Bundschuh, P., Väänänen, K. (2018) Hypertranscendence and algebraic independence of certain infinite products. Acta Arithmetica, 184 (1), 51-66. doi:10.4064/aa170528-16-12

### Hypertranscendence and algebraic independence of certain infinite products

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Author: Bundschuh, Peter1; Väänänen , Keijo2
Organizations: 1Mathematisches Institut, Universität zu Köln, Weyerta, 86-90, 50931 Köln, Germany
2Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.3 MB)
Language: English
Published: Polish Academy of Sciences, Institute of Mathematics, 2018
Publish Date: 2019-03-28
Description:

# 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 $$\mathbb{C}(z)$$.

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Series: Acta arithmetica
ISSN: 0065-1036
ISSN-E: 1730-6264
ISSN-L: 0065-1036
Volume: 184
Issue: 1
Pages: 51 - 66
DOI: 10.4064/aa170528-16-12