Lacunary Power Series and 𝑼𝒎-Numbers

Yıl 2019, , 365 - 370, 30.12.2019

Fatma Çalışkan

https://doi.org/10.18466/cbayarfbe.622670

Öz

Abstract

In this work, the values of certain lacunar power series with rational coefficients for 𝑈𝒎-number arguments were determined to be either in a particular algebraic number field or in the set of transcendental numbers under specific circumstances in the complex numbers field. The result was also applied on some of the lacunary power series with coefficients in an algebraic number field. Roth's theorem which is the essential result in Diophantine approximation to algebraic numbers was used to reach the present results.

Anahtar Kelimeler

Algebraic number field, power series, transcendental number, 𝑈_𝑚-number

Destekleyen Kurum

Scientific Research Projects Coordination Unit of Istanbul University

Proje Numarası

27422

Teşekkür

I sincerely thank to Prof. Dr. Bedriye M. ZEREN for important suggestions on this manuscript. This work was supported by Scientific Research Projects Coordination Unit of Istanbul University (project number 27422).

Kaynakça

• 1. Bugeaud, Y. 2003. Mahler's Classification of Numbers Compared with Koksma's, Acta Arithmetica; 110: 89-105.
• 2. Bugeaud, Y, Laurent, M. 2005. On Exponents of Homogeneous and Inhomegeneous Diophantine Approximation, Moscow Mathematical Journal; 5(4): 747-766.
• 3. Çalışkan, F. 2018. Transcendence of Some Power Series for Liouville Number Arguments, Proceedings Mathematical Sciences; 128(3): 128:29.
• 4. İçen, O Ş. 1973, Anhang zu den Arbeiten Über die Funktionswerte der 𝑝-Adischen Elliptischen Funktionen I und II, İstanbul Üniversitesi Fen Fakültesi Mecmuası. Seri A, 38: 25-35.
• 5. Hancl, J, Stepnicka, J. 2008. On the Transcendence of Some Infinite Series, Glasgow Mathematical Journal; 50: 33-37.
• 6. Koksma, J F. 1939. Über die Mahlersche Klasseneinteilung der Transzendenten Zahlen und die Approximation Komplexer Zahlen durch Algebraische Zahlen, Monatshefte für Mathematik Physics, 48: 176-789.
• 7. Mahler, K. 1932. Zur Approximation der Exponantialfunktion und des Logarithmus I, Journal für die Reine und Angewandte Mathematik, 166: 118-136.
• 8. Mahler, K. 1984. Some suggestions for further research, Bulletin Australian Mathematical Society, 29: 101-108.
• 9. Maillet, E. Introduction a la theorie des nombers transcendants et des proprietes arithmetiques des fonctions, Gauthier-Villars, Paris, 1906.
• 10. Marques,D, Ramirez, J, Silva, E. 2016. A note on Lacunary Power Series with Rational Coefficients, Bulletin Australian Mathematical Society, 93: 372-374.
• 11. Marques, D, Schleischitz, J. 2016. On a Problem Posed by Mahler, Journal of the Australian Mathematical Society, 100: 86-107.
• 12. Ram Murty, M, Saradha, N. 2007. Transcendental Values of the Digamma Function, Journal of Number Theory, 125: 298-318.
• 13. Roth, K F. 1955. Rational Approximations to Algebraic Numbers, Mathematika, 2: 1-20.
• 14. Schneider, Th. Einführung in die Transzendenten Zahlen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957.
• 15. Yılmaz, G. 1996. Arithmetical Properties of the Value of Some Power Series with Algebraic Coefficients Taken for 𝑈_𝑚-Numbers Arguments, İstanbul Üniversitesi Fen Fakültesi MatematikDergisi, 55-56: 111-144.
• 16. Wirsing, E. 1961. Approximation mit Algebraischen Zahlen Beschrankten Grades, Journal für die Reine und Angewandte Mathematik, 206: 67-77.
• 17. Wu, Q, Zhou, P. Transcendence of some Multivariate Power Series, Frontiers of Mathematics in China, 9(2): 425-430.