Some types of explicit continued fractions

Abstract

Being able to predict a pattern of a regular continued fraction is not only interesting in its own right but it sometimes yields more informations about that regular continued fraction. In the real number field and in the field of formal series over any base field, it is well-known that the termination of a regular continued fraction can be used to characterize rationality and is also known that any periodic regular continued fraction corresponds exactly to a quadratic irrational element. There are a number of researches about transcendental criteria via regular continued fractions. The major part of this thesis is devoted to the establishing of explicit formulae for continued fractions. First, identities for continued fractions with specific patterns, including palindromic patterns, are realized over an arbitrary field. Then by making use of these identities, explicit continued fractions representing the numbers expressible explicitly by certain series are obtained. These explicit continued fractions possess a beautiful property, that is, sequences of their partial quotients begin in arbitrarily long palindromes. By using a transcendental criterion of Adamczewski and Bugeaud in 2007, it can be concluded that the real numbers represented by these explicit continued fractions are transcendental. Besides explicit formulae for continued fractions, boundedness of the partial quotients of a continued fraction is of interest and is considered as the second main part in this thesis. In this part, a criterion of boundedness of the partial quotients of the regular continued fraction representing a linear fractional transformation of a formal series is given. Also, a fascinating example of rational numbers represented by regular continued fractions which their partial quotients are bounded by 5 is provided by proving a famous conjecture attributed to Zaremba for integers being of the form 2[superscript s] • 3[superscript t] where s,t are non-negative integers.