Algebraic Independence of the Values of Power Series and Their Derivatives Generated by Linear Recurrences

Abstract

Using a descent method, we construct certain power series generated by linear recurrences, each of which possesses the following property: The infinite set consisting of all its values and all the values of its derivatives of any order, at any nonzero algebraic numbers within its domain of existence, is algebraically independent. The main theorems of this paper assert that the power series of the form $\sum _{k=0}^{\infty }{z}^{e_k}$, where $\{e_k{\}}_{k\ge 0}$ is a linear recurrence with certain admissible properties, have this property. In particular, Main Theorem 1.16 provides a class of $\{e_k{\}}_{k\ge 0}$ which is simpler than ever before.