On the Zeros of Some Generalized Hypergeometric Functions

256
KI AND KIM
LEMMA. Let fŽ s. be a function which has a pole of order m 1 at
s , and let c ,...,c be arbitrary constants. Then either fŽ s. 1 m1
has
nonzero residue at s or there is a positie integer l m 1 such that
Ž s c . Ž s c . fŽ s. has nonzero residue at s .
1 l
From this lemma, it follows that if z 0, then for each k 1, 2, . . .
there is a nonnegative integer l such that the residue of
Ž a s. Ž a s.
Ž s l.Ž z.
b s b s
1 p s
Ž . Ž .
1 p
at s is not equal to zero. As a consequence, there is a non-negative
k
integer l such that if z 0, then the function
Ž . Ž .
Ž . Ž .
a l s a l s
1 p s
b l s b l s
1 p
Ž s.Ž z.
has a pole other than 0, 1, 2, . . . at which the residue is not equal to zero.
Since
F Žl. p p Ž a 1,...,a p; b 1,...,b p;
z.
Ž a1. l Ž ap
. l
pFpŽ a1l,...,a p l; b1 l,...,bp l; z . ,
b b
Ž . Ž .
1 l p l
we conclude that if a ,...,a are real, then some derivative of F Ž z. 1 p p p has
only a finite number of zeros in the sector argŽ z. 2 for each
0.
Now suppose that a ,...,a are real and that F Ž z. 1 p p p is a real entire
function, that is, its Maclaurin coefficients are all real. Then the above
Žl. argument implies that there is a non-negative integer l such that F Ž z.
p p
has only a finite number of negative real zeros, and hence Rolle’s theorem
implies that F Ž z. p p must have a finite number of negative real zeros. On
the other hand, Eq. Ž. 7 implies that F Ž z. p p has finitely many positive real
zeros. Consequently, F Ž z. has only a finite number of real zeros.
p p
THEOREM 2. Suppose that F Ž z. p p is a real entire function and a 1,...,a p
are real. Then F Ž z. has only a finite number of real zeros.
p p
COROLLARY. If F Ž z. p p is a real entire function and a 1,...,a p are real,
then R implies F.
As mentioned in the Introduction, Hille proved that if b ,...,b 0,
1 p
then I implies R, but he gave only a sketch of the proof. For completeness,