Indefinite sum

In mathematics the indefinite sum operator (also known as the antidifference operator), denoted by formula_1 or formula_2, is the linear operator, inverse of the forward difference operator formula_3. It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus
More explicitly, if formula_5, 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 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: formula_7.
Fundamental theorem of discrete calculus.
Indefinite sums can be used to calculate definite sums with the formula:
Definitions.
Faulhaber's formula.
provided that the right-hand side of the equation converges.
Mueller's formula.
If formula_14 then
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 formula_25 is a period of function formula_26 then
If formula_25 is an antiperiod of function formula_26, that is formula_30 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 formula_34 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 exponential functions.
Particularly,