A Transcendental Unbounded Continued Fraction Expansions over A Finite Field

Rima Ghorbel, Hassen Kthiri

Abstract


Let Fq be a finite field and Fq((X−1 )) the field of formal power series with coefficients in Fq. The purpose of this paper is to exhibit a family of transcendental continued fractions of formal power series over a finite field through some specific irregularities of its partial quotients

Keywords


Continued fraction, formal power series, transcendence, finite fields.

Full Text:

PDF

References


A. Baker, Continued fractions of trascendental numbers, Mathematika, 9 (1962), 1–8.

A. Khintchine, Continued fractions, Gosudarstv. Izdat. Tech-Teor. Lit. MoscowLeningrad, 2nd edition, 1949, (In Russian).

B. Adamczewski and Y. Bugea, On the Maillet-Baker continued fractions, J. Reine

Angew. Math., 606 (2007), 105–121.

B. Ammous, S. Driss and M. Hbaib, Continued Fractions and Transcendence of

Formal Power Series Over a Finite Field, Mediterranean Journal of Mathematics, 13

(2016), 527–536.

E. Maillet, Introduction la th´eorie des nombres transcendants et des proprits arithmtiques des fonctions, Gauthier-Villars, Paris, 1906.

J. Liouville, Sur des classes trs tendues de quantits dont la valeur n’est ni algbrique

ni mˆeme rductibles des rationnelles algbriques, J. Math. Pures Appl., 16 (1851),

–142.

H. Davenport and K. F. Roth, Rational approximations to algebraic numbers, Mathematika, 2 (1955), 160–167.

L.E. Baum and H.M. Sweet, Continued fractions of algebraic power series in characteristic 2, Ann. Math., 103 (1976), 593–610.

M. Hbaib, M. Mkaouar and K. Tounsi, Un critre de transcendance dans le corps des

s´eries formelles Fq((X−1

)), J. Number Theory, 116 (2006), 140–149.

M. Mkaouar, Fractions continues et s´eries formelles algbriques rduites, Port. Math.,

(2001).

M. Mkaouar, Transcendance de certaines fractions continues dans le corps des s´eries

formelles, J. Algebra, 281 (2004), 502–507.

O. Perron, Die Lehre von den Kettenbruchen, Teubner, Leipzig, 1929.

W.H. Mills and D.P. Robbins, Continued fractions for certain algebraic power series,

J. Number Theory, 23 (1986), 388–404.

W. Schmidt, On simultanous approximations of two algebraic numbers by rationals,

Acta Math., 119 (1967), 27–50.




DOI: https://doi.org/10.22342/jims.27.1.829.115-122

Refbacks

  • There are currently no refbacks.



Journal of the Indonesian Mathematical Society
Mathematics Department, Universitas Gadjah Mada
Senolowo, Sinduadi, Mlati, Sleman Regency, Special Region of Yogyakarta 55281, Telp. (0274) 552243
Email: jims.indoms@gmail.com


p-ISSN: 2086-8952 | e-ISSN: 2460-0245


Journal of the Indonesian Mathematical Society is licensed under a Creative Commons Attribution 4.0 International License

web statistics
View My Stats