## Hypertranscendence and algebraic independence of certain infinite products

### Volume 184 / 2018

Acta Arithmetica 184 (2018), 51-66
MSC: Primary 12H05; Secondary 11J81, 11J91, 34M15.
DOI: 10.4064/aa170528-16-12
Published online: 9 March 2018

#### Abstract

We study infinite products $F(z)=\prod_{j\ge0}p(z^{d^j})$, where $d\ge2$ is an integer and $p\in\mathbb{C}[z]$ with $p(0)=1$ has at least one zero not lying on the unit circle. In that case, $F$ is a transcendental function and we are mainly interested in conditions for its hypertranscendence. Moreover, we investigate finite sets of infinite products of type $F$ and show that, under certain natural assumptios, these functions and their first derivatives are algebraically independent over $\mathbb{C}(z)$.