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On the Zeros of Some Generalized Hypergeometric Functions
Journal of Mathematical Analysis and Applications 243, 249᎐260 Ž2000. doi:10.1006rjmaa.1999.6662, available online at http:rrwww.idealibrary.com on
On the Zeros of Some Generalized Hypergeometric Functions Haseo Ki Department of Mathematics, Yonsei Uni¨ ersity, Seoul 120-749, Korea E-mail: [email protected]
and Young-One Kim Department of Mathematics, Sejong Uni¨ ersity, Seoul 143-747, Korea E-mail: [email protected] Submitted by William F. Ames Received April 6, 1999
Let a1 , . . . , a p , b1 , . . . , bp be real constants with a1 , . . . , a p / 0, y1, y2, . . . and b1 , . . . , bp ) 0, and let p Fp Ž z . sp Fp Ž a1 , . . . , a p ; b1 , . . . , bp ; z .. It is shown that the following three conditions are equivalent to each other: Ži. p Fp Ž z . has only a finite number of zeros, Žii. p Fp Ž z . has only real zeros, and Žiii. the a j ’s can be re-indexed so that a1 s b1 q m1 , . . . , a p s bp q m p for some nonnegative integers m1 , . . . , m p . 䊚 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
p Fq Ž z . sp Fq Ž a1 , . . . , a p ; b 1 , . . . , bq ; z . s
⬁
Ý ns0
Ž a1 . n ⭈⭈⭈ Ž a p . n z n . Ž 1. Ž b1 . n ⭈⭈⭈ Ž bq . n n!
249 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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KI AND KIM
Here a1 , . . . , a p and b1 , . . . , bq Ž/ 0, y1, y2, . . . . are constant and Ž a. n s ⌫ Ž a q n.r⌫ Ž a., that is,
Ž a. 0 s 1 and Ž a. n s aŽ a q 1 . ⭈⭈⭈ Ž a q n y 1 .
Ž n s 1, 2, . . . . .
Observe that we must assume b1 , . . . , b1 / 0, y1, y2, . . . in order that the series may be well defined, and that the series reduces to a polynomial whenever some a j is equal to 0 or a negative integer. In 1929, Hille published a paper entitled Note on some hypergeometric series of higher order w3x. In this paper, he studied the zeros of the series Ž1. in the case where p F q and the parameters a1 , . . . , a p and b1 , . . . , bq are real. ŽNote that Ž1. defines an entire function if and only if p F q, provided a1 , . . . , a p / 0, y1, y2, . . . .. In particular, he proved that Ži. if a / 0, y1, y2, . . . , the necessary and sufficient condition for 1 F1Ž a; b; z . to have only a finite number of zeros is that a y b is a non-negative integer, and that Žii. if b1 , . . . , bp ) 0 and m1 , . . . , m p are non-negative integers, then p Fp Ž b1 q m1 , . . . , bp q m p ; b1 , . . . , bp ; z . has real zeros only. He proved Žii. in the special case where m1 s ⭈⭈⭈ s 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 Ž . p Fp a1 , . . . , a p ; b 1 , . . . , bp ; z . Ž a1 , . . . , a p ; b1 , . . . , bp ; z . has only a finite number of zeros. R. p Fp Ž a1 , . . . , a p ; b1 , . . . , bp ; z . has only real zeros. F.
p Fp
I. The a j ’s can be re-indexed so that a1 s b1 q m1 , . . . , a p s bp q m p for some non-negati¨ e integers m1 , . . . , m p . What Hille has proved are Ži. if p s 1 and a1 / 0, y1, y2, . . . , then F and I are equivalent, and Žii. if b1 , . . . , bp ) 0, then I implies R. In this paper, we first consider the general case when the parameters a1 , . . . , a p and b1 , . . . , bq are allowed to have complex values, and prove that if a1 , . . . , a p / 0, y1, y2, . . . , then F and I are equivalent ŽTheorem 1, Section 2.. Next, in Section 3, we apply a method due to Barnes to show that p Fp Ž z . is a real entire function and a1 , . . . , a p are real, then p Fp Ž z . has only a finite number of real zeros ŽTheorem 2.. As a result of Theorem 1 and Theorem 2 we obtain Theorem 3 which states that if a1 , . . . , a p Ž/ 0, y1, y2, . . . . are real and b1 , . . . , 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 ŽSection 4..
251
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
2. NUMBER OF ZEROS In this section, we will prove the following theorem. If a1 , . . . , a p / 0, y1, y2, . . . , then F and I are equi¨ alent.
THEOREM 1.
Proof. We first prove that I implies F. Let denote the differential operator z Ž drdz ., so that for each convergent power series Ý A n z n and a constant a g ⺓ we have ⬁
⬁
ns0
ns0
Ý Ž a q n. An z n s Ž a q . Ý
An z n .
If b / 0, y1, y2, . . . and m is a non-negative integer, then 1 F1
Ž b q m; b; z . s
⬁
Ý ns0
s s
Ž b q m. n z n Ž b . n n! ⬁
1
zn
Ý Ž b q n . Ž b q 1 q n . ⭈⭈⭈ Ž b q m y 1 q n . n! Ž b . m ns0 1
Ž b. m
Ž b q . Ž b q 1 q . ⭈⭈⭈ Ž b q m y 1 q .
