In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by or ,[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. 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: .
Indefinite sums can be used to calculate definite sums with the formula:[3]
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:
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 operator.
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.
Antidifferences of rational functions
- where , the generalized to real order Bernoulli polynomials.
- where is the polygamma function.
- where is the digamma function.
Antidifferences of exponential functions
Particularly,
Antidifferences of logarithmic functions
Antidifferences of hyperbolic functions
- where is the q-digamma function.
Antidifferences of trigonometric functions
- where is the q-digamma function.
- where is the normalized sinc function.
Antidifferences of inverse hyperbolic functions
Antidifferences of inverse trigonometric functions
Antidifferences of special functions
- where is the incomplete gamma function.
- where is the falling factorial.
- (see super-exponential function)
"Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1 Éric Delabaere, Ramanujan's Summation, Algorithms Seminar 2001–2002, F. Chyzak (ed.), INRIA, (2003), pp. 83–88.
- "Difference Equations: An Introduction with Applications", Walter G. Kelley, Allan C. Peterson, Academic Press, 2001, ISBN 0-12-403330-X
- Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations
- Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities
- S. P. Polyakov. Indefinite summation of rational functions with additional minimization of the summable part. Programmirovanie, 2008, Vol. 34, No. 2.
- "Finite-Difference Equations And Simulations", Francis B. Hildebrand, Prenctice-Hall, 1968