Series of reciprocal products with factors from linear recurrence sequences (Q1049755)

Let \(\{W_n\}_{n=1}^\infty\) be the sequence of integers such that \(W_1,W_2\in\mathbb Z\) and \(W_n=pW_{n-1}+qW_{n-2}\) holds for each \(n=3,4,\cdots\). Suppose that \(p\in\mathbb Z\) and \(q\in\{ 1,-1\}\). Assume that \(j,k,m\in\mathbb N\) and \(l\in\mathbb N_0\). The author describes some sufficient conditions for which the number \(\sum_{n=1}^\infty \frac{z^n}{\prod_{i=0}^{j-1}W_{(n+im)k+l}}\) belongs or does not belong to \(\mathbb Q(\sqrt{p^2+4q})\). The proof makes use the properties of the power series as well as the linear recurrences.