Geodesic
A note on the transcendence of infinite products
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 613-623 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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).
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).
DOI : 10.1007/s10587-012-0053-2
Classification : 11J81
Keywords: transcendence; infinite product 
@article{10_1007_s10587_012_0053_2,
     author = {Han\v{c}l, Jaroslav and Kolouch, Ond\v{r}ej and Pulcerov\'a, Simona and \v{S}t\v{e}pni\v{c}ka, Jan},
     title = {A note on the transcendence of infinite products},
     journal = {Czechoslovak Mathematical Journal},
     pages = {613--623},
     year = {2012},
     volume = {62},
     number = {3},
     doi = {10.1007/s10587-012-0053-2},
     mrnumber = {2984622},
     zbl = {1265.11078},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0053-2/}
}
TY  - JOUR
AU  - Hančl, Jaroslav
AU  - Kolouch, Ondřej
AU  - Pulcerová, Simona
AU  - Štěpnička, Jan
TI  - A note on the transcendence of infinite products
JO  - Czechoslovak Mathematical Journal
PY  - 2012
SP  - 613
EP  - 623
VL  - 62
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0053-2/
DO  - 10.1007/s10587-012-0053-2
LA  - en
ID  - 10_1007_s10587_012_0053_2
ER  - 
%0 Journal Article
%A Hančl, Jaroslav
%A Kolouch, Ondřej
%A Pulcerová, Simona
%A Štěpnička, Jan
%T A note on the transcendence of infinite products
%J Czechoslovak Mathematical Journal
%D 2012
%P 613-623
%V 62
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0053-2/
%R 10.1007/s10587-012-0053-2
%G en
%F 10_1007_s10587_012_0053_2
Hančl, Jaroslav; Kolouch, Ondřej; Pulcerová, Simona; Štěpnička, Jan. A note on the transcendence of infinite products. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 613-623. doi: 10.1007/s10587-012-0053-2

[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 | MR | Zbl

[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 | MR | Zbl

[3] Corvaja, P., Zannier, U.: Some new applications of the subspace theorem. Comp. Math. 131 (2002), 319-340. | DOI | MR | Zbl

[4] Erdős, P.: Some problems and results on the irrationality of the sum of infinite series. J. Math. Sci. 10 (1975), 1-7. | MR

[5] Genčev, M.: Evaluation of infinite series involving special products and their algebraic characterization. Math. Slovaca 59 (2009), 365-378. | DOI | MR | Zbl

[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 | MR | Zbl

[7] Hančl, J., Rucki, P., Šustek, J.: A generalization of Sándor's theorem using iterated logarithms. Kumamoto J. Math. 19 (2006), 25-36. | MR | Zbl

[8] Hančl, J., Štěpnička, J., Šustek, J.: Linearly unrelated sequences and problem of Erdős. Ramanujan J. 17 (2008), 331-342. | DOI | MR

[9] Kim, D., Koo, J. K.: On the infinite products derived from theta series I. J. Korean Math. Soc. 44 (2007), 55-107. | DOI | MR | Zbl

[10] Lang, S.: Algebra (3rd ed.). Graduate Texts in Mathematics. Springer, New York (2002). | MR

[11] Nyblom, M. A.: On the construction of a family of transcendental valued infinite products. Fibonacci Q. 42 (2004), 353-358. | MR | Zbl

[12] Tachiya, Y.: Transcendence of the values of infinite products in several variables. Result. Math. 48 (2005), 344-370. | DOI | MR | Zbl

[13] Zhou, P.: On the irrationality of a certain multivariable infinite product. Quaest. Math. 29 (2006), 351-365. | DOI | MR

[14] Zhu, Y. Ch.: Transcendence of certain infinite products. Acta Math. Sin. 43 (2000), 605-610 Chinese. English summary \MR 1825076. | MR | Zbl

Cité par Sources :