Continued Fractions and Linear Recurrences

Let t₀,t₁,t₂, ⋯ be a sequence of elements of a field F. We give a continued fraction algorithm for t₀x + t₁x² + t₂x³ + ⋯. 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 \[ ∑ j = 0^s {{ti + j}{λ _j}, 0 i s - 1,} \] for every positive integer s.