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 \(f\) refined at some set \(\mathcal G\) with values a 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}
Notation \( \OmSum \) 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 \( f_m(x) \).
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 &~=~& (s,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]
