A note on the transcendence of infinite products

Abstract

The paper deals with several criteria for the transcendence of infinite products of the form ∏n=1∞[bnaan]/bnaan where α > 1 is a positive algebraic number having a conjugate α* such that α ≠ |α*| > 1, {a n } n=1 ∞ and {b n } 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).