# §15.13 Zeros

Let $N(a,b,c)$ denote the number of zeros of $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits\!\left(1-z\right)|<\pi$. If $a$, $b$, $c$ are real, $a$, $b$, $c$, $c-a$, $c-b\neq 0,-1,-2,\dots$, and, without loss of generality, $b\geq a$, $c\geq a+b$ (compare (15.8.1)), then

 15.13.1 $N(a,b,c)=\begin{cases}0,&a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S),&a<0,c-a>0,\\ \left\lfloor-a\right\rfloor+\tfrac{1}{2}(1+S)+\left\lfloor a-c+1\right\rfloor S% ,&a<0,c-a<0,\\ \end{cases}$

where $S=\mathop{\mathrm{sign}\/}\nolimits\!\left(\mathop{\Gamma\/}\nolimits\!\left(a% \right)\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!% \left(c-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)\right)$.

If $a$, $b$, $c$, $c-a$, or $c-b\in\{0,-1,-2,\dots\}$, then $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ is not defined, or reduces to a polynomial, or reduces to $(1-z)^{c-a-b}$ times a polynomial.

For further information on the location of real zeros see Zarzo et al. (1995) and Dominici et al. (2013). A small table of zeros is given in Conde and Kalla (1981) and Segura (2008).