Algebraic independence of the values of the Hecke–Mahler series and its derivatives at algebraic numbers

Abstract

We show that the Hecke–Mahler series, the generating function of the sequence [Formula: see text] for [Formula: see text] real, has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if [Formula: see text] is a quadratic irrational number satisfying a suitable condition.