Indefinite sum

In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by $\sum _x $ or $$\Delta^{-1} $$,  is the linear operator, inverse of the forward difference operator $$\Delta $$. It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus


 * $$\Delta \sum_x f(x) = f(x) \, .$$

More explicitly, if $\sum_x f(x) = F(x) $, then


 * $$F(x+1) - F(x) = f(x) \, .$$

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: $$\Delta^{-1}=\frac1{e^D-1}$$.

Fundamental theorem of discrete calculus
Indefinite sums can be used to calculate definite sums with the formula:
 * $$\sum_{k=a}^b f(k)=\Delta^{-1}f(b+1)-\Delta^{-1}f(a)$$

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:

$$\sum_x = \int{} + \frac{1}{2} - \frac{1}{12}\Delta + \frac{1}{24}\Delta^2 - \frac{19}{720}\Delta^3 + \frac{3}{160}\Delta^4 - \cdots$$

In this formula, for instance, the term $$\tfrac12$$ represents an operator that divides the given function by two. The coefficients $$+\tfrac12$$, $$-\tfrac1{12}$$, etc., appearing in this formula are the Gregory coefficients, also called Laplace numbers. The coefficient in the term $$\Delta^{n-1}$$ is

$$\frac{\mathcal{C}_n}{n!}=\int_0^1 \binom{x}{n}\,dx$$

where the numerator $$\mathcal{C}_n$$ of the left hand side is called a Cauchy number of the first kind, although this name sometimes applies to the Gregory coefficients themselves.

Newton's formula

 * $$\sum_x f(x)=\sum_{k=1}^\infty \binom{x}k \Delta^{k-1} [f]\left (0\right)+C=\sum_{k=1}^{\infty}\frac{\Delta^{k-1}[f](0)}{k!}(x)_k+C$$


 * where $$(x)_k=\frac{\Gamma(x+1)}{\Gamma(x-k+1)}$$ is the falling factorial.

Faulhaber's formula

 * $$\sum _x f(x)= \sum_{n=1}^{\infty} \frac{f^{(n-1)} (0)}{n!} B_n(x) + C \, ,$$

Faulhaber's formula provides that the right-hand side of the equation converges.

Mueller's formula
If $$\lim_{x\to{+\infty}}f(x)=0,$$ then


 * $$\sum _x f(x)=\sum_{n=0}^\infty\left(f(n)-f(n+x)\right)+ C.$$

Euler–Maclaurin formula

 * $$\sum _x f(x)= \int_0^x f(t) dt - \frac12 f(x)+\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(x) + C$$

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

Let


 * $$F(x)=\sum _x f(x)+C$$

Then the constant C is fixed from the condition


 * $$\int_0^1 F(x) \, dx=0 $$

or


 * $$\int_1^2 F(x) \, dx=0 $$

Alternatively, Ramanujan's sum can be used:


 * $$\sum_{x \ge 1}^{\Re}f(x)=-f(0)-F(0)$$

or at 1


 * $$\sum_{x \ge 1}^{\Re}f(x)=-F(1)$$

respectively

Summation by parts
Indefinite summation by parts:
 * $$\sum_x f(x)\Delta g(x)=f(x)g(x)-\sum_x (g(x)+\Delta g(x)) \Delta f(x) $$


 * $$\sum_x f(x)\Delta g(x)+\sum_x g(x)\Delta f(x)=f(x)g(x)-\sum_x \Delta f(x)\Delta g(x) $$

Definite summation by parts:
 * $$\sum_{i=a}^b f(i)\Delta g(i)=f(b+1)g(b+1)-f(a)g(a)-\sum_{i=a}^b g(i+1)\Delta f(i)$$

Period rules
If $$T $$ is a period of function $$f(x)$$ then


 * $$\sum _x f(Tx)=x f(Tx) + C$$

If $$T $$ is an antiperiod of function $$f(x)$$, that is $$f(x+T)=-f(x)$$ then


 * $$\sum _x f(Tx)=-\frac12 f(Tx) + C$$

Alternative usage
Some authors use the phrase "indefinite sum" to describe a sum in which the numerical value of the upper limit is not given:


 * $$\sum_{k=1}^n f(k).$$

In this case a closed form expression F(k) for the sum is a solution of


 * $$F(x+1) - F(x) = f(x+1) $$

which is called the telescoping equation. It is the inverse of the backward difference $$\nabla$$ 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

 * $$\sum _x a = ax + C $$
 * From which $$a$$ can be factored out, leaving 1, with the alternative form $$x^0$$. From that, we have:
 * $$\sum _x x^0 = \ x$$
 * For the sum below, remember $$x=x^1$$


 * $$\sum _x x = \frac{x(x+1)}{2} + C$$
 * For positive integer exponents Faulhaber's formula can be used. For negative integer exponents,
 * $$\sum _x \frac{1}{x^a} = \frac{(-1)^{a+1}\psi^{(a+1)}(x)}{a!}+ C,\,a\in\mathbb{Z}$$
 * where $$\psi^{(n)}(x)$$ is the polygamma function can be used.
 * More generally,


