Linear independence of continued fractions with algebraic terms

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 $\mathbb{K}$.