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) 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.