Algebraic independence results for a certain family of power series, infinite products, and Lambert type series

Abstract

For a certain class of power series, infinite products, and Lambert type series, we establish a necessary and sufficient condition for the infinite set consisting of their values, as well as their derivatives of any order at any algebraic points except their poles and zeroes, to be algebraically independent. As its corollary, we construct an example of an infinite family of entire functions of two variables with the following property: Their values and their partial derivatives of any order at any distinct algebraic points with nonzero components are algebraically independent.