⬁
Ý ns0
zn n!
,
so that 1 F1Ž b q m; b; z . s P Ž z . e z for some polynomial P Ž z . of degree m. In general, if b1 , . . . , bp / 0, y1, y2, . . . and m1 , . . . , m p are non-negative integers, then we have p Fp
Ž b 1 q m 1 , . . . , bp q m p ; b 1 , . . . , bp ; z . p
s
m jy1
1
⬁
zn
Ł Ž b . Ł Ž bj q k q . Ý n! , js1 j m ks0 ns0 j
and hence p Fp Ž b1 q m1 , . . . , bp q m p ; b1 , . . . , bp ; z . s P Ž z . e z for some polynomial P Ž z . of degree m1 q ⭈⭈⭈ qm p . This proves that I implies F. Conversely, suppose that a1 , . . . , a p / 0, y1, y2, . . . and that p Fp Ž z . s Ž . has only a finite number of zeros. Then F p p a1 , . . . , a p ; b 1 , . . . , bp ; z Ž . p Fp z is an entire function of order 1 with finitely many zeros. Therefore Ž . p Fp z can be written in the form p Fp
Ž z . s Ž c0 q c1 z q ⭈⭈⭈ qc N z N . e ␣ z ,
Ž 2.
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where ␣ , c 0 , . . . , c N are constants with ␣ , c N / 0. It is well known w1, p. 60x and easy to see that p Fp Ž z . satisfies the differential equation d dz
Ž q b1 y 1 . ⭈⭈⭈ Ž q bp y 1 . y Ž q a1 . ⭈⭈⭈ Ž q a p . p Fp Ž z . s 0. Ž 3.
From Ž2. and Ž3., we obtain
Ž ␣ pq 1 y ␣ p . c N z Nq p q lower terms s 0, so that ␣ s 0 or 1. Since ␣ / 0, it follows that ␣ s 1, and hence Ž2. becomes ⬁
Ý ns0
⬁ zn Ž a1 . n ⭈⭈⭈ Ž a p . n z n s Ž c 0 q c1 z q ⭈⭈⭈ qc N z N . Ý . Ž b1 . n ⭈⭈⭈ Ž bp . n n! ns0 n!
Therefore we have c0 c1 cN Ž a1 . n ⭈⭈⭈ Ž a p . n 1 s q q ⭈⭈⭈ q n! Ž b1 . n ⭈⭈⭈ Ž bp . n n! Ž n y 1. ! ŽnyN.!
Ž n s N, N q 1, . . . . . Ž 4 . Let f Ž t . be the polynomial defined by f Ž t . s c 0 q c1 t q c 2 t Ž t y 1 . q ⭈⭈⭈ qc N t Ž t y 1 . ⭈⭈⭈ Ž t y N q 1 . . Then Ž4. implies that
Ž a1 . n ⭈⭈⭈ Ž a p . n s f Ž n. Ž b1 . n ⭈⭈⭈ Ž bp . n
Ž n s N, N q 1, . . . . ,
and hence we have
Ž a1 q n . ⭈⭈⭈ Ž a p q n . Ž a1 q n . ⭈⭈⭈ Ž a p q n . Ž a1 . n ⭈⭈⭈ Ž a p . n f Ž n. s Ž b1 q n . ⭈⭈⭈ Ž bp q n . Ž b1 q n . ⭈⭈⭈ Ž bp q n . Ž b1 . n ⭈⭈⭈ Ž bp . n s
Ž a1 . nq1 ⭈⭈⭈ Ž a p . nq1 s f Ž n q 1. Ž b1 . nq1 ⭈⭈⭈ Ž bp . nq1 Ž n s N, N q 1, . . . . ,
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
253
so that
Ž a1 q n . ⭈⭈⭈ Ž a p q n . f Ž n . s Ž b1 q n . ⭈⭈⭈ Ž bp q n . f Ž n q 1 . Ž n s N, N q 1, . . . . . Ž 5 . Since f Ž t . is a polynomial, Eq. Ž5. implies that
Ž a1 q t . ⭈⭈⭈ Ž a p q t . f Ž t . s Ž b1 q t . ⭈⭈⭈ Ž bp q t . f Ž t q 1 . .
Ž 6.
Now, using Ž6. and an induction, we will show that we can re-index a1 , . . . , a p so that a1 y b1 , . . . , a p y bp are non-negative integers. If p q N s p q deg f Ž t . s 1, then we must have p s 1 and N s 0, so that our assertion is trivial. Let p q N ) 1 and assume the induction hypothesis. Since b1 q t is a factor of the right-hand side of Ž6., it is a factor of the left-hand side also. Therefore b1 q t divides Ž a1 q t . ⭈⭈⭈ Ž a p q t ., or b1 q t divides f Ž t .. If b1 q t divides Ž a1 q t . ⭈⭈⭈ Ž a p q t ., then b1 s a k for some k. Re-index a1 , . . . , a p so that a1 s b1. Then Ž6. implies that
Ž a2 q t . ⭈⭈⭈ Ž a p q t . f Ž t . s Ž b 2 q t . ⭈⭈⭈ Ž bp q t . f Ž t q 1 . . Hence, by the induction hypothesis, we can re-index a2 , . . . , a p so that a2 y b 2 , . . . , a p y bp are non-negative integers. If b1 q t divides f Ž t ., then there is a polynomial f 1Ž t . of degree N y 1 such that f Ž t . s Ž b1 q t . f 1Ž t .. From Ž6., we obtain
Ž a1 q t . ⭈⭈⭈ Ž a p q t . f 1 Ž t . s Ž b 2 q t . ⭈⭈⭈ Ž bp q t . f Ž t q 1 . s Ž b 2 q t . ⭈⭈⭈ Ž bp q t . Ž b1 q t q 1 . f 1 Ž t q 1 . . Again, the induction hypothesis implies that we can re-index a1 , . . . , a p so that a1 y b1 y 1, a2 y b 2 , . . . , a p y bp are non-negative integers. This proves our assertion.
