The mathematical foundations of the asymptotic iteration method

Abstract

We introduce a new approach to the the asymptotic iteration method (AIM) by means of which we establish the standard AIM connection with the continued fractions technique and we develop a novel termination condition in terms of the approximants. With the help of this alternative termination condition and certain properties of continuous fractions, we derive a closed formula for the asymptotic function $\alpha$ of the AIM technique in terms of an infinite series. Furthermore, we show that such a series converges pointwise to $\alpha$ which, in turn, can be interpreted as a specific term of the minimal solution of a certain recurrence relation. We also investigate some conditions ensuring the existence of a minimal solution and hence, of the function $\alpha$ itself.