# 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.