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 mgroff100@hotmail.com

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You can view the z-transform as a formal power series, of course, and most of the theory survives. But your question is stated so vaguely that it is just impossible to answer what happens in your particular case. The words "limit" and "overall function description" in the case when the convergence radius is $0$ raise a red flag in my mind but you may still be fine. All depends on the details. – fedja Jan 30 2010 at 10:07
If the radius of convergence is 0, you cannot invert the $z$-transform. The formula for inversion involves the Cauchy formula on a circle enclosing the origin, and keeping within the radius of convergence. What is the $z$-transform good for if you can't invert it? – Anweshi Jan 30 2010 at 11:31
But yes, as fedja says, it is possible to treat it as a formal power series and do the manipulations. – Anweshi Jan 30 2010 at 11:33
I cannot see what an answer to this question might be... You can surely make it more concrete. – Mariano Suárez-Alvarez Jan 30 2010 at 23:11
@Anweshi The use of a non-invertible z-transform is that you may wish to perform some transformation in "z-transform space" before inverting, and that transformation may leave you with something having non-zero radius of convergence. – Dan Piponi Jan 30 2010 at 23:22