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A note on the transcendence of infinite products |
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| Authors: | Jaroslav Han?l Ond?ej Kolouch Simona Pulcerová Jan ?těpni?ka |
| Affiliation: | 1. Department of Mathematics and Centre of Excellence IT4Innovation, division of UO, Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03, Ostrava 1, Czech Republic 2. University of Ostrava, 30. dubna 22, 701 03, Ostrava 1, Czech Republic 3. Department of Mathematical Methods in Economics, Faculty of Economics, V?B-Technical University of Ostrava, Sokolská t?ída 33, 701 21, Ostrava 1, Czech Republic |
| Abstract: | The paper deals with several criteria for the transcendence of infinite products of the form $prodlimits_{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). |
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