In mathematics, a difference operator maps a function f(x) to another function f(x + a) − f(x + b).
The forward difference operator
When restricted to polynomial functions f, the forward difference operator is a delta operator, i.e., a shift-equivariant linear operator on polynomials that reduces degree by 1. For any polynomial function f we have
Note that only finitely many terms in the above sum are non-zero: Δk f = 0 if k is greater than the degree of f. Note also the formal similarity of this result and Taylor's theorem.
With p-adic numbers, the same identity is true not only of polynomial functions, but of continuous functions generally; that result is called Mahler's theorem.