Transcendental properties of the certain mix infinite products

Eiji Miyanohara
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 1, Pages 48–61
DOI: 10.7546/nntdm.2023.29.1.48-61
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Eiji Miyanohara
Tokyo, Japan

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.

Keywords

  • Infinite product
  • Transcendence
  • Infinite chains of Mahler functional equations

2020 Mathematics Subject Classification

  • 11J91
  • 11J87

References

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Manuscript history

  • Received: 10 August 2022
  • Revised: 3 January 2023
  • Accepted: 15 February 2023
  • Online First: 18 February 2023

Copyright information

Ⓒ 2023 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Miyanohara, E. (2023). Transcendental properties of the certain mix infinite products. Notes on Number Theory and Discrete Mathematics, 29(1), 48-61, DOI: 10.7546/nntdm.2023.29.1.48-61.

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