Irrationality of Certain Lambert Series
Irrationality of Certain Lambert Series
Let $q$ with $|q|>1$ be an integer in an algebraic number field $\mathbf{K}$ given below, and $\{a_n\}$ a periodic sequence in $\mathbf{K}$ of period two, not identically zero. Let $f(q)=\sum_{n=1}^{\infty}{a_n}/(1-q^n).$ We prove that (i) If $\mathbf{K}$ is either the rational number field $\mathbb{Q}$ or an imaginary quadratic field, then $f(q)\notin …