3. REALITY OF ZEROS We start this section with an asymptotic equality due to Barnes w1x. Let a1 , . . . , a p and b1 , . . . , bp be complex constants and assume that b1 , . . . , bp / 0, y1, y2, . . . . In w1, pp. 80᎐83x, it is shown that for each ⑀ ) 0 we
254
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have p
⌫ Ž aj .
Ł ⌫ Ž b . p Fp Ž a1 , . . . , a p ; b1 , . . . , bp ; z . js1 j s e z z Ý a jyÝ b j 1 q O
ž
1
ž // ž z
< z < ª ⬁,
2
y ⑀ . Ž 7.
/
Here arg is the principal branch of the argument. Hence p Fp Ž z . has only a finite number of zeros in the sector
Ž a1 . n ⭈⭈⭈ Ž a p . n g⺢ Ž b1 . n ⭈⭈⭈ Ž bp . n
Ž n s 0, 1, 2, . . . . ,
and if a1 , . . . , a p are real, then p Fp Ž z . has only a finite number of negative real zeros. If some a j is equal to zero or a negative integer, then p Fp Ž z . is a polynomial, and if the condition I holds, then p Fp Ž z . has only a finite number of zeros, by Theorem 1. Hence we may assume, without loss of generality, that a1 , . . . , a p / 0, y1, y2, . . . and that the condition I does not hold. Then the function ⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s .r⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s . has infinitely many poles all of which are different from 0, 1, 2, . . . . Let ␣ 1 , ␣ 2 , . . . denote the distinct poles of ⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s .r⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s .. For each k s 1, 2, . . . let R k Ž z . denote the residue of the function ⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . ⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s .
⌫ Ž ys . Ž yz .
s
at the pole s s ␣ k . Here Žyz . s is defined by s
Ž yz . s exp s log Ž yz . for s g ⺓ and z / 0; log is the principal branch of the logarithm. Then it is easy to see that R k Ž z .rŽyz . ␣ k is a polynomial of logŽyz . for each k s 1, 2, . . . . See w6, p. 288x also. Let A ) max
255
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
- ⬁. Then we have y1
H 2 i C
⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . 1
s
⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s . ⌫ Ž a1 . ⭈⭈⭈ ⌫ Ž a p .
p Fp
⌫ Ž b1 . ⭈⭈⭈ ⌫ Ž bp .
s
⌫ Ž ys . Ž yz . ds
Ž z. y
Ý
yK- ␣ k
Rk Ž z .
for Re z - 0. Let C2 denote the line s s yK y it, y⬁ - t - ⬁, and let ⑀ ) 0 be arbitrary. If
HC
⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . 1
⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s . s
HC
s
⌫ Ž ys . Ž yz . ds
⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . ⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s . 2
s
⌫ Ž ys . Ž yz . ds.
Moreover, we have
HC
⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . 2
⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s .
⌫ Ž ys . Ž yz . ds s O Ž < z
ž
arg Ž yz . F
2
y ⑀ , < z< ª ⬁ ,
/
so that ⌫ Ž a1 . ⭈⭈⭈ ⌫ Ž a p . ⌫ Ž b1 . ⭈⭈⭈ ⌫ Ž bp .
p Fp
Ž z. s
Ý
yK- ␣ k
ž
R k Ž z . q O Ž < z
arg Ž yz . F
2
y ⑀ , < z< ª ⬁ .
/
Ž 8.
From Ž8., it follows that if a1 , . . . , a p are real and if R k Ž z . / 0 for some k and z, then for each ⑀ ) 0 the function p Fp Ž z . has only a finite number of zeros in the sector
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KI AND KIM
LEMMA. Let f Ž s . be a function which has a pole of order m G 1 at s s ␣ , and let c1 , . . . , c my1 be arbitrary constants. Then either f Ž s . has nonzero residue at s s ␣ or there is a positi¨ e integer l F m y 1 such that Ž s y c1 . ⭈⭈⭈ Ž s y c l . f Ž s . has nonzero residue at s s ␣ . From this lemma, it follows that if z / 0, then for each k s 1, 2, . . . there is a nonnegative integer l such that the residue of ⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . ⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s .
⌫ Ž ys q l . Ž yz .
s
at s s ␣ k is not equal to zero. As a consequence, there is a non-negative integer l such that if z / 0, then the function ⌫ Ž a1 q l q s . ⭈⭈⭈ ⌫ Ž a p q l q s . ⌫ Ž b1 q l q s . ⭈⭈⭈ ⌫ Ž bp q l q s .
