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Let f(x+1) = g(f(x)). With g real-entire.
What if the ROC around the point x is less than 1 , say 1/4 ?
Suppose we can get to the point x+1 by analytic continuation.
Does the equation still hold there ?
In case of "depends" , on what exactly does it depend ?
What are the classical examples of succes and failure ?

Regards

Tommy1729