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Question: Why do we need to assume abs(z)<1 for evaluating hypergeometric series with abs(z)>1

Question:Why do we need to assume abs(z)<1 for evaluating hypergeometric series with abs(z)>1

Jesús Guillera 75 Maple
hypergeometric
March 30 2024
0

If we write

restart: alias(po=pochhammer):

assume(abs(z)<1);  F:=z->sum(po(1,n)^2/po(1/2,n)^2*z^n/(n+1),n=1..infinity);

Then the output of F(z) is a hypergeometric series, and

A=subs(z=4, F(z);

evalf(A)

gives the correct output. But, why do we need to assume that abs(z)<1 if we are evaluating at abs(z)>1?. Observe that if we do not assume it, then the output we get is wrong, why?
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