox-Barnes J-function. As an application, we show how the Laplace transform of the Kilbas-Saigo function can be conveniently written in terms of the Fox-Barnes J-function, even in cases where the usual series representation is not convergent. |
| title | A Generalization of the Fox H-function |
| topic | General Mathematics 33E20 (Primary) 44A10, 33B99 (Secondary) |
| url | https://arxiv.org/abs/2510.15920 |