An extension of algebraic independence of special values for non-lacunary power series
Abstract
We study the algebraic independence of special values of power series f(β−1), where β is a fixed Pisot or Salem number. In particular, we consider the case where f(X)=∑n≥ 0 t(n) Xw(n) is not a lacunary series and is not assumed to satisfy any special functional equation, such as a Mahler-type functional equation. In our main results, we give a new criterion of the algebraic independence of three values. Applying our main results, we prove that the following three values are algebraically independent: % ∑n=3∞⌊ ny⌋β−⌊ nlog log n⌋, ∑n=3∞β−⌊ nlog log n⌋, ∑n=1∞β−⌊ nlog n⌋, % where y is an arbitrary positive real number. Since our criterion is flexible, we have considerable freedom in choosing the coefficients (t(n))n≥ 0 and the exponents (w(n))n≥ 0.
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BibTeX
@misc{kaneko2026extension,
title = {{An extension of algebraic independence of special values for non-lacunary power series}},
author = {Hajime Kaneko and Satoru Oshima and Takafumi Tsurumaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15590},
eprint = {2609.15590},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15590},
}
RIS
TY - UNPB
TI - An extension of algebraic independence of special values for non-lacunary power series
AU - Hajime Kaneko
AU - Satoru Oshima
AU - Takafumi Tsurumaki
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15590
UR - https://arxiv.org/abs/2609.15590
AB - We study the algebraic independence of special values of power series f(β^−1), where β is a fixed Pisot or Salem number. In particular, we consider the case where f(X)=∑_n≥ 0 t(n) X^w(n) is not a lacunary series and is not assumed to satisfy any special functional equation, such as a Mahler-type functional equation. In our main results, we give a new criterion of the algebraic independence of three values. Applying our main results, we prove that the following three values are algebraically independent: % ∑_n=3^∞⌊ n^y⌋β^−⌊ n^log log n⌋, ∑_n=3^∞β^−⌊ n^log log n⌋, ∑_n=1^∞β^−⌊ n^log n⌋, % where y is an arbitrary positive real number. Since our criterion is flexible, we have considerable freedom in choosing the coefficients (t(n))_n≥ 0 and the exponents (w(n))_n≥ 0.
ER -