A note on G-operators of order 2
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913936933126144 |
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| author | Fischler, Stéphane Rivoal, Tanguy |
| author_facet | Fischler, Stéphane Rivoal, Tanguy |
| contents | It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a $G$-function solution of an inhomogeneous equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, as well as that of a $G$-function $f$ of differential order 2 over $\overline{\mathbb{Q}}(z)$, and such that $f$ and $f'$ are algebraically dependent over $\mathbb{C}(z)$. Our results apply more generally to Nilsson-Gevrey arithmetic series of order 0 that encompass $G$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_05572 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A note on G-operators of order 2 Fischler, Stéphane Rivoal, Tanguy Classical Analysis and ODEs It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a $G$-function solution of an inhomogeneous equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, as well as that of a $G$-function $f$ of differential order 2 over $\overline{\mathbb{Q}}(z)$, and such that $f$ and $f'$ are algebraically dependent over $\mathbb{C}(z)$. Our results apply more generally to Nilsson-Gevrey arithmetic series of order 0 that encompass $G$-functions. |
| title | A note on G-operators of order 2 |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2106.05572 |