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Article Dans Une Revue Acta Arithmetica Année : 2023

Transcendence and continued fraction expansion of values of Hecke-Mahler series

Yann Bugeaud
  • Fonction : Auteur
Michel Laurent
  • Fonction : Auteur
  • PersonId : 974960

Résumé

Let $\theta$ and $\rho$ be real numbers with $0 \le \theta, \rho < 1$ 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 $0 < |\beta|, |\beta \alpha^\theta | < 1$. This extends earlier results of Mahler (1929) and Loxton and van der Poorten (1977), who settled the case $\rho=0$. 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 $$ {(b-1)^2\over b} F_{\theta, \rho} \left({1\over b}, {1\over a}\right)+{\lfloor \theta+\rho\rfloor(b-1)\over b^2a}, $$ from which we derive a formula giving the irrationality exponent of $F_{\theta, \rho} (1/b, 1/a)$.

Dates et versions

hal-03620911 , version 1 (27-03-2022)

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Yann Bugeaud, Michel Laurent. Transcendence and continued fraction expansion of values of Hecke-Mahler series. Acta Arithmetica, 2023, ⟨10.4064/aa220323-18-1⟩. ⟨hal-03620911⟩
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