Old 296/496 Notes on Differences

These notes will attempt to collect a lot of information about polynomials and Stirling numbers and the calculus of finite differences. Let Pn denote the set of all polynomials of degree ≤ n. It is easy to see that this is a vector space over the reals (or the complex numbers if you like) of dimension n+ 1. The most familiar basis for Pn is: {1, x, x, . . . , x}. Any p ∈ Pn is almost instantaneously expressed in terms of this basis: