Difference operator

In mathematics, a difference operator assigns to a function f(x) another function f(x + a) - f(x + b). The forward difference operator

$\Delta f(x)=f(x+1)-f(x)$

occurs frequently in the calculus of finite differences. When restricted to polynomial functions f, the forward difference operator is a delta operator, i.e., a shift-equivariant operator on polynomials that reduces degree by 1. For any polynomial function f we have

$f(x)=\sum _{k=0}^{\infty }{\frac {(\Delta ^{k}f)(0)}{k!}}(x)_{k}$

where

$(x)_{k}=x(x-1)(x-2)\cdots (x-k+1)$

is the "falling factorial" or "lower factorial". (Warning: In the theory of special functions, one often sees the notation

$(x)_{k}=x(x+1)(x+2)\cdots (x+k-1),$

i.e., this is a "rising factorial" or "upper factorial". The former notation, however, is universal among combinatorialists.) With p-adic numbers, the same identity is true not only of polynomial functions, but of continuous functions generally; that results is called Mahler's theorem.