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.

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]

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]

Main article: Summation by parts

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. e.g.

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 inverse to 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 (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, 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]