In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by \sum _x or \Delta^{-1}, is the linear operator, inverse of the forward difference operator \Delta. It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus More explicitly, if, then If F(x) is a solution of this functional equation for a given f(x), then so is F(x)+C(x) for any periodic function C(x) with period 1. Therefore, each indefinite sum actually represents a family of functions. However, due to the Carlson's theorem, the solution equal to its Newton series expansion is unique up to an additive constant C. This unique solution can be represented by formal power series form of the antidifference operator:.

Fundamental theorem of discrete calculus

Indefinite sums can be used to calculate definite sums with the formula:

Definitions

Laplace summation formula

The Laplace summation formula allows the indefinite sum to be written as the indefinite integral plus correction terms obtained from iterating the difference operator, although it was originally developed for the reverse process of writing an integral as an indefinite sum plus correction terms. As usual with indefinite sums and indefinite integrals, it is valid up to an arbitrary choice of the constant of integration. Using operator algebra avoids cluttering the formula with repeated copies of the function to be operated on: In this formula, for instance, the term \tfrac12 represents an operator that divides the given function by two. The coefficients +\tfrac12,, etc., appearing in this formula are the Gregory coefficients, also called Laplace numbers. The coefficient in the term is where the numerator of the left hand side is called a Cauchy number of the first kind, although this name sometimes applies to the Gregory coefficients themselves.

Newton's formula

Faulhaber's formula

Faulhaber's formula provides that the right-hand side of the equation converges.

Mueller's formula

If then

Euler–Maclaurin formula

Choice of the constant term

Often the constant C in indefinite sum is fixed from the following condition. Let Then the constant C is fixed from the condition or Alternatively, Ramanujan's sum can be used: or at 1 respectively

Summation by parts

Indefinite summation by parts: Definite summation by parts:

Period rules

If T is a period of function f(x) then If T is an antiperiod of function f(x), that is then

Alternative usage

Some authors use the phrase "indefinite sum" to describe a sum in which the numerical value of the upper limit is not given: In this case a closed form expression F(k) for the sum is a solution of which is called the telescoping equation. It is the inverse of the backward difference \nabla operator. It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.

List of indefinite sums

This is a list of indefinite sums of various functions. Not every function has an indefinite sum that can be expressed in terms of elementary functions.

Antidifferences of rational functions

Antidifferences of exponential functions

Particularly,

Antidifferences of logarithmic functions

Antidifferences of hyperbolic functions

Antidifferences of trigonometric functions

Antidifferences of inverse hyperbolic functions

Antidifferences of inverse trigonometric functions

Antidifferences of special functions