TITLE: Notation for repeated composition of functions
QUESTION [9 upvotes]: I have a repeated composition of functions ${T_n}(z) = {\tau _0} \circ {\tau _1} \circ {\tau _2} \circ  \cdots  \circ {\tau _n}(z)$
By analogy with $\sum\limits_{i = 1}^n {} ,\prod\limits_{i = 1}^n {} ,\bigcup\limits_{i = 1}^n {} ,\bigcap\limits_{i = 1}^n {} ,$
I want to write ${T_n}(z) = \left( {\mathop  \circ \limits_{i = 0}^n {\tau _i}} \right)(z)$ or even ${T_n}(z) = {\mathop  \circ \limits_{i = 0}^n {\tau _i}} (z)$. Can I do this?

REPLY [4 votes]: Iterated summation and product are represented by the Greek uppercase letters sigma and pi. As we can see, there is a clear phoneticity here.
The only phonetic Greek equivalent of 'c' in 'composition' is kappa.
But as uppercase kappa resembles 'K', and is already used for continued fractions in some texts, we might use the lowercase kappa '$\kappa$' for our purpose:
$$\underset{i=1}{\overset{n}{\Huge{\kappa}}} f_i \left( x \right) \equiv f_n \left( f_{n-1} \left( \dots f_2 \left( f_1 \left( x \right) \right) \dots \right) \right)$$
Let's see an example where we can use this notation. Suppose we were to find the derivative of the composition of $n$ functions w.r.t. the variable they act on, say $x$ :
$\frac{d}{dx} f_n \left( f_{n-1} \left( \dots f_2 \left( f_1 \left( x \right) \right) \dots \right) \right)$
The derivative as computed using the chain rule recursively turns out to be,
$f_n^\prime \left( f_{n-1} \left( f_{n-2} \left( \dots f_2 \left( f_1 \left( x \right) \right) \dots \right) \right) \right) \cdot f_{n-1}^\prime \left( f_{n-2} \left( f_{n-3} \left( \dots f_2 \left( f_1 \left( x \right) \right) \dots \right) \right) \right) \dotsm f_2^\prime \left( f_1 \left( x \right) \right) \cdot f_1^\prime \left( x \right)$
But using the notations for iterative product and composition using $\prod$ and $\huge{\kappa}$ respectively, the expressions become much more compact:
$$\displaystyle{ \frac{d}{dx} \: \underset{i=1}{\overset{n}{\Huge{\kappa}}} f_i \left( x \right) = \prod_{i=1}^n f_i^\prime \left( \underset{j=1}{\overset{i-1}{\Huge{\kappa}}} f_j \left( x \right) \right) }$$