⌫ Ž ys . Ž yz .
s
has a pole other than 0, 1, 2, . . . at which the residue is not equal to zero. Since Žl. p Fp
Ž a1 , . . . , a p ; b1 , . . . , bp ; z . s
Ž a1 . l ⭈⭈⭈ Ž a p . l F Ž a q l, . . . , a p q l ; b1 q l, . . . , bp q l ; z . , Ž b1 . l ⭈⭈⭈ Ž bp . l p p 1
we conclude that if a1 , . . . , a p are real, then some derivative of p Fp Ž z . has only a finite number of zeros in the sector
257
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
we will give a detailed proof of the result. Our proof is based on the Polya᎐Schur theory of multiplier sequences. ´ A sequence ␥n4⬁ns0 of real numbers is said to be a multiplier sequence Žof the first kind. if it has the following property. M. If a0 q a1 z q ⭈⭈⭈ qa d z d is a real polynomial with real zeros only, then the polynomial
␥ 0 a0 q ␥ 1 a1 z q ⭈⭈⭈ q␥d a d z d also has real zeros only. It follows immediately from the definition that if ␥n4 and ␦n4 are multiplier sequences, then their termwise product ␥n ␦n4 is also a multiplier sequence. The multiplier sequences are completely characterized by the following theorem of Polya ´ and Schur w4x. THE POLYA ´ ᎐SCHUR THEOREM. A sequence ␥n4 of real numbers is a multiplier sequence if and only if the function ⌽Ž z. s
⬁
Ý ns0
␥n
zn 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 Polya᎐Schur theorem implies ´ that b q n4⬁ns 0 is a multiplier sequence, because ⬁
Ý
Ž b q n.
ns0
zn n!
s Ž z q b . e z s lim Ž z q b . 1 q kª⬁
ž
z k
k
/
.
Since the class of multiplier sequences is closed under termwise multiplication, it follows that for each positive integer m the sequence ⬁
Ž b q n . Ž b q 1 q n . ⭈⭈⭈ Ž b q m y 1 q n . 4 ns 0 is a multiplier sequence. Hence
½
Ž b q m. n Ž b. n
⬁
5 ½ s
ns0
1
Ž b. m
⬁
Ž b q n . Ž b q 1 q n . ⭈⭈⭈ Ž b q m y 1 q n .
is a multiplier sequence for each positive integer m.
5
ns0
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KI AND KIM
Now suppose that b1 , . . . , bp are positive real numbers and m1 , . . . , m p are nonnegative integers. Then it is clear that the sequence
½
Ž b1 q m1 . n ⭈⭈⭈ Ž bp q m p . n Ž b1 . n ⭈⭈⭈ Ž bp . n
⬁
5
ns0
is a multiplier sequence. Hence, by the Polya᎐Schur theorem, the function ´ p Fp
Ž b 1 q m 1 , . . . , bp q m p ; b 1 , . . . , bp ; z . s
⬁
Ý ns0
Ž b1 q m1 . n ⭈⭈⭈ Ž bp q m p . n z n n! Ž b1 . n ⭈⭈⭈ Ž bp . 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 a1 , . . . , a p Ž/ 0, y1, y2, . . . . are real and b1 , . . . , bp ) 0, then F, R, and I are equi¨ alent. Proof. We have just shown that if b1 , . . . , bp ) 0, then I implies R. 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 a1 , . . . , a p / 0, y1, y2, . . . in Theorem 1; otherwise the function p Fp Ž a1 , . . . , a p ; b1 , . . . , bp ; z . reduces to a polynomial, so that the implication F « I does not hold. Theorem 2 states that if p Fp Ž z . is a real entire function and a1 , . . . , a p are real, then p Fp Ž a1 , . . . , a p ; b1 , . . . , bp ; z . has only a finite number of real zeros. The following example, however, shows that this is not the case in general. EXAMPLE 1. The real entire function f Ž z . s2 F2 Ž i, yi; 1, 1; z . has infinitely many real zeros. This may be shown as follows. According to Ž8. of
259
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
Section 3 or the asymptotic equality given in w1, Sect. 9x, we have 2
y
e ye
f Ž yt . s
⌫ Ž i . ⌫ Ž y2 i . ⌫Ž1 y i. q
2
tyi Ž 1 q O Ž ty1 . .
⌫ Ž yi . ⌫ Ž 2 i . ⌫Ž1 q i.
2
t i Ž 1 q O Ž ty1 . . ,
as t ª ⬁ with t ) 0. Therefore there exist real constants A and B, with A ) 0, such that f Ž yt . s A cos Ž B q log t . q O Ž ty1 .
