, a difference operator
maps a function f
) 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 shift-equivariant linear operator on polynomials that reduces degree by 1. For any polynomial function f we have
is the "falling factorial
" or "lower factorial" and the empty product
defined to be 1.
In the theory of special functions
, the notation (x
is often used for rising factorials; the former notation, however, is universal among combinatorialists
.) In analysis with p-adic 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 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.