Linear independence of continued fractions with algebraic terms

TL;DR

This paper establishes conditions under which certain continued fractions with algebraic terms are linearly independent over a given number field, advancing understanding of algebraic independence in continued fractions.

Contribution

It provides new criteria for linear independence of continued fractions with algebraic entries over number fields.

Findings

Conditions for linear independence of continued fractions with algebraic terms

Extension of linear independence results to multiple continued fractions

Application to algebraic number theory and continued fraction analysis

Abstract

We give conditions on sequences of positive algebraic numbers $\{a_{n,j}\}_{n=1}^\infty$, $j=1,\dots ,M$ and number field $\mathbb K$ to ensure that the numbers defined by the continued fractions $[0;a_{1,j},a_{2,j},\dots ]$, $j=1,\dots ,M$ and $1$ are linearly independent over $\mathbb K$.