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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.06534 |
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| _version_ | 1866909895119339520 |
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| author | David, Sinnou Hirata-Kohno, Noriko Kawashima, Makoto |
| author_facet | David, Sinnou Hirata-Kohno, Noriko Kawashima, Makoto |
| contents | We prove new linear independence results for the values of generalized hypergeometric functions ${}_pF_q$ at several distinct algebraic points, over arbitrary algebraic number fields. Our approach combines constructions of type II Padé approximants with a novel non-vanishing argument for generalized Wronskians of Hermite type. The method applies uniformly across all parameter regimes. Even for $p = q+1$, we extend known results from single-point to multi-point settings over general number fields, in the both complex and $p$-adic settings. When $p < q+1$, we establish linear independence results over arbitrary number fields; and for $p > q+1$, we confirm that the values do not satisfy global linear relations in the $p$-adic setting. Our results generalize and strengthen earlier work by Chudnovsky's, Nesterenko, Sorokin, Delaygue and others, and demonstrate the flexibility of our Padé construction for families of contiguous hypergeometric values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06534 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear independence of values of hypergeometric functions and arithmetic Gevrey series David, Sinnou Hirata-Kohno, Noriko Kawashima, Makoto Number Theory We prove new linear independence results for the values of generalized hypergeometric functions ${}_pF_q$ at several distinct algebraic points, over arbitrary algebraic number fields. Our approach combines constructions of type II Padé approximants with a novel non-vanishing argument for generalized Wronskians of Hermite type. The method applies uniformly across all parameter regimes. Even for $p = q+1$, we extend known results from single-point to multi-point settings over general number fields, in the both complex and $p$-adic settings. When $p < q+1$, we establish linear independence results over arbitrary number fields; and for $p > q+1$, we confirm that the values do not satisfy global linear relations in the $p$-adic setting. Our results generalize and strengthen earlier work by Chudnovsky's, Nesterenko, Sorokin, Delaygue and others, and demonstrate the flexibility of our Padé construction for families of contiguous hypergeometric values. |
| title | Linear independence of values of hypergeometric functions and arithmetic Gevrey series |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.06534 |