On the irrationality measure for certain series
The main result of this paper is following. Theorem: Let \(K\) be either the rational number field \( \mathbb Q\) or an imaginary quadratic field. Let \(q\) be an integer with \(q\in K\) and \(\mid q\mid >1\). Assume that \(r\in K^\times\) such that \(\mid r\mid <\mid q\mid \). Suppose that \(l\) is an integer with \(1\leq l\). Let \(q^n\not= r^l\) for all \(n=1,2,\cdots\). Then \(\alpha =\sum_{n=1}^\infty \frac{r^n}{q^n-r^l}\notin K\). Further, for any \(\varepsilon >0\) the inequality \(\mid \alpha -\frac pq\mid \geq \frac 1{\mid q\mid^{6l^2+\varepsilon}}\) holds for all integers \(p,q\in K\) with the sufficiently large \(\mid q\mid\). The proof makes use the special integral of the Borwein's type.
scientific article; zbMATH DE number 5532496
scientific article