Zeros of Padé error functions for functions with smooth Maclaurin coefficients (Q1907506)

Let \(\pi_{n,m}\) be the \((n, m)\) Padé approximant for the function \(f\) defined by its Maclaurin series \(f= \sum_{k=0}^\infty a_k z^k\). This series is assumed to satisfy the Lubinsky smoothness condition, i.e., \(\eta_n= a_{n+1} a_{n-1}/ a^2_n\to \eta\neq \infty\) as \(n\to \infty\). This paper deals with the asymptotics of (a properly normalized form of) the error function \(e_{n,m}= f- \pi_{n,m}\) as \(n\to \infty\) with \(m\) fixed. Several results are given when \(\eta_n\) converges to \(\eta\), \(|\eta|\leq 1\) in a sufficiently smooth way. As a result, the asymptotic distribution of the zeros of the error are obtained.