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

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