On generalized Mittag-Leffler-type functions of two variables
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| Format: | Preprint |
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2025
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| author | Hasanov, Anvar Karimov, Erkinjon |
| author_facet | Hasanov, Anvar Karimov, Erkinjon |
| contents | We aim to study Mittag-Leffler type functions of two variables ${{D}_{1}}\left( x,y \right),...,{{D}_{5}}\left( x,y \right)$ by analogy with the Appell hypergeometric functions of two variables. Moreover, we targeted functions ${{E}_{1}}\left( x,y \right),$ $...,{{E}_{10}}\left( x,y \right)$ as limiting cases of the functions ${{D}_{1}}\left( x,y \right),$ $...,{{D}_{5}}\left( x,y \right)$ and studied certain properties, as well. Following Horn's method, we determine all possible cases of the convergence region of the function ${{D}_{1}}\left( x,y \right).$ Further, for a generalized hypergeometric function, ${{D}_{1}}\left( x,y \right)$ (two variable Mittag-Leffler-type function) integral representations of the Euler type have been proved. One-dimensional and two-dimensional Laplace transforms of the function are also defined. We have constructed a system of partial differential equations which is linked with the function ${{D}_{1}}\left( x,y \right)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_03918 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On generalized Mittag-Leffler-type functions of two variables Hasanov, Anvar Karimov, Erkinjon Classical Analysis and ODEs Analysis of PDEs 33E12, 33C60, 26A33, 47B38 We aim to study Mittag-Leffler type functions of two variables ${{D}_{1}}\left( x,y \right),...,{{D}_{5}}\left( x,y \right)$ by analogy with the Appell hypergeometric functions of two variables. Moreover, we targeted functions ${{E}_{1}}\left( x,y \right),$ $...,{{E}_{10}}\left( x,y \right)$ as limiting cases of the functions ${{D}_{1}}\left( x,y \right),$ $...,{{D}_{5}}\left( x,y \right)$ and studied certain properties, as well. Following Horn's method, we determine all possible cases of the convergence region of the function ${{D}_{1}}\left( x,y \right).$ Further, for a generalized hypergeometric function, ${{D}_{1}}\left( x,y \right)$ (two variable Mittag-Leffler-type function) integral representations of the Euler type have been proved. One-dimensional and two-dimensional Laplace transforms of the function are also defined. We have constructed a system of partial differential equations which is linked with the function ${{D}_{1}}\left( x,y \right)$. |
| title | On generalized Mittag-Leffler-type functions of two variables |
| topic | Classical Analysis and ODEs Analysis of PDEs 33E12, 33C60, 26A33, 47B38 |
| url | https://arxiv.org/abs/2501.03918 |