On the Zeros of Some Generalized Hypergeometric Functions

Journal of Mathematical Analysis and Applications 243, 249260 Ž 2000.
doi:10.1006jmaa.1999.6662, available online at http:www.idealibrary.com on
On the Zeros of Some Generalized
Hypergeometric Functions
Haseo Ki
Department of Mathematics, Yonsei Uniersity, Seoul 120-749, Korea
E-mail: haseo@bubble.yonsei.ac.kr
and
Young-One Kim
Department of Mathematics, Sejong Uniersity, Seoul 143-747, Korea
E-mail: kimyo@kunja.sejong.ac.kr
Submitted by William F. Ames
Received April 6, 1999
Let a 1,...,a p, b 1,...,bp be real constants with a 1,...,a p 0, 1, 2, . . . and
b ,...,b 0, and let F Ž z. F Ž a ,...,a ; b ,...,b ; z .
1 p p p p p 1 p 1 p . It is shown that the
following three conditions are equivalent to each other: Ž. i F Ž z. p p has only a finite
number of zeros, Ž ii. F Ž z. has only real zeros, and Ž iii. the a ’s can be re-indexed
p p j
so that a b m ,...,a b m for some nonnegative integers m ,...,m .
1 1 1 p p p 1 p
2000 Academic Press
Key Words: generalized hypergeometric function; multiplier sequence.
1. INTRODUCTION
This paper is concerned with the zeros of generalized hypergeometric
functions, namely the functions of the form
n
Ž a1. n Ž ap . z n
F Ž z. F Ž a ,...,a ; b ,...,b ; z. Ý
. Ž 1.
b b n!
p q p q 1 p 1 q
249
Ž . Ž .
n0 1 n q n
0022-247X00 $35.00
Copyright 2000 by Academic Press
All rights of reproduction in any form reserved.

250
KI AND KIM
Here a ,...,a and b ,...,b Ž 0, 1, 2, . . . . are constant and Ž a. 1 p 1 q n
Ž a n. Ž a . , that is,
Ž a. 0 1 and Ž a. n aŽ a1. Ž a n 1. Ž n 1,2,... . .
Observe that we must assume b 1,...,b10, 1, 2, . . . in order that the
series may be well defined, and that the series reduces to a polynomial
whenever some aj is equal to 0 or a negative integer.
In 1929, Hille published a paper entitled Note on some hypergeometric
series of higher order 3 . In this paper, he studied the zeros of the series Ž. 1
in the case where p q and the parameters a 1,...,a p and b 1,...,bq are
real. ŽNote that Ž 1. defines an entire function if and only if p q,
provided a ,...,a 0, 1, 2, . . . . . In particular, he proved that Ž i. 1 p
if
a 0, 1, 2, . . . , the necessary and sufficient condition for F Ž a; b; z.
1 1
to have only a finite number of zeros is that a b is a non-negative
integer, and that Ž ii. if b 1,...,bp 0 and m 1,...,m p are non-negative
integers, then F Ž b m ,...,b m ; b ,...,b ; z. p p 1 1 p p 1 p has real zeros only.
He proved Ž ii. in the special case where m1 m p,
but the same
method yields the general case.
The purpose of this paper is to generalize Hille’s results mentioned
above. For convenience, we introduce the following three conditions on
F Ž a ,...,a ; b ,...,b ; z . .
p p 1 p 1 p
F. F Ž a ,...,a ; b ,...,b ; z. p p 1 p 1 p has only a finite number of zeros.
R. F Ž a ,...,a ; b ,...,b ; z. has only real zeros.
p p 1 p 1 p
I. The a ’s can be re-indexed so that a b m ,...,a b m
j 1 1 1 p p p
for some non-negatie integers m ,...,m .
1 p
What Hille has proved are Ž. i if p 1 and a1 0, 1, 2, . . . , then F
and I are equivalent, and Ž ii. if b ,...,b 0, then I implies R. In this
1 p
paper, we first consider the general case when the parameters a 1,...,a p
and b 1,...,bq are allowed to have complex values, and prove that if
a ,...,a 0, 1, 2, . . . , then F and I are equivalent Ž
1 p
Theorem 1,
Section 2 . . Next, in Section 3, we apply a method due to Barnes to show
that F Ž z. is a real entire function and a ,...,a are real, then F Ž z. p p 1 p p p has
only a finite number of real zeros Ž Theorem 2 . . As a result of Theorem 1
and Theorem 2 we obtain Theorem 3 which states that if a ,...,a
1 p
Ž 0, 1, 2, . . . . are real and b 1,...,bp 0, then F, R, and I are
equivalent to each other. Finally, we conclude this paper with some
examples which show that the assumptions in our theorems cannot be
weakened Ž Section 4 . .

ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS 251
2. NUMBER OF ZEROS
In this section, we will prove the following theorem.
THEOREM 1. If a ,...,a 0, 1, 2,..., then F and I are equialent.
1 p
Proof. We first prove that I implies F. Let denote the differential
Ž . n
operator zddz , so that for each convergent power series Ý Anz and a
constant a we have
Ý Ž . n
n Ž
. Ý n
n
n0 n0
a n A z a A z .
If b 0, 1, 2, . . . and m is a non-negative integer, then
1F1Ž bm; b; z.
Ý
n0
Ž b m. n
n z
Ž b. n n!
n
1 z
Ý
Ž b n.Ž b1n. Ž b m 1 n.
Ž b. m n0
n!
n
1 z
Ž b .Ž b 1 . Ž b m 1 . Ý ,
Ž b. m n0 n!
Ž . Ž . z so that F bm; b; z P z e for some polynomial PŽ z. 1 1
of degree m.
In general, if b 1,...,bp 0, 1, 2, . . . and m 1,...,m p are non-negative
integers, then we have
pFpŽ b1m 1,...,bpm p; b 1,...,b p;
z.
m 1 n
p j
1 z
Ł Ł Ž bj k . Ý ,
b n!
Ž .
j1 j m k0
n0
j
Ž . Ž . z
and hence pFp b1m 1,...,bpm p; b 1,...,b p;
z P z e for some
polynomial PŽ z. of degree m1 m p.
This proves that I implies F.
Conversely, suppose that a ,...,a 0, 1, 2, . . . and that F Ž z. 1 p p p
F Ž a ,...,a ; b ,...,b ; z. p p 1 p 1 p has only a finite number of zeros. Then
F Ž z. p p is an entire function of order 1 with finitely many zeros. Therefore
F Ž z. can be written in the form
p p
Ž .
F z c cz c zN e z Ž . , Ž 2.
p p 0 1 N

252
KI AND KIM
where , c ,...,c are constants with , c 0. It is well known 1, p. 60
0 N N
and easy to see that F Ž z. satisfies the differential equation
p p
d
Ž b1 1. Ž bp 1. Ž a1. Ž ap. pFpŽ z. 0.
dz
Ž 3.
Ž. Ž.
From 2 and 3 , we obtain
p1 p c z Np lower terms 0,
Ž . N
Ž.
so that 0 or 1. Since 0, it follows that 1, and hence 2
becomes
Ž a . Ž a . z z
c cz c z .
n n
1 n p n N
Ý Ž 0 1 N . Ý
n0 Ž b1. n Ž bp. n! n0 n!
n
Therefore we have
Ž a1. n Ž ap . 1 c0 c1 c
n N
Ž b . b n! n! Ž n 1 . ! Ž n N . !
Ž .
1 n p n
Ž.
Let f t be the polynomial defined by
Ž n N, N 1,... . . Ž 4.
fŽ t. c0ct 1 c2tŽ t1. cNtŽ t1. Ž t N 1 . .
Ž.
Then 4 implies that
and hence we have
Ž a1. n Ž ap
. n
fŽ n. Ž nN, N 1,... . ,
b b
Ž . Ž .
1 n p n
Ž a n. Ž a n. Ž a n. Ž a n. Ž a1. Ž ap
.
fŽ n.
Ž b n. b n Ž b n. b n Ž b . b
1 p 1 p n n
Ž . Ž . Ž .
1 p 1 p 1 n p n
Ž a1. n1 Ž ap
. n1
fŽ n1. b b
Ž . Ž .
1 n1 p n1
Ž n N, N 1,... .
,

