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.Key wordsHypertranscendenceMahler-type functional equationstranscendence and algebraic independenceMahler’s method