The past couple of evengings have been 'catch up on lambda' evenings. So I only just stumbled across one of the most surreal threads I think I've ever read. Old computer science and technical books worth searching for certainly looked promising. It started will, and I will definately have to keep an eye out for "ACM Turing Award Lectures: The First Twenty Years". Then someone mentioned generating functions and type derivatives, and before too long I end up ambushed with
The |x| < 1 condition is a red herring in this case. It is quite standard to work with generating functions in the context of the ring of formal power series k[[x]] with coefficients in a field k. (You can generalize the heck out of this definition.) A formal power series is a polynomial except for the fact that we don't require the elements of its coefficient sequence to become zero after a certain point. We define addition in this ring in the obvious way and multiplication by convolution of the coefficient sequences, just like you would in a polynomial ring.Ouch! My poor head. I still need warning and a good runup when approaching this sort of thing.