so that
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS 253
Ž .
Ž a n. Ž a n. fŽ n. Ž b n. b n fŽ n1. 1 p 1 p
Ž. Ž.
Since f t is a polynomial, Eq. 5 implies that
Ž . Ž .
Ž n N, N 1, ... . . Ž 5.
Ž a t. a t fŽ t. Ž b t. b t fŽ t1 . . Ž 6.
1 p 1 p
Now, using Ž. 6 and an induction, we will show that we can re-index
a 1,...,a p so that a1b 1,...,a pbp are non-negative integers. If p N
p deg fŽ. t 1, then we must have p 1 and N 0, so that our
assertion is trivial.
Let p N 1 and assume the induction hypothesis. Since b1 t is a
factor of the right-hand side of Ž. 6 , it is a factor of the left-hand side also.
Therefore b t divides Ž a t. Ž a t . ,orb t divides fŽ t .
1 1 p 1
.
If b t divides Ž a t. Ž a t .
1 1 p , then b1 ak for some k. Re-index
a ,...,a so that a b . Then Ž. 6 implies that
1 p 1 1
Ž . Ž .
Ž a t. a t fŽ t. Ž b t. b t fŽ t1 . .
2 p 2 p
Hence, by the induction hypothesis, we can re-index a ,...,a so that
2 p
a b ,...,a b are non-negative integers.
2 2 p p
If b t divides fŽ. t , then there is a polynomial f Ž.
1 1 t of degree N 1
such that fŽ. t Ž b t. f Ž. t . From Ž. 6 , we obtain
1 1
Ž . Ž .
Ž a t. a t f Ž t. Ž b t. b t fŽ t1. 1 p 1 2 p
Ž .
Ž b t. b t Ž b t 1. f Ž t1 . .
2 p 1 1
Again, the induction hypothesis implies that we can re-index a ,...,a
1 p
so that a b 1, a b ,...,a b are non-negative integers. This
1 1 2 2 p p
proves our assertion.
3. REALITY OF ZEROS
We start this section with an asymptotic equality due to Barnes 1 . Let
a 1,...,a p and b 1,...,bp be complex constants and assume that b 1,...,bp 0, 1, 2, . . . . In 1, pp. 8083 ,
it is shown that for each 0we

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

ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS 255
. Then we have
1 Ž a1 s. Ž aps. s
H
Ž s.Ž z. ds
2 i b s b s
Ž . Ž .
C1 1 p
Ž a1. Ž ap
.
pFpŽ z. Ý RkŽ z.
b b
Ž . Ž .
1 p K k
for Re z 0.
Let C2 denote the line s K it, t , and let 0be
arbitrary. If argŽ z.Im logŽ z. , then
Moreover, we have
so that
Ž a1 s. Ž aps. s
H Ž s.Ž z. ds
b s b s
Ž . Ž .
C1 1 p
Ž a1 s. Ž aps. s
H Ž s.Ž z. ds.
b s b s
Ž . Ž .
C2 1 p
Ž a s. Ž a s.
H s z dsO z b s b s
1 p s K
Ž . Ž .
C2 1 p
Ž a . Ž a .
2
Ž .Ž . Ž .
ž /
argŽ z. , z ,
2
Ž . Ý Ž . Ž .
1 p
pFp z Rk z O
K z 1 p K k
Ž . Ž .
b b
ž /
argŽ z. , z . Ž 8.
2
From Ž. 8 , it follows that if a ,...,a are real and if R Ž z. 1 p k 0 for some
k and z, then for each 0 the function F Ž z. p p has only a finite number
of zeros in the sector argŽ z. , because R Ž z. Ž z . 1,
R Ž z. 2
1 2
Ž . z 2, . . . are polynomials of logŽ z . .
Remark. It seems hardly true that R Ž z. k 0 for all k and z, but the
authors were unable to prove that this is not the case.
To proceed further, we need the following lemma whose proof is almost
trivial.