Ž t ª ⬁, t ) 0 . ,
and this proves our assertion. Next, we consider the assumptions of Theorem 3, namely a1 , . . . , a p / 0, y1, y2, . . . and b1 , . . . , bp ) 0. The following example is about the condition a1 , . . . , a p / 0, y1, y2, . . . . EXAMPLE 2. Let b ) 0. Then
Ž x. s
⌫ Ž b. ⌫Ž b q x.
is a real entire function of order 1 and has negative real zeros only. Hence Ž b .y1 4⬁ Ž .4⬁ w n ns0 s n ns0 is a complex zero decreasing sequence by 2, Theorem 1.4x. In particular, it is a multiplier sequence. Let m be a positive 4⬁ integer. If we apply the multiplier sequence Ž b .y1 n ns0 to the polynomial m Ž1 y z . , then we obtain 1 F1Žym; b; z ., because m 1 F1 Ž ym; b; z . s
Ž ym . n z n
Ý Ž b. n ns0
n!
m
s
1
Ý ns0 Ž b . n
ž mn / Ž yz . . n
Therefore 1 F1Žym; b; z . has real zeros only for each m s 1, 2, . . . and b ) 0. Consequently, we must assume a1 , . . . , a p / 0, y1, y2, . . . in order to obtain the implication R « I in Theorem 3. Our last example is concerned with the condition b1 , . . . , bp ) 0. EXAMPLE 3. Let b - 0 and b / y1, y2, . . . . If m is a positive integer greater than yb, then 1 F1Ž b q m; b; z . has at least 2wŽ1 y b .r2x nonreal zeros. Proof. Let m be a positive integer greater than yb. Then Ž b q 4⬁ m.y1 n ns0 is a complex zero decreasing sequence. On the other hand, the proof of Theorem 1 shows that 1 F1Ž b q m; b; z . s P Ž z . e z for some real
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KI AND KIM
polynomial P Ž z . of degree m. Hence the number of nonreal zeros of 1 F1
Ž b q m; b; z . s
⬁
Ý ns0
Ž b q m. n z n Ž b . n n!
is at least that of 0 F1 Ž b; z . s
⬁
zn
1
Ý Ž b. n
ns0
n!
s
⬁
Ý ns0
Ž b q m. n z n . Ž b q m . n Ž b . n n! 1
Now our assertion follows from a well-known theorem of Hurwitz w5, 15.27x which states that if b - 0 and b / y1, y2, . . . , then 0 F1Ž b; z . has exactly 2wŽ1 y b .r2x 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 k Ž z . / 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 Ž1998᎐2000.. The second author was supported by the Korea Science and Engineering Foundation ŽKOSEF. through the Global Analysis Research Center ŽGARC. 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 Ž1907., 59᎐116. 2. T. Craven and G. Csordas, Complex zero decreasing sequences, Methods Appl. Anal. 2 Ž1995., 420᎐441. 3. E. Hille, Note on some hypergeometric series of higher order, J. London Math. Soc. 4 Ž1929., 50᎐54. ¨ 4. G. Polya zwei Arten von Faktorenfolgen in der Theorie der algebrais´ and J. Schur, Uber chen Gleichungen, J. Reine Angew. Math. 144 Ž1914., 89᎐113. 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 Ž1935., 286᎐293.
On the Zeros of Some Generalized Hypergeometric Functions Haseo Ki Department of Mathematics, Yonsei Uni¨ ersity, Seoul 120-749, Korea E-mail: [email protected]
and Young-One Kim Department of Mathematics, Sejong Uni¨ ersity, Seoul 143-747, Korea E-mail: [email protected] Submitted by William F. Ames Received April 6, 1999
Let a1 , . . . , a p , b1 , . . . , bp be real constants with a1 , . . . , a p / 0, y1, y2, . . . and b1 , . . . , bp ) 0, and let p Fp Ž z . sp Fp Ž a1 , . . . , a p ; b1 , . . . , bp ; z .. It is shown that the following three conditions are equivalent to each other: Ži. p Fp Ž z . has only a finite number of zeros, Žii. p Fp Ž z . has only real zeros, and Žiii. the a j ’s can be re-indexed so that a1 s b1 q m1 , . . . , a p s bp q m p for some nonnegative integers m1 , . . . , m p . 䊚 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
p Fq Ž z . sp Fq Ž a1 , . . . , a p ; b 1 , . . . , bq ; z . s
⬁
Ý ns0
Ž a1 . n ⭈⭈⭈ Ž a p . n z n . Ž 1. Ž b1 . n ⭈⭈⭈ Ž bq . n n!
249 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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Here a1 , . . . , a p and b1 , . . . , bq Ž/ 0, y1, y2, . . . . are constant and Ž a. n s ⌫ Ž a q n.r⌫ Ž a., that is,
Ž a. 0 s 1 and Ž a. n s aŽ a q 1 . ⭈⭈⭈ Ž a q n y 1 .
Ž n s 1, 2, . . . . .
Observe that we must assume b1 , . . . , b1 / 0, y1, y2, . . . in order that the series may be well defined, and that the series reduces to a polynomial whenever some a j is equal to 0 or a negative integer. In 1929, Hille published a paper entitled Note on some hypergeometric series of higher order w3x. In this paper, he studied the zeros of the series Ž1. in the case where p F q and the parameters a1 , . . . , a p and b1 , . . . , bq are real. ŽNote that Ž1. defines an entire function if and only if p F q, provided a1 , . . . , a p / 0, y1, y2, . . . .. In particular, he proved that Ži. if a / 0, y1, y2, . . . , the necessary and sufficient condition for 1 F1Ž a; b; z . to have only a finite number of zeros is that a y b is a non-negative integer, and that Žii. if b1 , . . . , bp ) 0 and m1 , . . . , m p are non-negative integers, then p Fp Ž b1 q m1 , . . . , bp q m p ; b1 , . . . , bp ; z . has real zeros only. He proved Žii. in the special case where m1 s ⭈⭈⭈ s 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 Ž . p Fp a1 , . . . , a p ; b 1 , . . . , bp ; z . Ž a1 , . . . , a p ; b1 , . . . , bp ; z . has only a finite number of zeros. R. p Fp Ž a1 , . . . , a p ; b1 , . . . , bp ; z . has only real zeros. F.
