indefinite sum

Recall that the finite difference operator Δ defined on the set of functionsMathworldPlanetmath is given by


The difference operator can be thought of as the discrete version of the derivative operator sending a function to its derivative (if it exists). With the derivative operation, there corresponds an inverse operation called the antiderivative, which, given a function f, finds its antiderivative F so that the derivative of F gives f. There is also a discrete analog of this inverse operation, and it is called the indefinite sum.

The indefinite sum of a function f: is the set of functions


This set is often denoted by Δ-1f or Σf, and any element in Δ-1f is called an indefinite sum of f.

Remark. Like the indefinite integral, the indefinite sum Δ-1 is shift invariant. This means that for any FΔ-1f, then F+cΔ-1f for any c. But, unlike the indefinite integral, the indefinite sum is also invariant by a shift of a periodic real function of period 1. Conversely, the difference of two indefinite sums of a function f is a periodic real function of period 1.

In the following discussion, we consider the indefinite sum of a function as a function.

Basic Properties

  1. 1.

    ΔΔ-1f=f, and Δ-1Δf=f modulo a real function of period 1.

  2. 2.

    Modulo a real number, and treating Δ-1 as an operator taking a function into a function, we see that Δ-1 is linear, that is,

    • Δ-1(rf)=rΔ-1f for any r, and

    • Δ-1(f+g)=Δ-1f+Δ-1g.

  3. 3.

    If F(x)=Δ-1f(x), then F(x+a)=Δ-1f(x+a).

  4. 4.

    If F=Δ-1f, then we see that

    F(a+1)-F(a) = f(a),
    F(a+2)-F(a+1) = f(a+1),
    F(x)-F(x-1) = f(x-1).

    where x-a is a positive integer. Summing these expressions, we get


    This is the discrete version of the fundamental theorem of calculusMathworldPlanetmathPlanetmath.

Below is a table of some basic functions and their indefinite sums (C is a real-valued periodic functionMathworldPlanetmath with period 1):

f(x) Δ-1f(x) Comment
r rx+C
x x(x-1)2+C
x2 x(x-1)(2x-1)6+C
x3 x2(x-1)24+C
xn Tn(x)+C See this link ( for detail
ax axa-1+C a1
(x)n (x)nn+1+C (x)n is the falling factorialDlmfMathworld of degree n
(xn) (xn+1)+C (xn):=(x)nn!
1x ψ(x)+C ψ(x) is the digamma functionMathworldPlanetmath
lnx lnΓ(x)+C Γ(x) is the gamma functionDlmfDlmfMathworldPlanetmath
sinx -cos(x-1/2)2sin(1/2)+C
cosx sin(x-1/2)2sin(1/2)+C

 References 1 C.Jordan.Calculus of Finite Differences,thirdedition.Chelsea,NewYork(1965)Titleindefinite sumCanonical nameIndefiniteSumDate of creation2013-03-22 17:35:14Last modified on2013-03-22 17:35:14OwnerCWoo (3771)Last modified byCWoo (3771)Numerical id20AuthorCWoo (3771)Entry typeDefinitionClassificationmsc 39A99Related topicFiniteDifference