OmSum

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

Keywords

Iterate, MathJax, MhSum, OmSum, Superfunction