On particular families of hyperquadratic continued fractions in power series fields of odd characteristic

TL;DR

This paper explores specific algebraic continued fractions in power series fields over finite fields of odd characteristic, providing explicit examples with controlled algebraic degree and irrationality measure, based on a special rational sequence.

Contribution

It introduces explicit algebraic continued fractions in power series fields over finite fields, linked to a unique rational sequence, and determines their algebraic degree and irrationality measure.

Findings

Explicit continued fractions satisfying algebraic equations of degree d

Irrationality measure of these fractions equals their algebraic degree

Based on a special finite rational sequence

Abstract

We discuss the form of certain algebraic continued fractions in the field of power series over $F_p$, where p is an odd prime number. This leads to give explicit continued fractions in these fields, satisfying an explicit algebraic equation of arbitrary degree $d\geq 2$ and having an irrationality measure equal to $d$. Our results are based on a mysterious finite sequence of rational numbers.