A note on the transcendence of infinite products

Jaroslav Hančl; Ondřej Kolouch; Simona Pulcerová; Jan Štěpnička

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 3, page 613-623
  • ISSN: 0011-4642

Abstract

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The paper deals with several criteria for the transcendence of infinite products of the form ∏ n = 1 ∞ [ b n α a n ] / b n α a n where α > 1 is a positive algebraic number having a conjugate α * such that α ≠ | α * | > 1 , { a n } n = 1 ∞ and { b n } n = 1 ∞ are two sequences of positive integers with some specific conditions. 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).

How to cite

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Hančl, Jaroslav, et al. "A note on the transcendence of infinite products." Czechoslovak Mathematical Journal 62.3 (2012): 613-623. <http://eudml.org/doc/246593>.

@article{Hančl2012,
abstract = {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 $\alpha \ne |\alpha ^*|>1$, $\lbrace a_n\rbrace _\{n=1\}^\infty $ and $\lbrace b_n\rbrace _\{n=1\}^\infty $ are two sequences of positive integers with some specific conditions. 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).},
author = {Hančl, Jaroslav, Kolouch, Ondřej, Pulcerová, Simona, Štěpnička, Jan},
journal = {Czechoslovak Mathematical Journal},
keywords = {transcendence; infinite product; transcendence; infinite product},
language = {eng},
number = {3},
pages = {613-623},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the transcendence of infinite products},
url = {http://eudml.org/doc/246593},
volume = {62},
year = {2012},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 613
EP - 623
AB - 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 $\alpha \ne |\alpha ^*|>1$, $\lbrace a_n\rbrace _{n=1}^\infty $ and $\lbrace b_n\rbrace _{n=1}^\infty $ are two sequences of positive integers with some specific conditions. 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).
LA - eng
KW - transcendence; infinite product; transcendence; infinite product
UR - http://eudml.org/doc/246593
ER -

References

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  1. Corvaja, P., Hančl, J., 10.4171/RLM/496, Atti Acad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 18 (2007), 295-303. (2007) Zbl1207.11075MR2318822DOI10.4171/RLM/496
  2. Corvaja, P., Zannier, U., 10.1007/BF02392563, Acta Math. 193 (2004), 175-191. (2004) Zbl1175.11036MR2134865DOI10.1007/BF02392563
  3. Corvaja, P., Zannier, U., 10.1023/A:1015594913393, Comp. Math. 131 (2002), 319-340. (2002) Zbl1010.11038MR1905026DOI10.1023/A:1015594913393
  4. Erdős, P., Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 (1975), 1-7. (1975) MR0539489
  5. Genčev, M., 10.2478/s12175-009-0133-4, Math. Slovaca 59 (2009), 365-378. (2009) Zbl1209.11067MR2505816DOI10.2478/s12175-009-0133-4
  6. Hančl, J., Nair, R., Šustek, J., 10.1016/S0019-3577(06)81034-7, Indag. Math., New Ser. 17 (2006), 567-581. (2006) Zbl1131.11048MR2320114DOI10.1016/S0019-3577(06)81034-7
  7. Hančl, J., Rucki, P., Šustek, J., A generalization of Sándor's theorem using iterated logarithms, Kumamoto J. Math. 19 (2006), 25-36. (2006) Zbl1220.11087MR2211630
  8. Hančl, J., Štěpnička, J., Šustek, J., 10.1007/s11139-008-9137-x, Ramanujan J. 17 (2008), 331-342. (2008) MR2456837DOI10.1007/s11139-008-9137-x
  9. Kim, D., Koo, J. K., 10.4134/JKMS.2007.44.1.055, J. Korean Math. Soc. 44 (2007), 55-107. (2007) Zbl1128.11037MR2283460DOI10.4134/JKMS.2007.44.1.055
  10. Lang, S., Algebra (3rd ed.), Graduate Texts in Mathematics. Springer, New York (2002). (2002) MR1878556
  11. Nyblom, M. A., On the construction of a family of transcendental valued infinite products, Fibonacci Q. 42 (2004), 353-358. (2004) Zbl1062.11048MR2110089
  12. Tachiya, Y., 10.1007/BF03323373, Result. Math. 48 (2005), 344-370. (2005) Zbl1133.11313MR2215585DOI10.1007/BF03323373
  13. Zhou, P., 10.2989/16073600609486169, Quaest. Math. 29 (2006), 351-365. (2006) MR2260768DOI10.2989/16073600609486169
  14. Zhu, Y. Ch., Transcendence of certain infinite products, Acta Math. Sin. 43 (2000), 605-610 Chinese. English summary 1825076. (2000) Zbl1005.11034MR1825076

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