Hypertranscendence and algebraic independence of certain infinite products

Abstract

<div lang="en" class="abs"><p class="abs_header">Abstract</p><p>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 \(\mathbb{C}(z)\).</p></div>