 * $$\sum _x x^a = \begin{cases}

- \zeta(-a, x+1) +C_1, &\text{if } a\neq-1 \\ \psi(x+1)+C_2, &\text{if } a=-1 \end{cases}$$
 * where $$\zeta(s,a)$$ is the Hurwitz zeta function and $$\psi(z)$$ is the Digamma function. $$C_1$$ and $$C_2$$ are constants which would normally be set to $$\zeta(-a)$$ (where $$\zeta(s)$$ is the Riemann zeta function) and the Euler–Mascheroni constant respectively. By replacing the variable $$a$$ with $$-a$$, this becomes the Generalized harmonic number. For the relation between the Hurwitz zeta and Polygamma functions, refer to Balanced polygamma function and Hurwitz zeta function.
 * From this, using $$\frac {\partial} {\partial a} \zeta (s,a) = -s\zeta(s+1,a)$$, another form can be obtained:
 * $$\sum _x x^a = \int_{0}^{x}-a\zeta(1-a, u+1)du +C, \text{ if } a\neq-1$$


 * $$\sum _x B_a(x)=(x-1)B_a(x)-\frac{a}{a+1} B_{a+1}(x)+C$$

Antidifferences of exponential functions

 * $$\sum _x a^x = \frac{a^x}{a-1} + C $$

Particularly,


 * $$\sum _x 2^x = 2^x + C $$

Antidifferences of logarithmic functions

 * $$\sum _x \log_b x = \log_b (x!) + C $$


 * $$\sum _x \log_b ax = \log_b (x!a^{x}) + C $$

Antidifferences of hyperbolic functions

 * $$\sum _x \sinh ax = \frac{1}{2} \operatorname{csch} \left(\frac{a}{2}\right) \cosh \left(\frac{a}{2} - a x\right) + C $$


 * $$\sum _x \cosh ax = \frac{1}{2} \operatorname{csch} \left(\frac{a}{2}\right) \sinh \left(ax-\frac{a}{2}\right) + C $$


 * $$\sum _x \tanh ax = \frac1a \psi _{e^a}\left(x-\frac{i \pi }{2 a}\right)+\frac1a \psi _{e^a}\left(x+\frac{i \pi }{2 a}\right)-x + C$$


 * where $$\psi_q(x)$$ is the q-digamma function.

Antidifferences of trigonometric functions

 * $$\sum _x \sin ax = -\frac{1}{2} \csc \left(\frac{a}{2}\right) \cos \left(\frac{a}{2}- ax \right) + C \,,\,\,a\ne 2n \pi $$


 * $$\sum _x \cos ax = \frac{1}{2} \csc \left(\frac{a}{2}\right) \sin \left(ax - \frac{a}{2}\right) + C \,,\,\,a\ne 2n \pi$$


 * $$\sum _x \sin^2 ax = \frac{x}{2} + \frac{1}{4} \csc (a) \sin (a-2ax) + C \, \,,\,\,a\ne n\pi$$


 * $$\sum _x \cos^2 ax = \frac{x}{2}-\frac{1}{4} \csc (a) \sin (a-2 a x) + C \,\,,\,\,a\ne n\pi$$


 * $$\sum_x \tan ax = i x-\frac1a \psi _{e^{2 i a}}\left(x-\frac{\pi }{2 a}\right) + C \,,\,\,a\ne \frac{n\pi}2$$


 * where $$\psi_q(x)$$ is the q-digamma function.


 * $$\sum_x \tan x=ix-\psi _{e^{2 i}}\left(x+\frac{\pi }{2}\right) + C = -\sum _{k=1}^{\infty } \left(\psi \left(k \pi -\frac{\pi }{2}+1-x\right)+\psi \left(k \pi -\frac{\pi }{2}+x\right)-\psi \left(k \pi -\frac{\pi }{2}+1\right)-\psi \left(k \pi -\frac{\pi }{2}\right)\right) + C$$


 * $$\sum_x \cot ax =-i x-\frac{i \psi _{e^{2 i a}}(x)}{a} + C \,,\,\,a\ne \frac{n\pi}2$$


 * $$\sum_x \operatorname{sinc} x=\operatorname{sinc}(x-1)\left(\frac{1}{2}+(x-1)\left(\ln(2)+\frac{\psi (\frac{x-1}{2})+\psi (\frac{1-x}{2})}{2}-\frac{\psi (x-1)+\psi (1-x)}{2}\right)\right) + C$$


 * where $$\operatorname{sinc} (x)$$ is the normalized sinc function.

Antidifferences of inverse hyperbolic functions

 * $$\sum_x \operatorname{artanh}\, a x =\frac{1}{2} \ln \left(\frac{\Gamma \left(x+\frac{1}{a}\right)}{\Gamma \left(x-\frac{1}{a}\right)}\right) + C$$

Antidifferences of inverse trigonometric functions

 * $$\sum_x \arctan a x = \frac{i}{2} \ln \left(\frac{\Gamma (x+\frac ia)}{ \Gamma (x-\frac ia)}\right)+C$$

Antidifferences of special functions

 * $$\sum _x \psi(x)=(x-1) \psi(x)-x+C $$


 * $$\sum _x \Gamma(x)=(-1)^{x+1}\Gamma(x)\frac{\Gamma(1-x,-1)}e+C$$


 * where $$\Gamma(s,x)$$ is the incomplete gamma function.


 * $$\sum _x (x)_a = \frac{(x)_{a+1}}{a+1}+C$$


 * where $$(x)_a$$ is the falling factorial.


 * $$\sum _x \operatorname{sexp}_a (x) = \ln_a \frac{(\operatorname{sexp}_a (x))'}{(\ln a)^x} + C $$
 * (see super-exponential function)