First-order factors of linear Mahler operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914131368476672 |
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| author | Chyzak, Frédéric Dreyfus, Thomas Dumas, Philippe Mezzarobba, Marc |
| author_facet | Chyzak, Frédéric Dreyfus, Thomas Dumas, Philippe Mezzarobba, Marc |
| contents | We develop and compare two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators$\ell_r M^r + \dots + \ell_1 M + \ell_0$where $\ell_0, \dots, \ell_r$ are polynomials in~$x$ and $Mx = x^b M$ for some integer $b \geq 2$. In other words, we give algorithms for finding all formal infinite product solutions of linear functional equations$\ell_r(x) f(x^{b^r}) + \dots + \ell_1(x) f(x^b) + \ell_0(x) f(x) = 0$. The first of our algorithms is adapted from Petkovšek's classical algorithm forthe analogous problem in the case of linear recurrences. The second one proceeds by computing a basis of generalized power series solutions of the functional equation and by using Hermite-Pad{é} approximants to detect those linear combinations of the solutions that correspond to first-order factors. We present implementations of both algorithms and discuss their use in combination with criteria from the literature to prove the differential transcendence of power series solutions of Mahler equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11545 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First-order factors of linear Mahler operators Chyzak, Frédéric Dreyfus, Thomas Dumas, Philippe Mezzarobba, Marc Symbolic Computation Number Theory Rings and Algebras We develop and compare two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators$\ell_r M^r + \dots + \ell_1 M + \ell_0$where $\ell_0, \dots, \ell_r$ are polynomials in~$x$ and $Mx = x^b M$ for some integer $b \geq 2$. In other words, we give algorithms for finding all formal infinite product solutions of linear functional equations$\ell_r(x) f(x^{b^r}) + \dots + \ell_1(x) f(x^b) + \ell_0(x) f(x) = 0$. The first of our algorithms is adapted from Petkovšek's classical algorithm forthe analogous problem in the case of linear recurrences. The second one proceeds by computing a basis of generalized power series solutions of the functional equation and by using Hermite-Pad{é} approximants to detect those linear combinations of the solutions that correspond to first-order factors. We present implementations of both algorithms and discuss their use in combination with criteria from the literature to prove the differential transcendence of power series solutions of Mahler equations. |
| title | First-order factors of linear Mahler operators |
| topic | Symbolic Computation Number Theory Rings and Algebras |
| url | https://arxiv.org/abs/2403.11545 |