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indefinite sum
Recall that the finite difference operator $\Delta$ defined on the set of functions $\mathbb{R}\to\mathbb{R}$ is given by
$\Delta f(x):=f(x+1)f(x).$ 
The difference operator can be thought of as the discrete version of the derivative operator sending a function to its derivative (if it exists). With the derivative operation, there corresponds an inverse operation called the antiderivative, which, given a function $f$, finds its antiderivative $F$ so that the derivative of $F$ gives $f$. There is also a discrete analog of this inverse operation, and it is called the indefinite sum.
The indefinite sum of a function $f:\mathbb{R}\to\mathbb{R}$ is the set of functions
$\{F:\mathbb{R}\to\mathbb{R}\mid\Delta F=f\}.$ 
This set is often denoted by $\Delta^{{1}}f$ or $\Sigma f$, and any element in $\Delta^{{1}}f$ is called an indefinite sum of $f$.
Remark. Like the indefinite integral, the indefinite sum $\Delta^{{1}}$ is shift invariant. This means that for any $F\in\Delta^{{1}}f$, then $F+c\in\Delta^{{1}}f$ for any $c\in\mathbb{R}$. But, unlike the indefinite integral, the indefinite sum is also invariant by a shift of a periodic real function of period $1$. Conversely, the difference of two indefinite sums of a function $f$ is a periodic real function of period $1$.
In the following discussion, we consider the indefinite sum of a function as a function.
Basic Properties
1. $\Delta\Delta^{{1}}f=f$, and $\Delta^{{1}}\Delta f=f$ modulo a real function of period $1$.
2. Modulo a real number, and treating $\Delta^{{1}}$ as an operator taking a function into a function, we see that $\Delta^{{1}}$ is linear, that is,

$\Delta^{{1}}(rf)=r\Delta^{{1}}f$ for any $r\in\mathbb{R}$, and

$\Delta^{{1}}(f+g)=\Delta^{{1}}f+\Delta^{{1}}g$.

3. If $F(x)=\Delta^{{1}}f(x)$, then $F(x+a)=\Delta^{{1}}f(x+a)$.
4. If $F=\Delta^{{1}}f$, then we see that
$\displaystyle F(a+1)F(a)$ $\displaystyle=$ $\displaystyle f(a),$ $\displaystyle F(a+2)F(a+1)$ $\displaystyle=$ $\displaystyle f(a+1),$ $\displaystyle\vdots$ $\displaystyle F(x)F(x1)$ $\displaystyle=$ $\displaystyle f(x1).$ where $xa$ is a positive integer. Summing these expressions, we get
$F(x)F(a)=\sum_{{i=1}}^{{xa}}f(a+i1).$ This is the discrete version of the fundamental theorem of calculus.
Below is a table of some basic functions and their indefinite sums ($C$ is a realvalued periodic function with period $1$):
$f(x)$  $\Delta^{{1}}f(x)$  Comment 

$r\in\mathbb{R}$  $rx+C$  
$x$  $\displaystyle{\frac{x(x1)}{2}+C}$  
$x^{2}$  $\displaystyle{\frac{x(x1)(2x1)}{6}+C}$  
$x^{3}$  $\displaystyle{\frac{x^{2}(x1)^{2}}{4}+C}$  
$x^{n}$  $T_{n}(x)+C$  See this link for detail 
$a^{x}$  $\displaystyle{\frac{a^{x}}{a1}+C}$  $a\neq 1$ 
$(x)_{n}$  $\displaystyle{\frac{(x)_{n}}{n+1}+C}$  $(x)_{n}$ is the falling factorial of degree $n$ 
$\displaystyle{\binom{x}{n}}$  $\displaystyle{\binom{x}{n+1}+C}$  $\displaystyle{\binom{x}{n}:=\frac{(x)_{n}}{n!}}$ 
$\displaystyle{\frac{1}{x}}$  $\psi(x)+C$  $\psi(x)$ is the digamma function 
$\ln{x}$  $\ln{\Gamma(x)}+C$  $\Gamma(x)$ is the gamma function 
$\sin{x}$  $\displaystyle{\frac{\cos(x1/2)}{2\sin(1/2)}+C}$  
$\cos{x}$  $\displaystyle{\frac{\sin(x1/2)}{2\sin(1/2)}+C}$ 
References
 1 C. Jordan. Calculus of Finite Differences, third edition. Chelsea, New York (1965)
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