# OmSum

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OmSum is notation for combination of functions suggested by James David Nixon .

$$\newcommand{\OmSum} {\mathop{\Large\Omega} }$$ He uses the compact notations $$\mathrm{OmSum}=\OmSum$$.

Operation OmSum appears as generalization of iterate; during the iteration, the shape of the iterand (iterated function) may change.

## Definition

Definition. For any set of numbered functions $$f$$ defined at some set $$\mathcal G$$ with values in the same set,

\begin{eqnarray} \OmSum_{j=n}^n f_j = f_n \label{o} \end{eqnarray}

for $$m>n$$, for any $$z\in \mathcal G$$,

\begin{eqnarray} \OmSum_{j=n}^m f_j ~ (z)= \OmSum_{j=n}^{m-1} f_j ~ (f_m(z)) \label{om} \end{eqnarray}

Operation $$\OmSum$$ has high priority. In the last expression of equation (\ref{om}), first the operation OmSum is performed; then, the resulting function is evaluated at the specified value, id est, at $$f_m(z)$$.

## Notations

Sometimes, the iterated function should be indicated with its argument (for example, if there is some additional parameter), \begin{eqnarray} f_j(z)=f_j(s,z) \end{eqnarray}

Then, the dummy argument can be indicated after the $$\bullet$$:

\begin{eqnarray} \OmSum_{j=n}^n f_j(s,x) \bullet x &~=~& z\!\mapsto\! f_n(s,z) \\ \OmSum_{j=n}^m f_j(s,x) \bullet x ~ (z)&~=~& \OmSum_{j=n}^{m-1} f_j(s,x)\bullet x ~ ~ (f_m(s,z)) \label{bul} \end{eqnarray}

Again, due to priority, in the right hand side of equation (\ref{bul}), first, expression
$$\displaystyle\OmSum_{j=n}^{m-1} f_j(s,z)\bullet z$$
should be calculated; then, the resulting function should be evaluated at argument specified; the dummy parameter $$x$$ should be replaced $$x \rightarrow f_m(s,z)$$