On the Zeros of Some Generalized Hypergeometric Functions

254
have
p Ž a .
KI AND KIM
j
Ł p p 1 p 1 p
j1 Ž bj
.
F Ž a ,...,a ; b ,...,b ; z.
ž ž / /
ž /
1
z Ýa jÝb ez j 1 O z, arg z . Ž 7.
z 2
Here arg is the principal branch of the argument. Hence F Ž z. p p has only a
finite number of zeros in the sector arg z 2 for each 0.
Next, we will show that if
Ž a1. n Ž ap
. n
Ž n 0,1,2,... . ,
b b
Ž . Ž .
1 n p n
and if a ,...,a are real, then F Ž z. 1 p p p has only a finite number of negative
real zeros. If some a is equal to zero or a negative integer, then F Ž z. j p p is a
polynomial, and if the condition I holds, then F Ž z. p p has only a finite
number of zeros, by Theorem 1. Hence we may assume, without loss of
generality, that a 1,...,a p 0, 1, 2, . . . and that the condition I does
not hold. Then the function Ž a s. Ž a s. Ž b s. Ž 1 p 1 bp
s. has infinitely many poles all of which are different from 0, 1, 2, . . . . Let
, , . . . denote the distinct poles of Ž a s. Ž a s. Ž 1 2 1 p b1
s. Ž b s . . For each k 1, 2, . . . let R Ž z. p k denote the residue of the
function
Ž a s. Ž a s.
Ž s.Ž z.
b s b s
1 p s
Ž . Ž .
1 p
Ž . s
at the pole s . Here z is defined by
k
s
z exp s log z
Ž . Ž .
for s and z 0; log is the principal branch of the logarithm. Then it
Ž . Ž . is easy to see that R z z k is a polynomial of logŽ z. k
for each
k 1, 2, . . . . See 6, p. 288 also.
Let A max Ima : j 1,..., p 4
j
, let K be an arbitrary positive real
number such that Re k K for all k 1, 2, . . . , and let C1 denote the
contour composed of the half line s t Ai, t K, the line
segment from K Ai to K Ai, and the half line s t Ai, K t