Quantitative linear independence of an infinite product and its derivatives (Q833076)

Let \(q\) be an integer with \(|q|>1\) and \[ H_q(z)=\prod_{n=1}^{\infty}\left(1+\frac{q^nz}{q^{2n}+1}\right), \qquad \Lambda_q=\prod_{n=1}^{\infty}\frac{q^{3n}}{(q^n-1)(q^{2n}+1)}. \] The authors establish a linear independence measure over \({\mathbb Q}\) for \(1, \Lambda_q\) and the values of \(H_q\) and its derivatives (up to some order) at a certain finite number of distinct non-zero rational points. This result is a quantitative and qualitative improvement of \textit{J.-P. Bézivin}'s main theorem from [Manuscr.~Math. 126, No.~1, 41-47 (2008; Zbl 1202.11039)]. As a consequence, the authors deduce irrationality measures for the values of the logarithmic derivative of \(H_q\) \[ \frac{H'_q(z)}{H_q(z)}=\sum_{n=1}^{\infty}\frac{q^n}{q^{2n}+q^nz+1} \] at suitable rational points.