Is there a multiple function composition operator?

Question

Is there a commonly-accepted operator which defines multiple function composition? I have not been able to find one on any of the related Wikipedia pages. In one of my proofs, I've been finding it very tiresome to continually write the following: $$ f_n\circ f_{n-1}\circ\cdots\circ f_0 $$ And I would really appreciate finding out if there is a common notation for what I'm trying to do. The above notation only works for a sequence of functions; I imagine there's probably a way of using this composition operator to compose over an arbitrary ordered set, sort of like $\bigcup_{i\in I}A_i$ allows a union over any (not necessarily countable) indexing set $I$, but without the ordering condition.

I'd use foldl1 (.), but that doesn't seem very appropriate for math homework.

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I feel like I should add why I feel like one is necessary, on top of the additional expressiveness we gain from being able to compose over arbitrary ordered sets. The problem itself defines $F_n$ as the above repeated-composition function; indeed, we can define $F_n$ in a recursive manner with $F_0=f_0$ and $F_{n+1}=f_{n+1}\circ F_n$. However, the issue remains because in my proof I frequently need to expand $F_n$ (for instance, calculating $F_n'$), in which case such an operator would come in handy.

Answer 1

If I were writing something in which I had to do a large number of these, the following is probably not quite what I would do: $$ \mathop{\bigcirc}^n_{k=0} f_k \quad \text{ or } \quad \mathop{\bigcirc}^0_{k=n} f_k \ . $$ Instead, I'd go over to tex.stackexchange.com and ask how to make this thing look respectable instead of like a workaround. I'd probably want it to be comparable in size and boldness to something like $\displaystyle\bigcap$ in $\displaystyle\bigcap_{k=0}^n A_k$ or to $\displaystyle\bigoplus$. Before the \begin{document} I'd put

\newcommand{\Circ}{blah blah blah}

with a capital C distinguishing it from \circ).