Definition:Summation/Indexed

Standard Number System
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.

Let $a, b \in \Z$ be integers.

Let $\left[{a \,.\,.\, b}\right]$ be the integer interval between $a$ and $b$.

Let $f : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a mapping.

The indexed summation of $f$ from $a$ to $b$ is recursively defined and denoted:


 * $\displaystyle \sum_{k \mathop = a}^b f(k) = \begin{cases} 0 & : b < a \\ \left( \displaystyle \sum_{k \mathop = a}^{b-1} f(k) \right) + f(b) & : b \geq a\end{cases}$

Abelian Group
Let $\left({G, +}\right)$ be a abelian group with identity element $0$.

Let $a, b \in \Z$ be integers.

Let $\left[{a \,.\,.\, b}\right]$ be the integer interval between $a$ and $b$.

Let $f : \left[{a \,.\,.\, b}\right] \to G$ be a mapping.

The indexed summation of $f$ from $a$ to $b$ is the indexed iteration of $+$ on $f$ from $a$ to $b$. That is:
 * $\displaystyle \sum_{k \mathop = a}^b f(k) = \begin{cases} 0 & : b < a \\ \left( \displaystyle \sum_{k \mathop = a}^{b-1} f(k) \right) + f(b) & : b \geq a\end{cases}$

Also see

 * Change of Variables in Indexed Summation
 * Definition:Indexed Product

Generalizations

 * Definition:Summation over Finite Set, as shown at Summation over Interval equals Indexed Summation
 * Definition:Indexed Iterated Binary Operation