Indefinite sum


In mathematics the indefinite sum operator, denoted by or, is the linear operator, inverse of the forward difference operator. It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus
More explicitly, if, then
If F is a solution of this functional equation for a given f, then so is F+C for any periodic function C with period 1. Therefore, each indefinite sum actually represents a family of functions. However 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

Newton's formula

Faulhaber's formula

provided 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 is a period of function then
If is an antiperiod of function, 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 for the sum is a solution of
which is called the telescoping equation. It is the inverse of the backward difference 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