15 Hypergeometric Function Properties 15.12 Asymptotic Approximations 15.14 Integrals

§15.13 Zeros

ⓘ

Keywords:: hypergeometric function, zeros
Notes:: See Runckel (1971).
Referenced by:: Erratum (V1.0.10) for References, Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/15.13
Addition (effective with 1.0.10):: A reference to Dominici et al. (2013) was added at the end of this section.
Addition (effective with 1.0.5):: The reference to Segura (2008) has been added at the end of this section.
See also:: Annotations for Ch.15

Let $N(a,b,c)$ denote the number of zeros of $F\left(a,b;c;z\right)$ in the sector $|\operatorname{ph}\left(1-z\right)|<\pi$ . 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: $\left\lfloor\NVar{x}\right\rfloor$ : 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 Encodings: TeX, pMML, png See also: Annotations for §15.13 and Ch.15

where $S=\operatorname{sign}\left(\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left% (c-a\right)\Gamma\left(c-b\right)\right)$ .

If $a$ , $b$ , $c$ , $c-a$ , or $c-b\in\{0,-1,-2,\dots\}$ , then $F\left(a,b;c;z\right)$ 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).