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

Abstract

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