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## Acta Arithmetica

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## Transcendence and continued fraction expansionof values of Hecke–Mahler series

### Volume 209 / 2023

Acta Arithmetica 209 (2023), 59-90 MSC: Primary 11J04; Secondary 11J70, 11J81. DOI: 10.4064/aa220323-18-1 Published online: 28 February 2023

#### Abstract

Let $\theta$ and $\rho$ be real numbers with $0 \le \theta , \rho \lt 1$ and $\theta$ irrational. We show that the Hecke–Mahler series $$F_{\theta , \rho } (z_1, z_2) = \sum _{k_1 \ge 1} \, \sum _{k_2 = 1}^{\lfloor k_1 \theta + \rho \rfloor } z_1^{k_1} z_2^{k_2},$$ where $\lfloor \cdot \rfloor$ denotes the integer part function, takes transcendental values at any algebraic point $(\beta , \alpha )$ with $0 \lt |\beta |$, $|\beta \alpha ^\theta | \lt 1$. This extends earlier results of Mahler (1929) and Loxton and van der Poorten (1977), who settled the case of $\rho =0$. Furthermore, for positive integers $b$ and $a$, with $b \ge 2$ and $a$ congruent to $1$ modulo $b-1$, we give the continued fraction expansion of the number $$\frac {(b-1)^2}{b} F_{\theta , \rho } \left (\frac {1}{b}, \frac {1}{a}\right ),$$ from which we derive a formula giving the irrationality exponent of $F_{\theta , \rho } (\frac 1b, \frac 1a)$.

#### Authors

• Yann BugeaudI.R.M.A., UMR 7501
Université de Strasbourg et CNRS
67084 Strasbourg, France
and
Institut Universitaire de France
e-mail
• Michel LaurentAix-Marseille Université, CNRS
Institut de Mathématiques de Marseille
13288 Marseille, France
e-mail

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