Irrationality results for reciprocal sums of certain Lucas numbers (Q1322605)

The authors prove the irrationality of the series \(\sum^ \infty_{n = 0} \varepsilon^ n / V_{2^ n}\), where \(\varepsilon \in \{\pm 1\}\) and \(V_{2^ n}\) are elements of a non-degenerate sequence of Lucas numbers \(\{V_ n\}_{n \in \mathbb{N}_ 0}\) satisfying the binary recurrence relation \(V_{n+2} = A_ 1 V_{n+1} + A_ 2 V_ n\), \(V_ 0 = 2\), \(V_ 1 = A_ 1\) with \(A_ 1, A_ 2 \in \mathbb{Z}\) and \(A^ 2_ 1 + 4A_ 2 \neq 0\). This extends results of \textit{C. Badea} [Glasg. Math. J. 29, 221-228 (1987; Zbl 0629.10027)] and \textit{R. André-Jeannin} [Fibonacci Q. 29, 132-136 (1991; Zbl 0725.11034)]. The proof is based on the study of certain dynamic properties of the map \(z \to z^ 2\) on the unit circle.