# Bounds on the lengths of certain series expansions

Research paper by Yanapat Tongron, Narakorn Rompurk Kanasri Vichian Laohakosol

Indexed on: 11 Dec '18Published on: 10 Dec '18Published in: Journal of physics. Conference series

#### Abstract

In the real number field, there are several unique series expansions for each A ∈ (0, 1). Of interest are the Sylvester and alternating Sylvester series expansions since both expansions are finite if and only if A is rational. We obtain upper bounds on the length of rational A ∈ (0, 1) and lower bound on the length of certain classes of rational numbers. In the power series fields, let ##IMG## [http://ej.iop.org/images/1742-6596/1132/1/012001/JPCS_1132_1_012001_ieqn1.gif] {${{\mathbb{F}}}_{q}$} denote the finite field of q elements, let p ( x ) be an irreducible polynomial in ##IMG## [http://ej.iop.org/images/1742-6596/1132/1/012001/JPCS_1132_1_012001_ieqn2.gif] {${{\mathbb{F}}}_{q}[x]$} , and let ##IMG## [http://ej.iop.org/images/1742-6596/1132/1/012001/JPCS_1132_1_012001_ieqn3.gif] {${{\mathbb{F}}}_{q}((p(x)))$} , respectively, ##IMG## {\${{\mathbb{F}}}_{q}((1/...}