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A note on the transcendence of infinite products. (English) Zbl 1265.11078
Summary: The paper deals with several criteria for the transcendence of infinite products of the form \(\prod _{n=1}^\infty {[b_n\alpha ^{a_n}]}/{b_n\alpha ^{a_n}}\) where \(\alpha >1\) is a positive algebraic number having a conjugate \(\alpha ^*\) such that \(\alpha \not =| \alpha ^{*}| >1\), \(\{a_n\}_{n=1}^{\infty }\) and \(\{b_n\}_{n=1}^{\infty }\) are two sequences of positive integers with some specific conditions. The proofs are based on the recent theorem of [P. Corvaja and U. Zannier [Acta Math. 193, No. 2, 175–191 (2004; Zbl 1175.11036)], which relies on the subspace theorem.

MSC:
11J81 Transcendence (general theory)
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[1] P. Corvaja, J. Hančl: A transcendence criterion for infinite products. Atti Acad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 18 (2007), 295–303. · Zbl 1207.11075 · doi:10.4171/RLM/496
[2] P. Corvaja, U. Zannier: On the rational approximations to the powers of an algebraic number: solution of two problems of Mahler and Mendès France. Acta Math. 193 (2004), 175–191. · Zbl 1175.11036 · doi:10.1007/BF02392563
[3] P. Corvaja, U. Zannier: Some new applications of the subspace theorem. Comp. Math. 131 (2002), 319–340. · Zbl 1010.11038 · doi:10.1023/A:1015594913393
[4] P. Erdos: Some problems and results on the irrationality of the sum of infinite series. J. Math. Sci. 10 (1975), 1–7. · doi:10.1007/BF00541025
[5] M. Genčev: Evaluation of infinite series involving special products and their algebraic characterization. Math. Slovaca 59 (2009), 365–378. · Zbl 1209.11067 · doi:10.2478/s12175-009-0133-4
[6] J. Hančl, R. Nair, J. Šustek: On the Lebesgue measure of the expressible set of certain sequences. Indag. Math., New Ser. 17 (2006), 567–581. · Zbl 1131.11048 · doi:10.1016/S0019-3577(06)81034-7
[7] J. Hančl, P. Rucki, J. Šustek: A generalization of Sándor’s theorem using iterated logarithms. Kumamoto J. Math. 19 (2006), 25–36.
[8] J. Hančl, J. Štěpnička, J. Šustek: Linearly unrelated sequences and problem of Erdos. Ramanujan J. 17 (2008), 331–342. · Zbl 1242.11049 · doi:10.1007/s11139-008-9137-x
[9] D. Kim, J.K. Koo: On the infinite products derived from theta series I. J. Korean Math. Soc. 44 (2007), 55–107. · Zbl 1128.11037 · doi:10.4134/JKMS.2007.44.1.055
[10] S. Lang: Algebra (3rd ed.). Graduate Texts in Mathematics. Springer, New York, 2002.
[11] M.A. Nyblom: On the construction of a family of transcendental valued infinite products. Fibonacci Q. 42 (2004), 353–358. · Zbl 1062.11048
[12] Y. Tachiya: Transcendence of the values of infinite products in several variables. Result. Math. 48 (2005), 344–370. · Zbl 1133.11313 · doi:10.1007/BF03323373
[13] P. Zhou: On the irrationality of a certain multivariable infinite product. Quaest. Math. 29 (2006), 351–365. · Zbl 1117.11040 · doi:10.2989/16073600609486169
[14] Y. Ch. Zhu: Transcendence of certain infinite products. Acta Math. Sin. 43 (2000), 605–610. (In Chinese. English summary.) · Zbl 1005.11034
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