# MhSum

MhSum is notation for combination of functions suggested by
James David Nixon, in analogy with operation OmSum ^{[1]}.

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

## 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]