I have searched the internet for methods on solving functional equations, unfortunately, most of them consist mainly of substituting values for the variables or on guessing solutions. I think those methods sound like either an impossible task due to the infinite number of possible substitutions or a leap of faith due also to the infinite number of guesses there can be.
How could one for example solve:
$$f(1+x)=1+f(x)^2$$
I have tried many different ideas on this, from trying guesses to applying integrals, to eliminating the $1$ by calculating $f(2+x)$ and subtracting equations, then trying to simplify the answer through sums or products, but nothing seems to work.
I think this is a rather interesting subject because $f(x)=f(x-1)^2$ has a solution that is very easy to find but somehow adding a $1$ makes it a much harder problem. Are there ways to solve this equation or even a polynomial functional equation?
[Math] What are some standard methods for solving functional equations
functional-equationsfunctions
Best Answer
As for iterating polynomials, there are two general forms of polynomials that we currently know how to iterate. They are polynomials of the form $$P(x)=a(x-b)^d+b$$ and $$T(x)=\cos(k\arccos(x))$$ and yes, I know the second one doesn't look like a polynomial, but for integer values of $k$, it is a polynomial where the inverse cosine is defined, and can be extended to other values (accurately, for our purposes) using the hyperbolic cosine function. For more info, see Chebyshev Polynomials.
The iteration formulas for each of these is given by $$P^n(x)=a^{\frac{b^n-1}{b-1}}(x-b)^{d^n}+b$$ and $$T^n(x)=\cos(k^n\arccos(x))$$
In case you were wondering how this applies to your problem of finding $f$ given that $$f(1+n)=(Q\circ f)(n)$$ where $Q$ is a polynomial, here's how: if you assign a value for $f(0)$, say $y_0$, then you can say that $$f(n)=(Q^n\circ f)(0)=Q^n(y_0)$$ which allows you to find $f:\mathbb Z\to\mathbb Z$.