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closed form for sequence z=z^2+c & generating function
a closed form for the recursively defined (this is the term you are looking for) mandelbrot sequence is not known. this is often the case with recursively defined functions, so it is a cause for celebration when closed forms are found.
because of this difficulty, it is sometimes quite a chore to determine what the "regions of convergence" are. as the mandelbrot set demonstrates so stunningly, the nature of these sets can be quite complex, even if they admit of a simple description. for these kinds of tasks, a high-speed computer is your friend....
to give another example, the value to which "Newton's method" converges (if it does) depends on an initial guess (a choice of intial value). the boundaries of the regions of convergence, here as well, display a similar kind of behavior as the madelbrot set does, with large regions of "stable" behavior, and unpredictable behavior along the boundaries.
to answer your question about the "real version" of the madelbrot generating function, to get an idea of the behavior there, you might want to look here, and learn more about Julia sets (based on re-iterated polynomials).
