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Transcendence and continued fraction expansionof values of Hecke–Mahler series

Volume 209 / 2023

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


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 $F_{\theta , \rho } (\frac 1b, \frac 1a)$.


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

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