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arXiv 1907.07572math.NTmath.CO
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θ函数与某些无穷乘积定义的自动序列

Automatic sequences defined by Theta functions and some infinite products

Shuo Li

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摘要：

设$p(x) \in C(x)$是满足条件$p(0)=1$的有理函数，$q$是大于$1$的整数，本文考虑无穷乘积$$f(x)=\prod_{s=0}^{\infty}f(x^{q^{s}})=\sum_{i=0}^{\infty}c_ix^i,$$的幂展开，研究序列$(c_i)_{i \in \mathbf{N}}$何时是$q$ -自动的.主要结果是：对于给定的整数$q \geq 2$和$d \geq 0$，存在定义在有理数域$\mathbf{Q}$上的n阶多项式$d$，使得$\prod_{s=0}^{\infty}f(x^{q^{s}})=\sum_{i=1}^{\infty}c_ix^i$是$q$ -自动幂级数.

英文摘要：

Let $p(x) \in C(x)$ be a rational function satisfying the condition $p(0)=1$ and $q$ an integer larger than $1$, in this article we will consider the power expansion of the infinite product $$f(x)=\prod_{s=0}^{\infty}f(x^{q^{s}})=\sum_{i=0}^{\infty}c_ix^i,$$ and study when the sequence $(c_i)_{i \in \mathbf{N}}$ is $q$-automatic. The main result is that for given integers $q \geq 2$ and $d \geq 0$, there exist finitely many polynomials of degree $d$ defined over the field of rational numbers $\mathbf{Q}$, such that $\prod_{s=0}^{\infty}f(x^{q^{s}})=\sum_{i=1}^{\infty}c_ix^i$ is a $q$-automatic power series.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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