Algebraic independence of the values of the Hecke-Mahler series and its derivatives at algebraic numbers

Taka Aki Tanaka; Yusuke Tanuma

doi:10.1142/S1793042118501440

Algebraic independence of the values of the Hecke-Mahler series and its derivatives at algebraic numbers

Taka Aki Tanaka, Yusuke Tanuma

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We show that the Hecke-Mahler series, the generating function of the sequence {[nω]}n=1∞ for ω real, has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if ω is a quadratic irrational number satisfying a suitable condition.

Original language	English
Pages (from-to)	2369-2384
Number of pages	16
Journal	International Journal of Number Theory
Volume	14
Issue number	9
DOIs	https://doi.org/10.1142/S1793042118501440
Publication status	Published - 2018 Oct 1

Keywords

Algebraic independence
Hecke-Mahler series
Mahler's method

ASJC Scopus subject areas

Algebra and Number Theory

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10.1142/S1793042118501440

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abstract = "We show that the Hecke-Mahler series, the generating function of the sequence {[nω]}n=1∞ for ω real, has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if ω is a quadratic irrational number satisfying a suitable condition.",

keywords = "Algebraic independence, Hecke-Mahler series, Mahler's method",

author = "Tanaka, {Taka Aki} and Yusuke Tanuma",

note = "Funding Information: The authors are grateful to the anonymous referee for careful reading and insightful comments that improved this paper. This work was supported by JSPS KAKENHI Grant Number 15K04792. Publisher Copyright: {\textcopyright} 2018 World Scientific Publishing Company.",

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AU - Tanaka, Taka Aki

AU - Tanuma, Yusuke

N1 - Funding Information: The authors are grateful to the anonymous referee for careful reading and insightful comments that improved this paper. This work was supported by JSPS KAKENHI Grant Number 15K04792. Publisher Copyright: © 2018 World Scientific Publishing Company.

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N2 - We show that the Hecke-Mahler series, the generating function of the sequence {[nω]}n=1∞ for ω real, has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if ω is a quadratic irrational number satisfying a suitable condition.

AB - We show that the Hecke-Mahler series, the generating function of the sequence {[nω]}n=1∞ for ω real, has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if ω is a quadratic irrational number satisfying a suitable condition.

KW - Algebraic independence

KW - Hecke-Mahler series

KW - Mahler's method

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Algebraic independence of the values of the Hecke-Mahler series and its derivatives at algebraic numbers

Abstract

Keywords

ASJC Scopus subject areas

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