Irrationality of certain sums of series. (Q926273)

Let \(q\) be a rational integer with \(|q|>1\). The main result of this paper is the linear independence over the field of rational numbers of the numbers \(1\) and \(H^{(\ell)}(a_i)\) for \(1\leq i\leq t\) and \(0\leq \ell\leq s\), when \(H\) is the entire function of one complex variable defined by the infinite product \[ H(x)=\prod_{n\geq 1} \left(1+\frac{q^nx}{q^{2n}+1}\right) \] and where \(a_1,\dots,a_t\) are non--zero rational numbers satisfying some assumption of non--degeneracy. As a corollary, the irrationality of the series \[ \sum_{n\geq 1}\frac{q^n}{q^{2n}+aq^n+1} \] for \(a\) a rational number follows, under the assumption that \((q^\ell+1)^2\not= a^2 q^\ell\) for all \(\ell\in\mathbb Z\), \(\ell\not=0\).