# OmSum

OmSum is notation for combination of functions suggested by
James David Nixon ^{[1]}.

\( \newcommand{\OmSum} {\mathop{\Large\Omega} } \) 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.

## Contents

## 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
\(x \rightarrow f_m(s,z) \)

## References

- ↑ https://arxiv.org/abs/1910.05111v1 James David Nixon. \(Δy=e^sy\) or: How I Learned to Stop Worrying and Love the Γ-function [Submitted on 1 Oct 2019]