A note on the transcendence of infinite products

This paper has been elaborated in the framework of the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 supported by Operational Programme ‘Research and Development for Innovations’ funded by Structural Funds of the European Union and state budget of the Czech Republic and by grants no. ME09017, P201/12/2351 and MSM 6198898701.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).Web of Science62362361