15 Hypergeometric Function Properties 15.12 Asymptotic Approximations 15.14 Integrals

§15.13 Zeros

Notes:: See Runckel (1971).
Keywords:: hypergeometric function
Referenced by:: Other Changes
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Addition (effective with 1.0.5):: The reference to Segura (2008) has been added at the end of this subsection.
Reported 2012-07-30

Let 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 , , are real, , , , , $c-b\neq 0,-1,-2,\dots$ , 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 , : real or complex parameter, : real or complex parameter, : real or complex parameter, : number of zeros and
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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 , , , , 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) and Segura (2008).

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