product functions

[bo198214]

despite that simple condition , its not quite easy to me how to find

f(z)/f(0) = 1 + a x + b x^2 + ... = (1 + a' x)(1 + b'x^2) ...

with both expressions converging in the same domain ( or all of C ).

Ya basically this is a polynomial approximation different from the usual powerseries one, which is easiest to handle.
I was aware of the different domain of convergence already when discussing the Mittag-Leffler expansion/star. This is a polynomial approximatino that converges in the star-region, instead of a disk for the powerseries approximation.

I wonder what approximation implies what region of convergence.
For example is there a polynomial approximation that implies a quadratic region of convergence? But perhaps this more a question for mathoverflow.