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Lacunary Power Series and 𝑼𝒎-Numbers

Year 2019, Volume: 15 Issue: 4, 365 - 370, 30.12.2019
https://doi.org/10.18466/cbayarfbe.622670

Abstract

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.

Supporting Institution

Scientific Research Projects Coordination Unit of Istanbul University

Project Number

27422

Thanks

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).

References

  • 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.
Year 2019, Volume: 15 Issue: 4, 365 - 370, 30.12.2019
https://doi.org/10.18466/cbayarfbe.622670

Abstract

Project Number

27422

References

  • 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.
There are 17 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Fatma Çalışkan 0000-0001-7869-870X

Project Number 27422
Publication Date December 30, 2019
Published in Issue Year 2019 Volume: 15 Issue: 4

Cite

APA Çalışkan, F. (2019). Lacunary Power Series and 𝑼𝒎-Numbers. Celal Bayar University Journal of Science, 15(4), 365-370. https://doi.org/10.18466/cbayarfbe.622670
AMA Çalışkan F. Lacunary Power Series and 𝑼𝒎-Numbers. CBUJOS. December 2019;15(4):365-370. doi:10.18466/cbayarfbe.622670
Chicago Çalışkan, Fatma. “Lacunary Power Series and 𝑼𝒎-Numbers”. Celal Bayar University Journal of Science 15, no. 4 (December 2019): 365-70. https://doi.org/10.18466/cbayarfbe.622670.
EndNote Çalışkan F (December 1, 2019) Lacunary Power Series and 𝑼𝒎-Numbers. Celal Bayar University Journal of Science 15 4 365–370.
IEEE F. Çalışkan, “Lacunary Power Series and 𝑼𝒎-Numbers”, CBUJOS, vol. 15, no. 4, pp. 365–370, 2019, doi: 10.18466/cbayarfbe.622670.
ISNAD Çalışkan, Fatma. “Lacunary Power Series and 𝑼𝒎-Numbers”. Celal Bayar University Journal of Science 15/4 (December 2019), 365-370. https://doi.org/10.18466/cbayarfbe.622670.
JAMA Çalışkan F. Lacunary Power Series and 𝑼𝒎-Numbers. CBUJOS. 2019;15:365–370.
MLA Çalışkan, Fatma. “Lacunary Power Series and 𝑼𝒎-Numbers”. Celal Bayar University Journal of Science, vol. 15, no. 4, 2019, pp. 365-70, doi:10.18466/cbayarfbe.622670.
Vancouver Çalışkan F. Lacunary Power Series and 𝑼𝒎-Numbers. CBUJOS. 2019;15(4):365-70.