p Fp
I. The a j ’s can be re-indexed so that a1 s b1 q m1 , . . . , a p s bp q m p for some non-negati¨ e integers m1 , . . . , m p . What Hille has proved are Ži. if p s 1 and a1 / 0, y1, y2, . . . , then F and I are equivalent, and Žii. if b1 , . . . , bp ) 0, then I implies R. In this paper, we first consider the general case when the parameters a1 , . . . , a p and b1 , . . . , bq are allowed to have complex values, and prove that if a1 , . . . , a p / 0, y1, y2, . . . , then F and I are equivalent ŽTheorem 1, Section 2.. Next, in Section 3, we apply a method due to Barnes to show that p Fp Ž z . is a real entire function and a1 , . . . , a p are real, then p Fp Ž z . has only a finite number of real zeros ŽTheorem 2.. As a result of Theorem 1 and Theorem 2 we obtain Theorem 3 which states that if a1 , . . . , a p Ž/ 0, y1, y2, . . . . are real and b1 , . . . , 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 ŽSection 4..
251
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
2. NUMBER OF ZEROS In this section, we will prove the following theorem. If a1 , . . . , a p / 0, y1, y2, . . . , then F and I are equi¨ alent.
THEOREM 1.
Proof. We first prove that I implies F. Let denote the differential operator z Ž drdz ., so that for each convergent power series Ý A n z n and a constant a g ⺓ we have ⬁
⬁
ns0
ns0
Ý Ž a q n. An z n s Ž a q . Ý
An z n .
If b / 0, y1, y2, . . . and m is a non-negative integer, then 1 F1
Ž b q m; b; z . s
⬁
Ý ns0
s s
Ž b q m. n z n Ž b . n n! ⬁
1
zn
Ý Ž b q n . Ž b q 1 q n . ⭈⭈⭈ Ž b q m y 1 q n . n! Ž b . m ns0 1
Ž b. m
Ž b q . Ž b q 1 q . ⭈⭈⭈ Ž b q m y 1 q .
⬁
Ý ns0
zn n!
,
so that 1 F1Ž b q m; b; z . s P Ž z . e z for some polynomial P Ž z . of degree m. In general, if b1 , . . . , bp / 0, y1, y2, . . . and m1 , . . . , m p are non-negative integers, then we have p Fp
Ž b 1 q m 1 , . . . , bp q m p ; b 1 , . . . , bp ; z . p
s
m jy1
1
⬁
zn
Ł Ž b . Ł Ž bj q k q . Ý n! , js1 j m ks0 ns0 j
and hence p Fp Ž b1 q m1 , . . . , bp q m p ; b1 , . . . , bp ; z . s P Ž z . e z for some polynomial P Ž z . of degree m1 q ⭈⭈⭈ qm p . This proves that I implies F. Conversely, suppose that a1 , . . . , a p / 0, y1, y2, . . . and that p Fp Ž z . s Ž . has only a finite number of zeros. Then F p p a1 , . . . , a p ; b 1 , . . . , bp ; z Ž . p Fp z is an entire function of order 1 with finitely many zeros. Therefore Ž . p Fp z can be written in the form p Fp
Ž z . s Ž c0 q c1 z q ⭈⭈⭈ qc N z N . e ␣ z ,
Ž 2.
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where ␣ , c 0 , . . . , c N are constants with ␣ , c N / 0. It is well known w1, p. 60x and easy to see that p Fp Ž z . satisfies the differential equation d dz
Ž q b1 y 1 . ⭈⭈⭈ Ž q bp y 1 . y Ž q a1 . ⭈⭈⭈ Ž q a p . p Fp Ž z . s 0. Ž 3.
From Ž2. and Ž3., we obtain
Ž ␣ pq 1 y ␣ p . c N z Nq p q lower terms s 0, so that ␣ s 0 or 1. Since ␣ / 0, it follows that ␣ s 1, and hence Ž2. becomes ⬁
Ý ns0
⬁ zn Ž a1 . n ⭈⭈⭈ Ž a p . n z n s Ž c 0 q c1 z q ⭈⭈⭈ qc N z N . Ý . Ž b1 . n ⭈⭈⭈ Ž bp . n n! ns0 n!
Therefore we have c0 c1 cN Ž a1 . n ⭈⭈⭈ Ž a p . n 1 s q q ⭈⭈⭈ q n! Ž b1 . n ⭈⭈⭈ Ž bp . n n! Ž n y 1. ! ŽnyN.!
