Zeros of hypergeometric functions with complex variables

Question

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

Answer 1

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...

Answer 2

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) 573-590.

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.