Two Constructive Results in Continued Fractions

In this paper we retrieve from the classical literature what we feel are two of the most basic results of the analytic theory of continued fractions: Van Vleck’s theorem and the parabola theorem. There is a remarkable resefnblance between these two theorems which has never been adequately explained. Indeed, there are general theorems which subsume them both, but only in these two cases are explicit error bounds known. We also give a simplified proof of an extension of Thron’s improvement of the Henrici–Pfluger bound for Stieltjes fractions and apply it to two further cases. Here we make use of Van Vleck fractions instead of the parabola theorem.