In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by
,[1][2] 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.
, 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:
Indefinite sums can be used to calculate definite sums with the formula:[3] 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:[4]
In this formula, for instance, the term
represents an operator that divides the given function by two.
etc., appearing in this formula are the Gregory coefficients, also called Laplace numbers.
of the left hand side is called a Cauchy number of the first kind, although this name sometimes applies to the Gregory coefficients themselves.
[4] Faulhaber's formula provides that the right-hand side of the equation converges.
then[5] 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[6][7] Indefinite summation by parts: Definite summation by parts: If
is a period of function
then 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.
[8] It is the inverse of the backward difference
It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.
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.