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

$\begingroup$ 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 ${} > 1$. $\endgroup$– Gerald EdgarCommented Apr 26, 2022 at 14:02

$\begingroup$ no a is strictly positive $\endgroup$– Assinisa HamidataCommented May 22, 2022 at 12:43
2 Answers
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,
Uber die Nullstellen der hypergeometrischen Reihe, Math. Ann., 37 (1890) 573590.
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:
Van Vleck, Edward B. A determination of the number of real and imaginary roots of the hypergeometric series. Trans. Amer. Math. Soc. 3 (1902), no. 1, 110–131.
For complex parameters, only partial results are available.