A note on the transcendence of infinite products

Abstract

The paper deals with several criteria for the transcendence of infinite products of the form $$\prod\limits_{n = 1}^\infty {[{b_n}{a^{{a_n}}}]/{b_n}{a^{{a_n}}}} $$ where α > 1 is a positive algebraic number having a conjugate α* such that α ≠ |α*| > 1, {a n } =1 ∞ and {b n } =1 ∞ are two sequences of positive integers with some specific conditions. The proofs are based on the recent theorem of Corvaja and Zannier which relies on the Subspace Theorem (P.Corvaja, U.Zannier: On the rational approximation to the powers of an algebraic number: solution of two problems of Mahler and Mend`es France, Acta Math. 193, (2004), 175–191).