Recurrence relation given by a polynomial

Question

Consider non-homogeneous, first order, nonlinear recurrence $$a(n+1)=P(a(n)),$$ where $P$ is a given polynomial.

Are there any general solutions to such a problem? How about generalization to exponential polynomials?

Answer 1

With few exceptions, this does not have a closed-form general solution, even when $P$ is a quadratic polynomial.