Difference operator
In
mathematics, a
difference operator maps a
function f(
x) to another function
f(
x + a) −
f(
x + b).
The forward difference operator

occurs frequently in the calculus of finite differences, where it plays a role formally similar to that of the
derivative. Difference equations can often be solved with techniques very similar to those for solving
differential equations.
When restricted to polynomial functions f, the forward difference operator is a delta operator, i.e., a shiftequivariant linear operator on polynomials that reduces degree by 1. For any polynomial function f we have
where
is the "
falling factorial" or "lower factorial" and the
empty product (
x)
_{0} defined to be 1.
(
Warning: In the theory of
special functions, the notation (
x)
_{k} is often used for rising factorials; the former notation, however, is universal among
combinatorialists.) In analysis with
padic numbers, the assumption that
f is a polynomial function can be weakened all the way to the assumption that
f is merely continuous. That is
Mahler's theorem.
Note that only finitely many terms in the above sum are nonzero: Δ^{k} f = 0 if k is greater than the degree of f. Note also the formal similarity of this result and Taylor's theorem.
With padic numbers, the same identity is true not only of polynomial functions, but of continuous functions generally; that result is called Mahler's theorem.