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: .
Fundamental theorem of discrete calculus
Indefinite sums can be used to calculate definite sums with the formula:[3]
Definitions
Laplace summation formula
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. The coefficients , , etc., appearing in this formula are the
Gregory coefficients, also called Laplace numbers. The coefficient in the term is[4]
where the numerator 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]
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.
List of indefinite sums
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
From which can be factored out, leaving 1, with the alternative form . From that, we have:
For the sum below, remember
For positive integer exponents
Faulhaber's formula can be used. For negative integer exponents,
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: .
Fundamental theorem of discrete calculus
Indefinite sums can be used to calculate definite sums with the formula:[3]
Definitions
Laplace summation formula
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. The coefficients , , etc., appearing in this formula are the
Gregory coefficients, also called Laplace numbers. The coefficient in the term is[4]
where the numerator 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]
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.
List of indefinite sums
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
From which can be factored out, leaving 1, with the alternative form . From that, we have:
For the sum below, remember
For positive integer exponents
Faulhaber's formula can be used. For negative integer exponents,