# Article

Full entry | PDF   (0.2 MB)
Keywords:
transcendence; infinite product
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. \endgraf 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 Mendès France, Acta Math. 193, (2004), 175–191).
References:
[1] Corvaja, P., Hančl, J.: 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. DOI 10.4171/RLM/496 | MR 2318822 | Zbl 1207.11075
[2] Corvaja, P., Zannier, U.: 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. DOI 10.1007/BF02392563 | MR 2134865 | Zbl 1175.11036
[3] Corvaja, P., Zannier, U.: Some new applications of the subspace theorem. Comp. Math. 131 (2002), 319-340. DOI 10.1023/A:1015594913393 | MR 1905026 | Zbl 1010.11038
[4] Erdős, P.: Some problems and results on the irrationality of the sum of infinite series. J. Math. Sci. 10 (1975), 1-7. MR 0539489
[5] Genčev, M.: Evaluation of infinite series involving special products and their algebraic characterization. Math. Slovaca 59 (2009), 365-378. DOI 10.2478/s12175-009-0133-4 | MR 2505816 | Zbl 1209.11067
[6] Hančl, J., Nair, R., Šustek, J.: On the Lebesgue measure of the expressible set of certain sequences. Indag. Math., New Ser. 17 (2006), 567-581. DOI 10.1016/S0019-3577(06)81034-7 | MR 2320114 | Zbl 1131.11048
[7] Hančl, J., Rucki, P., Šustek, J.: A generalization of Sándor's theorem using iterated logarithms. Kumamoto J. Math. 19 (2006), 25-36. MR 2211630 | Zbl 1220.11087
[8] Hančl, J., Štěpnička, J., Šustek, J.: Linearly unrelated sequences and problem of Erdős. Ramanujan J. 17 (2008), 331-342. DOI 10.1007/s11139-008-9137-x | MR 2456837
[9] Kim, D., Koo, J. K.: On the infinite products derived from theta series I. J. Korean Math. Soc. 44 (2007), 55-107. DOI 10.4134/JKMS.2007.44.1.055 | MR 2283460 | Zbl 1128.11037
[10] Lang, S.: Algebra (3rd ed.). Graduate Texts in Mathematics. Springer, New York (2002). MR 1878556
[11] Nyblom, M. A.: On the construction of a family of transcendental valued infinite products. Fibonacci Q. 42 (2004), 353-358. MR 2110089 | Zbl 1062.11048
[12] Tachiya, Y.: Transcendence of the values of infinite products in several variables. Result. Math. 48 (2005), 344-370. DOI 10.1007/BF03323373 | MR 2215585 | Zbl 1133.11313
[13] Zhou, P.: On the irrationality of a certain multivariable infinite product. Quaest. Math. 29 (2006), 351-365. DOI 10.2989/16073600609486169 | MR 2260768
[14] Zhu, Y. Ch.: Transcendence of certain infinite products. Acta Math. Sin. 43 (2000), 605-610 Chinese. English summary \MR 1825076. MR 1825076 | Zbl 1005.11034

Partner of