List of indefinite sums

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This is a list of indefinite sums (also known as antidifferences) of various functions. An indefinite sum $\sum _{x}f(x)$ is the inverse of the forward difference operator $\mathrm{\Delta }$, defined as $\mathrm{\Delta }f(x)=f(x+1)-f(x)$. It satisfies the relation

$\mathrm{\Delta }\sum _{x}f(x)=f(x).$

The operator is defined only up to an additive periodic function with period 1.

Antidifferences of rational functions

For positive integer exponents, Faulhaber's formula can be used. Note that $x$ in the result of Faulhaber's formula must be replaced with $x-1$ due to the offset, as Faulhaber's formula finds ${\mathrm{\nabla }}^{-1}$ rather than ${\mathrm{\Delta }}^{-1}$.

For negative integer exponents, the indefinite sum is closely related to the polygamma function:

$\sum _x\frac{1}{x^a}=\frac{(-1{)}^{a-1}{\psi }^{(a-1)}(x)}{(a-1)!}+C,a\in \mathbb{\mathbb{N}}$

For fractions not listed in this section, one may use the polygamma function with partial fraction decomposition. More generally,

$\sum _x{x}^a=\{\begin{array}{ll}\frac{B_{a+1}(x)}{a+1}+C,& \text{if }a\ne -1\\ \psi (x)+C,& \text{if }a=-1\end{array}=\{\begin{array}{ll}-\zeta (-a,x)+C,& \text{if }a\ne -1\\ \psi (x)+C,& \text{if }a=-1\end{array}$

where $B_a(x)$ are the Bernoulli polynomials, $\zeta (s,a)$ is the Hurwitz zeta function, and $\psi (z)$ is the digamma function. This is related to the generalized harmonic numbers.

As the generalized harmonic numbers use reciprocal powers, $a$ must be substituted for $-a$, and the most common form uses the inverse of the backward difference offset:

${\mathrm{\nabla }}^{-1}{x}^a=H_x^{(-a)}=\zeta (-a)-\zeta (-a,x+1).$

Here, $\zeta (-a)$ is the constant $C$.

$\frac{\partial }{\partial x}\left(\sum _x{x}^a\right)=B_a(x)=-a\zeta (1-a,x).$

This relationship can be expressed via the inverse backward difference operator as:

$\frac{\partial }{\partial x}\left({\mathrm{\nabla }}^{-1}{x}^a\right){|}_{x=0}=-a\zeta (1-a,x+1){|}_{x=0}=-a\zeta (1-a)=B_a.$

$\sum _x\frac{z^x}{(x+a{)}^s}=-z^x\mathrm{\Phi }(z,s,x+a)+C,$

which generalizes the generalized harmonic numbers as $z\mathrm{\Phi }(z,s,a+1)-z^{x+1}\mathrm{\Phi }(z,s,x+1+a)$ when taking ${\mathrm{\nabla }}^{-1}$.

Antidifferences of exponential functions

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

Antidifferences of logarithmic functions

$\sum _x{\log }_{b}x={\log }_b\mathrm{\Gamma }(x)+C$

$\sum _x{\log }_{b}ax={\log }_b(a^{x-1}\mathrm{\Gamma }(x))+C$

Antidifferences of hyperbolic functions

where ${\psi }_q(x)$ is the q-digamma function.

Antidifferences of trigonometric functions

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

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

$\sum _x\tan ax=ix-\frac{1}{a}{\psi }_{e^{2ia}}(x-\frac{\pi }{2a})+C,a\ne \frac{n\pi }{2}$

where ${\psi }_q(x)$ is the q-digamma function.

$$\begin{array}{rl}\sum _x\tan x& =ix-{\psi }_{e^{2i}}(x+\frac{\pi }{2})+C\\ & =-\sum _{k=1}^{\mathrm{\infty }}(\psi (k\pi -\frac{\pi }{2}+1-x)+\psi (k\pi -\frac{\pi }{2}+x)\\ & -\psi (k\pi -\frac{\pi }{2}+1)-\psi (k\pi -\frac{\pi }{2}))+C\end{array}:<math>\sum _x\cot ax=-ix-\frac{i{\psi }_{e^{2ia}}(x)}{a}+C,a\ne \frac{n\pi }{2}$$ The antidifference of the normalized sinc function can be obtained by applying the Abel–Plana formula presented in Candelpergher with the shift $x\mapsto x-1$, the condition $F(0)=0$, and recurrence of $F(x+1)-F(x)=f(x)$. Using the reflection formula for the digamma function, this simplifies to: $$\begin{array}{rl}\sum _x\mathrm{sinc}x& =\mathrm{sinc}(x-1)(\frac{1}{2}+(x-1)\\ & \times (\ln (2)+\frac{\psi (\frac{x-1}{2})+\psi (\frac{1-x}{2})}{2}\\ & -\frac{\psi (x-1)+\psi (1-x)}{2}))+\frac{1}{2}+C\end{array}$$

Period rules

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

$\sum _{x}f(Tx)=xf(Tx)+C.$

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

$\sum _{x}f(Tx)=-\frac{1}{2}f(Tx)+C.$

Antidifferences of special functions

where $\mathrm{\Gamma }(s,x)$ is the incomplete gamma function.

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

where $(x{)}_a$ is the falling factorial.

(see super-exponential function)