Indefinite sum

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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 calculus[edit]

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


Laplace summation formula[edit]

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

Newton's formula[edit]

where is the falling factorial.

Faulhaber's formula[edit]

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

Mueller's formula[edit]

If then[6]

Euler–Maclaurin formula[edit]

Choice of the constant term[edit]

Often the constant C in indefinite sum is fixed from the following condition.


Then the constant C is fixed from the condition


Alternatively, Ramanujan's sum can be used:

or at 1


Summation by parts[edit]

Indefinite summation by parts:

Definite summation by parts:

Period rules[edit]

If is a period of function then

If is an antiperiod of function , that is then

Alternative usage[edit]

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 sums[edit]

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[edit]

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

Antidifferences of exponential functions[edit]


Antidifferences of logarithmic functions[edit]

Antidifferences of hyperbolic functions[edit]

where is the q-digamma function.

Antidifferences of trigonometric functions[edit]

where is the q-digamma function.

Antidifferences of inverse hyperbolic functions[edit]

Antidifferences of inverse trigonometric functions[edit]

Antidifferences of special functions[edit]

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

See also[edit]


  1. ^ Indefinite Sum at
  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 reading[edit]