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In mathematics the indefinite sum operator (also known as the antidifference operator), denoted by or ,[1][2][3] 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 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 calculusEdit

Indefinite sums can be used to calculate definite sums with the formula:[4]

 

DefinitionsEdit

Laplace summation formulaEdit

 
where   are the Cauchy numbers of the first kind.[5][citation needed]

Newton's formulaEdit

 
where   is the falling factorial.

Faulhaber's formulaEdit

 

provided that the right-hand side of the equation converges.

Mueller's formulaEdit

If   then[6]

 

Euler–Maclaurin formulaEdit

 

Choice of the constant termEdit

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[7][8]

Summation by partsEdit

Indefinite summation by parts:

 
 

Definite summation by parts:

 

Period rulesEdit

If   is a period of function   then

 

If   is an antiperiod of function  , that is   then

 

Alternative usageEdit

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.[9] 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 sumsEdit

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 functionsEdit

 
 
 
where  , the generalized to real order Bernoulli polynomials.
 
where   is the polygamma function.
 
where   is the digamma function.

Antidifferences of exponential functionsEdit

 

Particularly,

 

Antidifferences of logarithmic functionsEdit

 
 

Antidifferences of hyperbolic functionsEdit

 
 
 
where   is the q-digamma function.

Antidifferences of trigonometric functionsEdit

 
 
 
 
 
where   is the q-digamma function.
 
 

Antidifferences of inverse hyperbolic functionsEdit

 

Antidifferences of inverse trigonometric functionsEdit

 

Antidifferences of special functionsEdit

 
 
where   is the incomplete gamma function.
 
where   is the falling factorial.
 
(see super-exponential function)

See alsoEdit

ReferencesEdit

  1. ^ Indefinite Sum at PlanetMath.org.
  2. ^ On Computing Closed Forms for Indefinite Summations. Yiu-Kwong Man. J. Symbolic Computation (1993), 16, 355-376
  3. ^ "If Y is a function whose first difference is the function y, then Y is called an indefinite sum of y and denoted Δ−1y" Introduction to Difference Equations, Samuel Goldberg
  4. ^ "Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1
  5. ^ Bernoulli numbers of the second kind on Mathworld
  6. ^ Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations Archived 2011-06-17 at the Wayback Machine (note that he uses a slightly alternative definition of fractional sum in his work, i.e. inverse to backwards difference, hence 1 as the lower limit in his formula)
  7. ^ Bruce C. Berndt, Ramanujan's Notebooks Archived 2006-10-12 at the Wayback Machine, Ramanujan's Theory of Divergent Series, Chapter 6, Springer-Verlag (ed.), (1939), pp. 133–149.
  8. ^ Éric Delabaere, Ramanujan's Summation, Algorithms Seminar 2001–2002, F. Chyzak (ed.), INRIA, (2003), pp. 83–88.
  9. ^ Algorithms for Nonlinear Higher Order Difference Equations, Manuel Kauers

Further readingEdit