Transcendence and continued fraction expansionof values of Hecke–Mahler series
Volume 209 / 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 $F_{\theta , \rho } (\frac 1b, \frac 1a)$.
