# zbMATH — the first resource for mathematics

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)
##### Keywords:
transcendence; infinite product
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.