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 $$\frac{(b-1{)}^2}{b}{F}_{\theta ,\rho }\left(\frac{1}{b},\frac{1}{a}\right)+\frac{\lfloor \theta +\rho \rfloor (b-1)}{b^{2}a},$$ from which we derive a formula giving the irrationality exponent of $F_{\theta ,\rho }(1/b,1/a)$.

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