On the zeros of some generalized hypergeometric functions

Let a(1),...,a(p), b(1),..., b(p) be real constants with a(1),...,a(p) not equal 0, -1, -2,... and b(1),..., b(p) > 0, and let F-p(p)(z) =F-p(p)(a(1), ...,a(p); b(1),...,b(p); z). It is shown that the following three condition s are equivalent to each other: (i)(p)F-p(z) has only a finite number of ze ros, (ii),F,(z) has only real zeros, and (iii) the a(j)'s can be re-indexed so that a(1) = b(1) + m(1),..., a(p) = b(p) + m(p) for some nonnegative in tegers m(1),..., m(p). (C) 2000 Academic Press.