Continued Fractions and Linear Recurrences

Abstract

Let ${t_0},{t_1},{t_2}, \cdots$ be a sequence of elements of a field F. We give a continued fraction algorithm for ${t_0}x + {t_1}{x^2} + {t_2}{x^3} + \cdots$. If our sequence satisfies a linear recurrence, then the continued fraction algorithm is finite and produces this recurrence. More generally the algorithm produces a nontrivial solution of the system \[ \sum \limits _{j = 0}^s {{t_{i + j}}{\lambda _j},\quad 0 \leqslant i \leqslant s - 1,} \] for every positive integer s.