# 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.*

### Look at other dictionaries:

**Difference polynomials**— In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg s polynomials, and the Stirling… … Wikipedia**Operator messaging**— is the term, similar to Text Messaging and Voice Messaging, applying to an answering service call center who focuses on one specific scripting style that has grown out of the alphanumeric pager history. Contents 1 Early history 2 Message Center… … Wikipedia**Operator (sternwheeler)**— Operator on Skeena River 1911 Career (Canada) … Wikipedia**Difference of Gaussian**— Helligkeitsänderung einer Kante Verlauf der 2. Ableitung an der Kante Der Marr Hildreth Operator oder Laplacian of Gaussian (LoG) ist eine spezielle Form eines diskreten Laplace Filters … Deutsch Wikipedia**Difference of Gaussians**— In computer vision, Difference of Gaussians is a grayscale image enhancement algorithm that involves the subtraction of one blurred version of an original grayscale image from another, less blurred version of the original. The blurred images are… … Wikipedia**Operator product expansion**— Contents 1 2D Euclidean quantum field theory 2 General 3 See also 4 External links 2D Euclidean quantum field theory … Wikipedia**Difference Engine**— ▪ calculating machine an early calculating machine, verging on being the first computer, designed and partially built during the 1820s and 30s by Charles Babbage (Babbage, Charles). Babbage was an English mathematician and inventor; he… … Universalium**Finite difference**— A finite difference is a mathematical expression of the form f(x + b) − f(x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences… … Wikipedia**Delta operator**— In mathematics, a delta operator is a shift equivariant linear operator on the vector space of polynomials in a variable over a field that reduces degrees by one. To say that is shift equivariant means that if … Wikipedia**Lag operator**— In time series analysis, the lag operator or backshift operator operates on an element of a time series to produce the previous element. For example, given some time series:X= {X 1, X 2, dots },then :, L X t = X {t 1} for all ; t > 1,where L is… … Wikipedia