# Zeros of hypergeometric functions with complex variables

Let $$z$$ be a complex number and let $$a,b,c > 0$$. I would like to know the zeros of the following hypergeometric function: $$_{2}F_{1} (a,b; c :z )=\sum_{k=0}^{+ \infty} \frac{(a)_{k}(b)_{k}}{(c)_{k}} \frac{z^{k}}{k!}$$ i.e, i want to identify the set $$A$$ such $$_{2}F_{1} (a,b; c :z )=0$$ for all $$z \in A$$. For instance, consider $$t$$ as real number where $$_{2}F_{1} (a,b; c :t )=0$$, it is obvious in that case that $$t$$ would be a negative real number. Is there a paper or an article or a suggestion salving that problem

• The radius of convergence is $1$. (Unless the series terminates, say $a=-n$ as in Carlo's answer.) If you consider analytic continuation (as suggested by Alexandre) it could be a multivalued function. And you could have a zero which is real and ${} > 1$. Commented Apr 26, 2022 at 14:02
• no a is strictly positive Commented May 22, 2022 at 12:43

Much of the literature addresses the case that $$a=-n$$ is a negative integer, see Real zeros of $$2F1$$ hypergeometric polynomials (2013) and Zeros of the hypergeometric polynomial $$F(-n,b;c;z)$$ (2008). Even that case has not been fully solved...
The hypergeometric function is not a function in the usual meaning of this word: it is multivalued. For real parameters $$a,b,c$$, the question about real zeros of real branches was investigated by Klein,
He determined exactly, how many zeros are there on the intervals $$(0,1)$$, $$(1,+\infty)$$ and $$(-\infty,0)$$. Later A. Hurwitz gave simpler proofs, and E. van Vleck generalized Klein's result to complex zeros: