- 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**:$Delta\; f(x)=f(x+1)-f(x),$occurs frequently in the calculus offinite difference s, where it plays a role formally similar to that of thederivative , but used in discrete circumstances.Difference equation s can often be solved with techniques very similar to those for solvingdifferential equation s.This similarity led to the development oftime scale calculus . Analogously we can have the**backward difference operator**:$abla\; f(x)=f(x)-f(x-1).,$

When restricted to

polynomial functions "f", the forward difference operator is adelta operator , i.e., ashift-equivariant linear operator on polynomials that reduces degree by 1.**"n"-th difference**The "n"th forward difference of a function "f"("x") is given by

:$Delta^n\; [f]\; (x)=\; sum\_\{k=0\}^n\; \{n\; choose\; k\}\; (-1)^\{n-k\}\; f(x+k)$

where $\{n\; choose\; k\}$ is the

binomial coefficient . Forward differences applied to asequence are sometimes called thebinomial transform of the sequence, and, as such, have a number of interesting combinatorial properties.Forward differences may be evaluated using the

Nörlund-Rice integral . The integral representation for these types of series is interesting because the integral can often be evaluated usingasymptotic expansion orsaddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large "n".**Newton series**The

**Newton series**or**Newton forward difference equation**, named afterIsaac Newton , is the relationship:$f(x+a)=sum\_\{k=0\}^inftyfrac\{Delta^k\; [f]\; (a)\}\{k!\}(x)\_k=\; sum\_\{k=0\}^infty\; \{x\; choose\; k\}\; Delta^k\; [f]\; (a)$

which holds for any

polynomial function "f" and for some, but not all,analytic function s. Here,:$\{x\; choose\; k\}$

is the

binomial coefficient , and:$(x)\_k=x(x-1)(x-2)cdots(x-k+1)$

is the "

falling factorial " or "lower factorial" and theempty product ("x")_{0}defined to be 1. Note also the formal similarity of this result toTaylor's theorem ; this is one of the observations that lead to the idea ofumbral calculus .In analysis with

p-adic number s,Mahler's theorem states that the assumption that "f" is a polynomial function can be weakened all the way to the assumption that "f" is merely continuous.Carlson's theorem provides necessary and sufficient conditions for a Newton series to be unique, if it exists. However, a Newton series will not, in general, exist.The Newton series, together with the

Stirling series and theSelberg series , is a special case of the generaldifference series , all of which are defined in terms of scaled forward differences.**Rules for finding the difference applied to a combination of functions**Analogous to rules for finding the derivative, we have:

***Constant rule**: If "c" is aconstant , then :$riangle\; c\; =\; 0$

***Linearity**: if "a" and "b" areconstant s,:$riangle\; (a\; f\; +\; b\; g)\; =\; a\; ,\; riangle\; f\; +\; b\; ,\; riangle\; g$All of the above rules apply equally well to any difference operator, including $abla$ as to $riangle$.

*: :$riangle\; (f\; g)\; =\; f\; ,\; riangle\; g\; +\; g\; ,\; riangle\; f\; +\; riangle\; f\; ,\; riangle\; g$:$abla\; (f\; g)\; =\; f\; ,\; abla\; g\; +\; g\; ,\; abla\; f\; -\; abla\; f\; ,\; abla\; g$Product rule

*::$abla\; left(\; frac\{f\}\{g\}\; ight)\; =\; frac\{1\}\{g\}\; det\; egin\{bmatrix\}\; abla\; f\; abla\; g\; \backslash \; f\; g\; end\{bmatrix\}\; left(\; det\; \{egin\{bmatrix\}\; g\; abla\; g\; \backslash \; 1\; 1\; end\{bmatrix\; ight)^\{-1\}$::or:$ablaleft(\; frac\{f\}\{g\}\; ight)=\; frac\; \{g\; ,\; abla\; f\; -\; f\; ,\; abla\; g\}\{g\; cdot\; (g\; -\; abla\; g)\}$:$riangleleft(\; frac\{f\}\{g\}\; ight)=\; frac\; \{g\; ,\; riangle\; f\; -\; f\; ,\; riangle\; g\}\{g\; cdot\; (g\; +\; riangle\; g)\}$Quotient rule *

**Summation rules**::$sum\_\{n=a\}^\{b\}\; riangle\; f(n)\; =\; f(b+1)-f(a)$:$sum\_\{n=a\}^\{b\}\; abla\; f(n)\; =\; f(b)-f(a-1)$**Generalizations**Difference operator generalizes to

Möbius inversion over apartially ordered set .**As a convolution operator**Via the formalism of

incidence algebra s, difference operators and other Möbius inversion can be represented byconvolution with a function on the poset, called theMöbius function μ; for the difference operator, μ is the sequence $(1,-1,0,dots)$.**ee also***

Newton polynomial

*Table of Newtonian series

*Lagrange polynomial

*Gilbreath's conjecture **References***citation

first1 = Philippe | last1 = Flajolet

authorlink2 = Robert Sedgewick (computer scientist) | first2 = Robert | last2 = Sedgewick

url = http://www-rocq.inria.fr/algo/flajolet/Publications/mellin-rice.ps.gz

title = Mellin transforms and asymptotics: Finite differences and Rice's integrals

journal = Theoretical Computer Science

volume = 144 | issue = 1–2 | year = 1995 | pages = 101–124

doi = 10.1016/0304-3975(94)00281-M.

*Wikimedia Foundation.
2010.*