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,

ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS 257
we will give a detailed proof of the result. Our proof is based on the
PolyaSchur ´ theory of multiplier sequences.
4 A sequence of real numbers is said to be a multiplier sequence Ž
nn0
of
the first kind. if it has the following property.
M. If a az a zd 0 1 d is a real polynomial with real zeros only,
then the polynomial
also has real zeros only.
a az a z d
0 0 1 1 d d
It follows immediately from the definition that if 4 and 4
n n are
multiplier sequences, then their termwise product 4 n n is also a multi-
plier sequence.
The multiplier sequences are completely characterized by the following
theorem of Polya ´ and Schur 4 .
THE POLYASCHUR ´ THEOREM. A sequence 4 n of real numbers is a
multiplier sequence if and only if the function
n z
Ž z. Ý n
n0 n!
can be uniformly approximated on compact sets in the complex plane by a
sequence of real polynomials all of whose zeros are real and of the same sign.
Let b be a positive real number. Then the PolyaSchur ´ theorem implies
4 that b n is a multiplier sequence, because
n0
z z
Ž b n. Ž z b. e lim Ž z b. 1 .
n! k
k
n k
z
Ý ž /
n0
Since the class of multiplier sequences is closed under termwise multiplication,
it follows that for each positive integer m the sequence
Ž b n.Ž b1n. Ž b m 1 n. 4 n0
is a multiplier sequence. Hence
½ 5
½ 5
Ž b m. n
1
Ž b n.Ž b1n. Ž b m 1 n.
Ž b. Ž b.
n n0
m n0
is a multiplier sequence for each positive integer m.

258
KI AND KIM
Now suppose that b ,...,b are positive real numbers and m ,...,m
1 p 1 p
are nonnegative integers. Then it is clear that the sequence
½ Ž b1m1. n Ž bpmp. 5 n
Ž b . Ž b .
1 n p n n0
is a multiplier sequence. Hence, by the PolyaSchur ´ theorem, the function
pFpŽ b1m 1,...,bpm p; b 1,...,b p;
z.
Ý
Ž 1 1. Ž p p.
Ž . Ž .
b m b m z
n n
b b n!
n0 1 n p n
can be uniformly approximated on compact sets in the complex plane by a
sequence of real polynomials all of whose zeros are real and of the same
sign; in particular, it has only real zeros. This completes the proof.
THEOREM 3. If a ,...,a Ž 0, 1, 2, . . . .
1 p are real and b 1,...,bp 0,
then F, R, and I are equialent.
Proof. We have just shown that if b ,...,b 0, then I implies R.
1 p
Now, our theorem is an immediate consequence of Theorem 1 and the
corollary to Theorem 2.
4. EXAMPLES
In this section, we exhibit some examples which show that we cannot
weaken the assumptions of the theorems in this paper without altering
their conclusions. First of all, we remark that we must assume the
condition that a 1,...,a p 0, 1, 2, . . . in Theorem 1; otherwise the
function F Ž a ,...,a ; b ,...,b ; z. p p 1 p 1 p reduces to a polynomial, so that the
implication F I does not hold.
Theorem 2 states that if F Ž z. p p is a real entire function and a 1,...,a p
are real, then F Ž a ,...,a ; b ,...,b ; z. p p 1 p 1 p has only a finite number of real
zeros. The following example, however, shows that this is not the case in
general.
Ž . Ž .
EXAMPLE 1. The real entire function f z F i, i;1,1; z has in-
2 2
Ž.
finitely many real zeros. This may be shown as follows. According to 8 of
n

ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS 259
Section 3 or the asymptotic equality given in 1, Sect. 9 , we have
Ž . Ž . i 1
2 i 2i
f Ž t. t 1 OŽ t
2
.
e e Ž 1 i.
Ž .
Ž . Ž . i 1
i 2i
t Ž 1 OŽ t . . ,
2
Ž 1 i.
as t with t 0. Therefore there exist real constants A and B, with
A 0, such that
f t A cos B log t O t1 Ž . Ž . Ž . Ž t , t 0 . ,
and this proves our assertion.
Next, we consider the assumptions of Theorem 3, namely a 1,...,a p
0, 1, 2, . . . and b 1,...,bp 0. The following example is about the
condition a 1,...,a p 0, 1, 2, . . . .
EXAMPLE 2. Let b 0. Then
Ž x.
Ž b.
Ž b x.
is a real entire function of order 1 and has negative real zeros only. Hence
Ž . 14 Ž .4 b n is a complex zero decreasing sequence by
n n0 n0
2,
Theorem 1.4 .
In particular, it is a multiplier sequence. Let m be a positive
Ž . 14 integer. If we apply the multiplier sequence b n n0to
the polynomial
Ž . m 1 z , then we obtain F Ž m; b; z . , because
1 1
m n Ž m. z m
n 1
m n
1 1Ž . Ý Ý ž n / Ž .
n0 Ž b. n n! n0 Ž b.
n
F m; b; z z .
Therefore F Ž m; b; z. 1 1
has real zeros only for each m 1, 2, . . . and
b 0. Consequently, we must assume a 1,...,a p 0, 1, 2, . . . in order
to obtain the implication R I in Theorem 3.
Our last example is concerned with the condition b 1,...,bp 0.
EXAMPLE 3. Let b 0 and b 1, 2,... .If m is a positive integer
greater than b, then F Ž bm; b; z. has at least 2Ž 1 b. 2 1 1
nonreal
zeros.
Proof. Let m be a positive integer greater than b. Then Žb
. 14 m n n0is
a complex zero decreasing sequence. On the other hand, the
Ž . Ž . z
proof of Theorem 1 shows that F bm; b; z P z e for some real
1 1

260
KI AND KIM
Ž .
polynomial P z of degree m. Hence the number of nonreal zeros of
is at least that of
F Ž bm; b; z. Ý
1 1
n0
Ž b m. n
n z
Ž b. n n!
n n
1 z 1 Ž b m. n z
F Ž b; z. Ý Ý
.
Ž b. n! Ž b m. Ž b. n!
0 1
n0 n n0
n n
Now our assertion follows from a well-known theorem of Hurwitz 5, 15.27
which states that if b 0 and b 1, 2, . . . , then F Ž b; z. 0 1 has exactly
2Ž 1b. 2 nonreal zeros.
ACKNOWLEDGMENTS
Professor Askey informed us that Theorem 3 of this paper is not known. We truly thank
him for this. Professor Gasper conveyed many valuable facts to us. We thank him. We thank
Professor Csordas for pointing out the correct theorem on the complex zero decreasing
sequences. In the original version of this paper, we asserted that R Ž z. k 0 for some k and
z, and the referee pointed out that our proof of this may be invalid. We deeply thank the
referee for this as well as for the valuable comments. The authors acknowledge the financial
support of the Korea Research Foundation in the program year of Ž 19982000 . . The second
author was supported by the Korea Science and Engineering Foundation Ž KOSEF. through
the Global Analysis Research Center Ž GARC. at Seoul National University.
REFERENCES
1. E. W. Barnes, The asymptotic expansion of integral functions defined by generalised
hypergeometric series, Proc. London Math. Soc. Ž 2. 5 Ž 1907 . , 59116.
2. T. Craven and G. Csordas, Complex zero decreasing sequences, Methods Appl. Anal. 2
Ž 1995 . , 420441.
3. E. Hille, Note on some hypergeometric series of higher order, J. London Math. Soc. 4
Ž 1929 . , 5054.
4. G. Polya ´ and J. Schur, Uber ¨ zwei Arten von Faktorenfolgen in der Theorie der algebraischen
Gleichungen, J. Reine Angew. Math. 144 Ž 1914 . , 89113.
5. G. N. Watson, ‘‘Theory of Bessel Functions,’’ 2nd ed., Cambridge Univ. Press, London,
1952.
6. E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, J.
London Math. Soc. 10 Ž 1935 . , 286293.