Recursive application of a function : Symbol of

Question

I need to apply a function $f(x)$ recursively/repeatedly for n times; how do I express it (mathematically) ?

Is their a mathematical symbol which denotes $f(x)$ applied n times ie
$g(x,n)$ =$f(f(\dots_{ times \quad n}(x)))\dots)$

** With a variable parameter **
$h[x,1,n)]$ =$f(1,f(2,\dots_{ times \quad n}(n,x)))\dots)$
here the variable parameter say i varies from 1 to n for each successive application of $f()$ starting with 1 for the first (outermost) instance to n for the innermost.

In this context would using " $h[x,i,j]$ " be correct or may be $f_{i,j}(x)$ or $_ {_i}f_j(x,n)$?

$f^n(x)$ is a common notation. Sometimes it can be confused with $\bigl(f(x)\bigr)^n$, in such cases, make a remark what is meant. — Daniel Fischer, Jul 27 '13 at 17:52

score 5 · Accepted Answer · 2013-07-27T17:57:08.450

5

Some common notations for this are

$$ f^n(x)$$

$$ f^{(n)}(x)$$

$$ f^{\circ n}(x)$$

$$\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}}$$

If you use either of the first two, the first time it appears it's probably worth explaining that you are using it for repeated composition, just to make sure the reader knows what you mean. Maybe the same is true for the third as well.

If you only ever use the notation once in a paper (e.g. because you use your $g(x,n)$ everywhere else), the last choice is probably best.

edited Jul 27 '13 at 17:57

answered Jul 27 '13 at 17:53

Thanks The last line answers my question. – ARi Jul 27 '13 at 18:03
I have also seen $f^{[n]}$ – GEdgar Jul 27 '13 at 18:50
But how to use it with a variable parameter ? – ARi Jul 27 '13 at 19:08

Recursive application of a function : Symbol of

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