ABSTRACT
Abstract It is shown that the Gauss hypergeometric series F (−n, 2m; 2m+ n+ 1; z) has roots outside the unit disk.
It is known in [10] that the normed conjugate product of gamma functions such as
pi Γ(1− ix)Γ(1 + ix) = 2
pi
Π∞n=1(1 + x2/n2) (1)
is an infinitely divisible density. In the process showing the infinite divisibility of the probability distribution with density (1), a family of polynomials with roots outside the unit disk appeared. From the infinite divisibility of the above probability distribution and from numerical analysis of roots of the terminating hypergeometric series, we conjectured that the following density function consisting of the normed conjugate product of gamma functions is an infinitely divisible density:
c
∣∣∣∣Γ(m+ ix)Γ(m) ∣∣∣∣2 = cΠ∞k=0(1 + x2/(m+ k)2) (m ∈ N) (2)
(cf. [1. 6.1.25]) In this case the Gauss hypergeometric series F (−n, 2m; 2m+ n + 1; z) appears in general form and it is much more complicated than the case m = 1. In this paper we show that the Gauss hypergeometric series F (−n, 2m; 2m+n+ 1; z) has roots outside the unit disk. This result will play an important part in the proof of the infinite divisibility of the probability distribution with density function (2).
