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Journals / Hacettepe Journal of Mathematics and Statistics / 2020 / Cilt: 49 - Sayı: 1

On Mahler’s S-numbers, T-numbers, and U-numbers

Gülcan KEKEÇ
Pages
45–55
Publication Date
2020-02-06
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Then we consider these series for certain Liouville number arguments and obtain similar results.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this work, we consider some power series with algebraic coefficients from a certain algebraic number field, whose radiuses of convergence are infinite.
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Key Findings & Data:We show that under certain conditions these series take transcendental values at non-zero algebraic number arguments, and we determine the classes to which these transcendental values belong in Mahler's classification.
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Conclusion & Impact:Then we consider these series for certain Liouville number arguments and obtain similar results.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Mahler’s classification of the transcendental numbers · Mahler’s S−number · Mahler’s T−number

Abstract

In this work, we consider some power series with algebraic coefficients from a certain algebraic number field, whose radiuses of convergence are infinite. We show that under certain conditions these series take transcendental values at non-zero algebraic number arguments, and we determine the classes to which these transcendental values belong in Mahler's classification. Then we consider these series for certain Liouville number arguments and obtain similar results.

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Cite This Article

KEKEÇ, G. (2020). On Mahler’s S-numbers, T-numbers, and U-numbers. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 1, 45–55. https://ajindex.com/articles/669292

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References (15)

  1. A. Baker, On Mahler’s classification of transcendental numbers, Acta Math. 111, 97–120, 1964
  2. Y. Bugeaud, Approximation by algebraic numbers, Cambridge Tracts in Mathematics 160, Cambridge University Press, Cambridge, 2004. bibitemk J.F. Koksma, Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen, Monatsh. Math. Phys. 48, 176–189, 1939
  3. W.J. LeVeque, On Mahler’s U−numbers, J. London Math. Soc. 28, 220–229, 1953
  4. W.J. LeVeque, Topics in Number Theory Volume II, Addison-Wesley Publishing, 1956
  5. K. Mahler, Zur Approximation der Exponentialfunktion und des Logarithmus I, II, J. Reine Angew. Math. 166, 118–150, 1932
  6. M.H. Oryan, On power series and Mahler’s U−numbers, Math. Scand. 65, 143–151, 1989
  7. M.H. Oryan, On power series and Mahler’s U−numbers, İstanbul Üniv. Fen Fak. Mecm. Ser. A, 47, 117–125, 1990
  8. M.H. Oryan, Über gewisse Potenzreihen, deren Funktionswerte für Argumente aus der Menge der Liouvilleschen Zahlen U-Zahlen vom Grade$\leq m$ sind, İstanbul Üniv. Fen Fak. Mecm. Ser. A, 47, 15–34, 1990
  9. M.H. Oryan, On the power series and Liouville numbers, Doga Mat. 14 (2), 79–90, 1990
  10. O. Perron, Irrationalzahlen, Walter de Gruyter & Co., Berlin, 1960
  11. T. Schneider, Einführung in die transzendenten Zahlen, Springer-Verlag, Berlin- Göttingen-Heidelberg, 1957
  12. E. Wirsing, Approximation mit algebraischen Zahlen beschränkten Grades, J. Reine Angew. Math. 206, 67–77, 1961
  13. B.M. Zeren, Über die Transzendenz der Werte einiger schnell konvergenter Poten- zreihen für algebraische Argumente, İstanbul Tek. Üniv. Bül. 38, 473–496, 1985
  14. B.M. Zeren, Über eine Klasse von verallgemeinerten Lückenreihen, deren Werte für algebraische Argumente transzendent, aber keine U−Zahlen sind I, İstanbul Üniv. Fen Fak. Mat. Derg. 50, 79–99, 1991
  15. B.M. Zeren, Über eine Klasse von verallgemeinerten Lückenreihen, deren Werte für algebraische Argumente transzendent, aber keine U−Zahlen sind III, İstanbul Üniv. Fen Fak. Mat. Derg. 50, 147–158, 1991.