Is it always allowed to perform a z transform with radius of convergence 0? I'm looking for a way to use limits of a generating function or related (as the power approaches a certain value, using only the overall function description without relying on a function for individual coefficients) in recursive equations. One possibility seems to be to use a specific value as a dummy variable. However, I would prefer to use a z-transform.
More background: I am attempting to extract the coefficient that is the middle in a series. Not much information is known about the series; I am working in the general case and its radius of convergence could be zero. My approach was to attempt to use an integral and differentiation similar to the z-transform. This should allow one to extract coefficients using differences of manipulated generating functions, one at-a-time. However, it seems that I can't use a contour integral if the r.o.c. is zero with garaunteed correctness.
I guess I'm really asking if there is any method that would allow one to extract coefficients, like taking the limit as the function approaches a given coefficient. I'd like to be able to handle this procedure in a recursive fashion, and it seems that this is impossible.
I don't want to waste anyone's time, but any help would be greatly appreciated. Email is welcomed at firstname.lastname@example.org