Ž n s N, N q 1, . . . . . Ž 4 . Let f Ž t . be the polynomial defined by f Ž t . s c 0 q c1 t q c 2 t Ž t y 1 . q ⭈⭈⭈ qc N t Ž t y 1 . ⭈⭈⭈ Ž t y N q 1 . . Then Ž4. implies that
Ž a1 . n ⭈⭈⭈ Ž a p . n s f Ž n. Ž b1 . n ⭈⭈⭈ Ž bp . n
Ž n s N, N q 1, . . . . ,
and hence we have
Ž a1 q n . ⭈⭈⭈ Ž a p q n . Ž a1 q n . ⭈⭈⭈ Ž a p q n . Ž a1 . n ⭈⭈⭈ Ž a p . n f Ž n. s Ž b1 q n . ⭈⭈⭈ Ž bp q n . Ž b1 q n . ⭈⭈⭈ Ž bp q n . Ž b1 . n ⭈⭈⭈ Ž bp . n s
Ž a1 . nq1 ⭈⭈⭈ Ž a p . nq1 s f Ž n q 1. Ž b1 . nq1 ⭈⭈⭈ Ž bp . nq1 Ž n s N, N q 1, . . . . ,
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
253
so that
Ž a1 q n . ⭈⭈⭈ Ž a p q n . f Ž n . s Ž b1 q n . ⭈⭈⭈ Ž bp q n . f Ž n q 1 . Ž n s N, N q 1, . . . . . Ž 5 . Since f Ž t . is a polynomial, Eq. Ž5. implies that
Ž a1 q t . ⭈⭈⭈ Ž a p q t . f Ž t . s Ž b1 q t . ⭈⭈⭈ Ž bp q t . f Ž t q 1 . .
Ž 6.
Now, using Ž6. and an induction, we will show that we can re-index a1 , . . . , a p so that a1 y b1 , . . . , a p y bp are non-negative integers. If p q N s p q deg f Ž t . s 1, then we must have p s 1 and N s 0, so that our assertion is trivial. Let p q N ) 1 and assume the induction hypothesis. Since b1 q t is a factor of the right-hand side of Ž6., it is a factor of the left-hand side also. Therefore b1 q t divides Ž a1 q t . ⭈⭈⭈ Ž a p q t ., or b1 q t divides f Ž t .. If b1 q t divides Ž a1 q t . ⭈⭈⭈ Ž a p q t ., then b1 s a k for some k. Re-index a1 , . . . , a p so that a1 s b1. Then Ž6. implies that
Ž a2 q t . ⭈⭈⭈ Ž a p q t . f Ž t . s Ž b 2 q t . ⭈⭈⭈ Ž bp q t . f Ž t q 1 . . Hence, by the induction hypothesis, we can re-index a2 , . . . , a p so that a2 y b 2 , . . . , a p y bp are non-negative integers. If b1 q t divides f Ž t ., then there is a polynomial f 1Ž t . of degree N y 1 such that f Ž t . s Ž b1 q t . f 1Ž t .. From Ž6., we obtain
Ž a1 q t . ⭈⭈⭈ Ž a p q t . f 1 Ž t . s Ž b 2 q t . ⭈⭈⭈ Ž bp q t . f Ž t q 1 . s Ž b 2 q t . ⭈⭈⭈ Ž bp q t . Ž b1 q t q 1 . f 1 Ž t q 1 . . Again, the induction hypothesis implies that we can re-index a1 , . . . , a p so that a1 y b1 y 1, a2 y b 2 , . . . , a p y bp are non-negative integers. This proves our assertion.
3. REALITY OF ZEROS We start this section with an asymptotic equality due to Barnes w1x. Let a1 , . . . , a p and b1 , . . . , bp be complex constants and assume that b1 , . . . , bp / 0, y1, y2, . . . . In w1, pp. 80᎐83x, it is shown that for each ⑀ ) 0 we
254
KI AND KIM
have p
⌫ Ž aj .
Ł ⌫ Ž b . p Fp Ž a1 , . . . , a p ; b1 , . . . , bp ; z . js1 j s e z z Ý a jyÝ b j 1 q O
ž
1
ž // ž z
< z < ª ⬁,
2
y ⑀ . Ž 7.
/
Here arg is the principal branch of the argument. Hence p Fp Ž z . has only a finite number of zeros in the sector
Ž a1 . n ⭈⭈⭈ Ž a p . n g⺢ Ž b1 . n ⭈⭈⭈ Ž bp . n
Ž n s 0, 1, 2, . . . . ,
and if a1 , . . . , a p are real, then p Fp Ž z . has only a finite number of negative real zeros. If some a j is equal to zero or a negative integer, then p Fp Ž z . is a polynomial, and if the condition I holds, then p Fp Ž z . has only a finite number of zeros, by Theorem 1. Hence we may assume, without loss of generality, that a1 , . . . , a p / 0, y1, y2, . . . and that the condition I does not hold. Then the function ⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s .r⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s . has infinitely many poles all of which are different from 0, 1, 2, . . . . Let ␣ 1 , ␣ 2 , . . . denote the distinct poles of ⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s .r⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s .. For each k s 1, 2, . . . let R k Ž z . denote the residue of the function ⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . ⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s .
⌫ Ž ys . Ž yz .
s
at the pole s s ␣ k . Here Žyz . s is defined by s
Ž yz . s exp s log Ž yz . for s g ⺓ and z / 0; log is the principal branch of the logarithm. Then it is easy to see that R k Ž z .rŽyz . ␣ k is a polynomial of logŽyz . for each k s 1, 2, . . . . See w6, p. 288x also. Let A ) max
255
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
- ⬁. Then we have y1
H 2 i C
⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . 1
s
⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s . ⌫ Ž a1 . ⭈⭈⭈ ⌫ Ž a p .
p Fp
⌫ Ž b1 . ⭈⭈⭈ ⌫ Ž bp .
s
⌫ Ž ys . Ž yz . ds
Ž z. y
Ý
yK- ␣ k
Rk Ž z .
for Re z - 0. Let C2 denote the line s s yK y it, y⬁ - t - ⬁, and let ⑀ ) 0 be arbitrary. If
HC
⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . 1
⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s . s
HC
s
⌫ Ž ys . Ž yz . ds
⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . ⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s . 2
s
⌫ Ž ys . Ž yz . ds.
