| Title:
|
A note on the transcendence of infinite products (English) |
| Author:
|
Hančl, Jaroslav |
| Author:
|
Kolouch, Ondřej |
| Author:
|
Pulcerová, Simona |
| Author:
|
Štěpnička, Jan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
613-623 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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). (English) |
| Keyword:
|
transcendence |
| Keyword:
|
infinite product |
| MSC:
|
11J81 |
| idZBL:
|
Zbl 1265.11078 |
| idMR:
|
MR2984622 |
| DOI:
|
10.1007/s10587-012-0053-2 |
| . |
| Date available:
|
2012-11-10T20:59:50Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143013 |
| . |
| Reference:
|
[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. Zbl 1207.11075, MR 2318822, 10.4171/RLM/496 |
| Reference:
|
[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. Zbl 1175.11036, MR 2134865, 10.1007/BF02392563 |
| Reference:
|
[3] Corvaja, P., Zannier, U.: Some new applications of the subspace theorem.Comp. Math. 131 (2002), 319-340. Zbl 1010.11038, MR 1905026, 10.1023/A:1015594913393 |
| Reference:
|
[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 |
| Reference:
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[5] Genčev, M.: Evaluation of infinite series involving special products and their algebraic characterization.Math. Slovaca 59 (2009), 365-378. Zbl 1209.11067, MR 2505816, 10.2478/s12175-009-0133-4 |
| Reference:
|
[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. Zbl 1131.11048, MR 2320114, 10.1016/S0019-3577(06)81034-7 |
| Reference:
|
[7] Hančl, J., Rucki, P., Šustek, J.: A generalization of Sándor's theorem using iterated logarithms.Kumamoto J. Math. 19 (2006), 25-36. Zbl 1220.11087, MR 2211630 |
| Reference:
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[8] Hančl, J., Štěpnička, J., Šustek, J.: Linearly unrelated sequences and problem of Erdős.Ramanujan J. 17 (2008), 331-342. MR 2456837, 10.1007/s11139-008-9137-x |
| Reference:
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[9] Kim, D., Koo, J. K.: On the infinite products derived from theta series I.J. Korean Math. Soc. 44 (2007), 55-107. Zbl 1128.11037, MR 2283460, 10.4134/JKMS.2007.44.1.055 |
| Reference:
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[10] Lang, S.: Algebra (3rd ed.).Graduate Texts in Mathematics. Springer, New York (2002). MR 1878556 |
| Reference:
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[11] Nyblom, M. A.: On the construction of a family of transcendental valued infinite products.Fibonacci Q. 42 (2004), 353-358. Zbl 1062.11048, MR 2110089 |
| Reference:
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[12] Tachiya, Y.: Transcendence of the values of infinite products in several variables.Result. Math. 48 (2005), 344-370. Zbl 1133.11313, MR 2215585, 10.1007/BF03323373 |
| Reference:
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[13] Zhou, P.: On the irrationality of a certain multivariable infinite product.Quaest. Math. 29 (2006), 351-365. MR 2260768, 10.2989/16073600609486169 |
| Reference:
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[14] Zhu, Y. Ch.: Transcendence of certain infinite products.Acta Math. Sin. 43 (2000), 605-610 Chinese. English summary \MR 1825076. Zbl 1005.11034, MR 1825076 |
| . |
