§15.13 Zeros

Notes:: See Runckel (1971).
Keywords:: hypergeometric function
Permalink:: http://dlmf.nist.gov/15.13

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

Symbols:: $\left\lfloor x\right\rfloor$ : floor of $x$ , $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $N(a,b,c)$ : number of zeros and $S$
Permalink:: http://dlmf.nist.gov/15.13.E1
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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). A small table of zeros is given in Conde and Kalla (1981).