Simultaneous rational approximants for a pair of functions with smooth Maclaurin series coefficients (Q1907502)

Let \(f(z) = \sum^\infty_{j = 0} a_j z^j\) and \(g(z) = \sum^\infty_{j = 0} b_j z^j\) be formal power series for which the quantities \(a_{j + 1} a_{j - 1}/a^2_j\) and \((b_j/b_{j + 1}) / (a_j / a_{j + 1})\) have a prescribed asymptotic behaviour as \(j \to \infty\). The author obtains the asymptotic behaviour as \(l \to \infty\) of the \((l - s,r,s)\), \(l,r,s \in \mathbb{N}\), Hermite-Padé approximant to \((f,g)\) and the associated determinants.