A Transcendental Unbounded Continued Fraction Expansions over A Finite Field

Rima Ghorbel,Hassen Kthiri

doi:10.22342/jims.27.1.829.115-122

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

Highlights

In [5, 1], Maillet and Baker studied the real number x = [a0, a1, . . .] wherei≥0 is the sequence of partial quotients of x such that an = an+1 = . . . = an+λ(n)−1, for infinitely many positive integers n where λ(n) is a sequence of integers verifying some increasing properties
Adamczewski and Bugeaud [3] suggested a new transcendence criteria for continued fractions by using the Schmidt subspace Theorem given in [15] where the author showed that if an irrational number is very well approximated by quadratic numbers it is quadratic or transcendental
In 2004, Mkaouar [12] gave a similar result to the Baker one [1] concerning the transcendence of formal series over a finite field

Summary

Introduction

Key words and Phrases: Continued fraction, formal power series, transcendence, finite fields. . .] where (ai)i≥0 is the sequence of partial quotients of x such that an = an+1 = . Xα3 + α + X = 0 has a continued fraction expansion with partial quotients of bounded degree. In 1986, Mills and Robbins [14] provided an example of algebraic formal series over F2(X) whose sequence of partial quotients is unbounded.

Objectives

Results

Conclusion