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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### Abstract

Let and be two multiplicatively independent positive integers and be an integer with . Let be a finite set of integers. Nishioka proved that for any algebraic number with the infinite products () are algebraically independent over . As her result, for example, the transcendence of 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 with () is either rational or transcendental.

In this paper, we prove that the infinite product with is either rational or transcendental. Moreover, we give sufficient conditions that is transcendental.

### Keywords

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

### 2020 Mathematics Subject Classification

- 11J91
- 11J87

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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.