Algebraic Independence of Infinite Products and Their Derivatives

Abstract

For fixed rational integers q > 1, and for non-constant polynomials P with P(0) = 1 and with algebraic coefficients, we consider the infinite product $A_{q}(z) =\prod _{k\geq 0}P({z}^{{q}^{k} })$. Using Mahler’s transcendence method, we prove results on the algebraic independence over $\mathbb{Q}$ of the numbers $A_{q}(\alpha ),A_{q}^\prime(\alpha ),A_{q}^{\prime\prime}(\alpha ),\ldots$ at algebraic points α with 0 < | α | < 1. The basic analytic ingredient of the proof is the hypertranscendence of the function A _q (z), and we provide sufficient criteria for it.