A linear independence criterion for certain infinite series with polynomial orders
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916951306010624 |
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| author | Kudo, Shinya |
| author_facet | Kudo, Shinya |
| contents | Let $q$ be a Pisot or Salem number. Let $f_j(x)$ $(j=1,2,\dots)$ be integer-valued polynomials of degree $\ge2$ with positive leading coefficients, and let $\{a_j (n)\}_{n\ge1}$ $(j=1,2,\dots)$ be sequences of algebraic integers in the field $\mathbb{Q}(q)$ with suitable growth conditions. In this paper, we investigate linear independence over $\mathbb{Q}(q)$ of the numbers \begin{equation*} 1,\qquad \sum_{n=1}^{\infty} \frac{a_j (n)}{q^{f_j (n)}} \quad (j=1,2,\dots). \end{equation*} In particular, when $a_j(n)$ $(j=1,2,\dots)$ are polynomials of $n$, we give a linear independence criterion for the above numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04801 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A linear independence criterion for certain infinite series with polynomial orders Kudo, Shinya Number Theory Let $q$ be a Pisot or Salem number. Let $f_j(x)$ $(j=1,2,\dots)$ be integer-valued polynomials of degree $\ge2$ with positive leading coefficients, and let $\{a_j (n)\}_{n\ge1}$ $(j=1,2,\dots)$ be sequences of algebraic integers in the field $\mathbb{Q}(q)$ with suitable growth conditions. In this paper, we investigate linear independence over $\mathbb{Q}(q)$ of the numbers \begin{equation*} 1,\qquad \sum_{n=1}^{\infty} \frac{a_j (n)}{q^{f_j (n)}} \quad (j=1,2,\dots). \end{equation*} In particular, when $a_j(n)$ $(j=1,2,\dots)$ are polynomials of $n$, we give a linear independence criterion for the above numbers. |
| title | A linear independence criterion for certain infinite series with polynomial orders |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.04801 |