# A note on the transcendence of infinite products

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## 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 }} _{ 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).

## Keywords

transcendence infinite product## MSC 2010

11J81## Preview

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© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2012