Continued fractions and linear recurrences

Let to, t1, t2, be a sequence of elements of a field F. We give a continued fraction algorithm for tox + tlx2 + 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 E ti+j j 0O < i < s-1, j=O for every positive integer s. 1. Let tO, tl, t2, be a sequence of elements of a field F. Set