MhSum

MhSum is notation for combination of functions suggested by James David Nixon, in analogy with operation OmSum .

\( \newcommand{\OmSum}{\mathop{\Large\Omega}} \newcommand{\MhSum}{\mathop{\Large\mho}} \) He uses the compact notations \(~ \mathrm{OmSum}\!=\!\OmSum ~\), \(~ \mathrm{MhSum}\!=\!\MhSum ~\).

Operations OmSum and mhSum appear as generalization of iterate; during the iteration, the shape of the iterand (iterated function) may change.

Definition

For any set of numbered functions \(f\) defined at some set \(\mathcal G\) with values in the same set,

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

for \(m>n\), for any \(z\in \mathcal G\), \begin{eqnarray} \MhSum_{j=n}^m f_j ~ (z)= f_m\! \left(\MhSum_{j=n}^{m-1} f_j ~ (z)\!\right) \end{eqnarray}

The last statement can be rewritten also as follows:

for \(m>n\), \begin{eqnarray} \MhSum_{j=n}^m f_j ~ &=& ~ z \!\mapsto\! f_m\!\left(\MhSum_{j=n}^{m-1} f_j ~ (z)\!\right) \end{eqnarray}

Analogy with iterates

In the special case, when \(f\) does not depend on its subscript \(f_j=f_n=f\), the MhSum becomes just iterate of function \(f\):

\begin{eqnarray} \MhSum_{j=n}^m f_j ~ &=&\OmSum_{j=n}^m f_j ~ ~=~ f^{m-n}~ = z \!\mapsto\! f\!\left(\MhSum_{j=n}^{m-1} f ~ (z)\!\right) \end{eqnarray}

The \(k\)th iterate \(f^k\) of function \(f\) can be written as follows: \begin{eqnarray} f^0(z)&=&z \\ f^1(z)&=&f(z) \\ ..\\ f^k(z)&=&f(f^{k-1}(z))=F(k+G(z)) \end{eqnarray} were \(F\) is superfunction of \(f\) and \(G=F^{-1}\) is the abelfunction.

The challenging goal is generalization of the definition of \(\MhSum\) to the non-integer values of subscript, in analogy with iterate. Some analogies of the superfunction and the abelfunction should be defined, constructed and implemented for the numerical evaluation.

Keywords

Iterate, MathJax, MhSum, OmSum, Superfunction