The mathematical foundations of the asymptotic iteration method

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