On non-holonomicity, transcendence and $p$-adic valuations
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917876087128064 |
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| author | Cobeli, Cristian Prunescu, Mihai Zaharescu, Alexandru |
| author_facet | Cobeli, Cristian Prunescu, Mihai Zaharescu, Alexandru |
| contents | Let $ν_q(n)$ be the p-adic valuation of $n$. We show that the power series with coefficients $ν_q(n)$, respectively $ν_p(n)(\mathrm{ mod\;} k)$, are non-holonomic and not algebraic in characteristic 0. We find infinitely many rational numbers and infinitely many algebraic irrational numbers for which the values of these series are transcendental. We apply these results to some $p$-automatic sequences, one of them being the period-doubling sequence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_16517 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On non-holonomicity, transcendence and $p$-adic valuations Cobeli, Cristian Prunescu, Mihai Zaharescu, Alexandru Number Theory Primary 11B37, Secondary 11J81 Let $ν_q(n)$ be the p-adic valuation of $n$. We show that the power series with coefficients $ν_q(n)$, respectively $ν_p(n)(\mathrm{ mod\;} k)$, are non-holonomic and not algebraic in characteristic 0. We find infinitely many rational numbers and infinitely many algebraic irrational numbers for which the values of these series are transcendental. We apply these results to some $p$-automatic sequences, one of them being the period-doubling sequence. |
| title | On non-holonomicity, transcendence and $p$-adic valuations |
| topic | Number Theory Primary 11B37, Secondary 11J81 |
| url | https://arxiv.org/abs/2412.16517 |