In mathematics, a difference operator maps a function f(x) to another function f(x + a) − f(x + b). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

The forward difference operator

occurs frequently in the calculus of finite differences, where it plays a role formally similar to that of the derivative, but used in discrete circumstances. Difference equations can often be solved with techniques very similar to those for solving differential equations. Analogously we can have the backward difference operator There are two subfields of mathematics that concern themselves with finite differences. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

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 In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, a delta operator is a shift-equivariant linear operator Q on the vector space of polynomials in a variable x that reduces degrees by one. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

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 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. In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ... In arithmetic, the empty product, or nullary product, is the result of multiplying no numbers. ... In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ... In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ... Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria. ... The p-adic number systems were first described by Kurt Hensel in 1897. ... In the notation of combinatorialists, which conflicts with that used in the theory of special functions, the Pochhammer symbol denotes the falling factorial: Denote by Δ the forward difference operator defined by Then we have so that the relationship between the operator Δ and this polynomial sequence is much like...

Note that only finitely many terms in the above sum are non-zero: 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. In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ...