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15 Hypergeometric FunctionProperties

§15.13 Zeros

Let N(a,b,c) denote the number of zeros of F(a,b;c;z) in the sector |ph(1-z)|<π. If a, b, c are real, a, b, c, c-a, c-b0,-1,-2,, and, without loss of generality, ba, ca+b (compare (15.8.1)), then

15.13.1 N(a,b,c)={0,a>0,-a+12(1+S),a<0,c-a>0,-a+12(1+S)+a-c+1S,a<0,c-a<0,

where S=sign(Γ(a)Γ(b)Γ(c-a)Γ(c-b)).

If a, b, c, c-a, or c-b{0,-1,-2,}, then F(a,b;c;z) 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).