We show that the theory of divided differences is a natural tool for the study of linearly recurrent sequences. The divided differences functional associated with a monic polynomial w on degree n + 1 yields a vector space isomorphism between the space of polynomials of degree at most equal to n and the space of linearly recurrent sequences f that satisfy the difference equation w(E)f=0 where E is the usual shift operator. Using such isomorphisms, we can translate problems about recurrent sequences into simple problems about polynomials. We present here a new approach to the theory of divided differences, using only generating functions and elementary linear algebra, which clarifies the connections of divided differences with rational functions, polynomial interpolation, residues, and partial fractions decompositions.