Algebraic independence of certain entire functions of two variables generated by linear recurrences
Volume 202 / 2022
Acta Arithmetica 202 (2022), 303-336
MSC: Primary 11J81; Secondary 11J85.
DOI: 10.4064/aa191206-31-8
Published online: 10 March 2022
Abstract
We construct an entire function of two variables having the property that its values and its partial derivatives of any order at any distinct algebraic points are algebraically independent. Such an entire function is obtained as a corollary to the main theorem of this paper, asserting the algebraic independence of the values and the partial derivatives of certain series of two variables and infinite products at any distinct algebraic points. This result is proved by providing an invertible linear relation between the values treated in the main theorem and those of a wider class of functions including Mahler functions.
