  Our overall goal is to unify and extend some results in the literature
related to the approximation of generating functions of finite and infinite
sequences over a field by rational functions. In our approach, numerators play
a significant role. We revisit a theorem of Niederreiter on (i) linear
complexities and (ii) '$n^{th}$ minimal polynomials' of an infinite sequence,
proved using partial quotients. We prove (i) and its converse from first
principles and generalise (ii) to rational functions where the denominator need
not have minimal degree. We prove (ii) in two parts: firstly for geometric
sequences and then for sequences with a jump in linear complexity. The basic
idea is to decompose the denominator as a sum of polynomial multiples of two
polynomials of minimal degree; there is a similar decomposition for the
numerators. The decomposition is unique when the denominator has degree at most
the length of the sequence. The proof also applies to rational functions
related to finite sequences, generalising a result of Massey. We give a number
of applications to rational functions associated to sequences.
