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So I am working on a problem that involves an expression that has a "nested" sigma notation. Maybe "nested" isn't the correct word, because you start with an input (of your choosing) and the output becomes the input for the next iteration. So it's like a recursive sequence.

I am almost positive that this has been studied before. Not because I've heard of it, but because it's too simple of an idea to not have been studied before. (I could be wrong.)

If, instead of a series, we had a function, then this function "nested" 3 times would look like $f(f(f(x)))$. I am looking to generalize this to any type of function nested any number of times.

To give a simple example of what I mean: if we have $f(x)=ax+b$, then, $f(x)$ "nested" $N$ times with starting value $X$ can be generalized as $$Xa^N+b((1-a^N)/(1-a))$$ If anybody can provide with generalizations (or links to generalizations) for trigonometric, exponential, quadratic, polynomial, etc. functions, like the one I have provided, it would be of most help. Thank you.

EDIT: Also, if these types of generalizations have been made for series (in sigma notation would be even better) that would just be, as you say, "icing on the cake". :) Thanks.

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That kind of problem is studied in the discipline known as Discrete Dynamical Systems. Given a set $X$ (usually a subset of $\mathbb{R}^n$, a manifold,...) and a function $f\colon X\to X$, the $n$-th iterate of $f$ is defined as the composition of $f$ with itself $n$ times, and is represented as $$ f^n(x)=f(f(\dots f(x)\dots))=f\circ f\circ\dots\circ f(x). $$ The main purpose is the study of the behavior of the (forward) orbits of points $x\in X$, defined as the sequence $\{f^n(x)\}_{n=0}^\infty$. Only in special cases as your example $f(x)=ax+b$ is it possible to obtain a closed form for $f^n(x)$.

There are many books on Discrete Dynamical Systems. I recommend A First Course in Chaotic Dynamical Systems and An Introduction to Chaotic Dynamical Systems, both by Bob Devaney.

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Your answer was most helpful, especially the links to books on this specific subject. Thank you. –  Hautdesert Jul 3 '11 at 22:43

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