Transcendental properties of the certain mix infinite products

Abstract

Let $k$ and $l$ be two multiplicatively independent positive integers and $b$ be an integer with $b\ge2$. Let $S$ be a finite set of integers. Nishioka proved that for any algebraic number $\alpha$ with $0<|\alpha|<1$ the infinite products $\prod_{y=0}^{\infty}(1-{\alpha}^{d^{y}})$ ($d=2,3,\ldots$) are algebraically independent over $\mathbb{Q}$. As her result, for example, the transcendence of $\prod_{y=0}^{\infty}(1-\frac{1}{{b}^{2^{y}}})\prod_{y=0}^{\infty}(1-\frac{1}{{b}^{3^{y}}})$ is deduced. On the other hand, Tachiya, Amou–Väänänen investigated the certain infinite products which satisfy infinite chains of Mahler functional equation. The special case of the result of Tachiya shows that the infinite product $\prod_{y\ge0}(1+\sum_{i=1}^{k-1} \frac{\tau(i,y)}{b^{ik^y}})$ with $\tau(i,y)\in S$ ($1\le i\le k-1, y\ge0$) is either rational or transcendental. In this paper, we prove that the infinite product $\prod_{y\ge0}(1+\sum_{i=1}^{k-1} \frac{\tau(i,y)}{b^{ik^y}})\prod_{y\ge0}(1+\sum_{j=1}^{l-1} \frac{\delta(j,y)}{b^{jl^y}})$ with $\tau(i,y),\delta(j,y) \in S$ $(1\le i\le k-1, 1\le j\le l-1, y\ge0)$ is either rational or transcendental. Moreover, we give sufficient conditions that $\prod_{y\ge0}(1+\sum_{i=1}^{k-1} \frac{\tau(i,y)}{b^{ik^y}})\prod_{y\ge0}(1+\sum_{j=1}^{l-1} \frac{\delta(j,y)}{b^{jl^y}})$ is transcendental.