Moreover, we have
HC
⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . 2
⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s .
⌫ Ž ys . Ž yz . ds s O Ž < z
ž
arg Ž yz . F
2
y ⑀ , < z< ª ⬁ ,
/
so that ⌫ Ž a1 . ⭈⭈⭈ ⌫ Ž a p . ⌫ Ž b1 . ⭈⭈⭈ ⌫ Ž bp .
p Fp
Ž z. s
Ý
yK- ␣ k
ž
R k Ž z . q O Ž < z
arg Ž yz . F
2
y ⑀ , < z< ª ⬁ .
/
Ž 8.
From Ž8., it follows that if a1 , . . . , a p are real and if R k Ž z . / 0 for some k and z, then for each ⑀ ) 0 the function p Fp Ž z . has only a finite number of zeros in the sector
256
KI AND KIM
LEMMA. Let f Ž s . be a function which has a pole of order m G 1 at s s ␣ , and let c1 , . . . , c my1 be arbitrary constants. Then either f Ž s . has nonzero residue at s s ␣ or there is a positi¨ e integer l F m y 1 such that Ž s y c1 . ⭈⭈⭈ Ž s y c l . f Ž s . has nonzero residue at s s ␣ . From this lemma, it follows that if z / 0, then for each k s 1, 2, . . . there is a nonnegative integer l such that the residue of ⌫ Ž a1 q s . ⭈⭈⭈ ⌫ Ž a p q s . ⌫ Ž b1 q s . ⭈⭈⭈ ⌫ Ž bp q s .
⌫ Ž ys q l . Ž yz .
s
at s s ␣ k is not equal to zero. As a consequence, there is a non-negative integer l such that if z / 0, then the function ⌫ Ž a1 q l q s . ⭈⭈⭈ ⌫ Ž a p q l q s . ⌫ Ž b1 q l q s . ⭈⭈⭈ ⌫ Ž bp q l q s .
⌫ Ž ys . Ž yz .
s
has a pole other than 0, 1, 2, . . . at which the residue is not equal to zero. Since Žl. p Fp
Ž a1 , . . . , a p ; b1 , . . . , bp ; z . s
Ž a1 . l ⭈⭈⭈ Ž a p . l F Ž a q l, . . . , a p q l ; b1 q l, . . . , bp q l ; z . , Ž b1 . l ⭈⭈⭈ Ž bp . l p p 1
we conclude that if a1 , . . . , a p are real, then some derivative of p Fp Ž z . has only a finite number of zeros in the sector
257
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
we will give a detailed proof of the result. Our proof is based on the Polya᎐Schur theory of multiplier sequences. ´ A sequence ␥n4⬁ns0 of real numbers is said to be a multiplier sequence Žof the first kind. if it has the following property. M. If a0 q a1 z q ⭈⭈⭈ qa d z d is a real polynomial with real zeros only, then the polynomial
␥ 0 a0 q ␥ 1 a1 z q ⭈⭈⭈ q␥d a d z d also has real zeros only. It follows immediately from the definition that if ␥n4 and ␦n4 are multiplier sequences, then their termwise product ␥n ␦n4 is also a multiplier sequence. The multiplier sequences are completely characterized by the following theorem of Polya ´ and Schur w4x. THE POLYA ´ ᎐SCHUR THEOREM. A sequence ␥n4 of real numbers is a multiplier sequence if and only if the function ⌽Ž z. s
⬁
Ý ns0
␥n
zn 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 Polya᎐Schur theorem implies ´ that b q n4⬁ns 0 is a multiplier sequence, because ⬁
Ý
Ž b q n.
ns0
zn n!
s Ž z q b . e z s lim Ž z q b . 1 q kª⬁
ž
z k
k
/
.
Since the class of multiplier sequences is closed under termwise multiplication, it follows that for each positive integer m the sequence ⬁
Ž b q n . Ž b q 1 q n . ⭈⭈⭈ Ž b q m y 1 q n . 4 ns 0 is a multiplier sequence. Hence
½
Ž b q m. n Ž b. n
⬁
5 ½ s
ns0
1
Ž b. m
⬁
Ž b q n . Ž b q 1 q n . ⭈⭈⭈ Ž b q m y 1 q n .
is a multiplier sequence for each positive integer m.
5
ns0
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KI AND KIM
Now suppose that b1 , . . . , bp are positive real numbers and m1 , . . . , m p are nonnegative integers. Then it is clear that the sequence
½
Ž b1 q m1 . n ⭈⭈⭈ Ž bp q m p . n Ž b1 . n ⭈⭈⭈ Ž bp . n
⬁
5
ns0
is a multiplier sequence. Hence, by the Polya᎐Schur theorem, the function ´ p Fp
Ž b 1 q m 1 , . . . , bp q m p ; b 1 , . . . , bp ; z . s
⬁
Ý ns0
Ž b1 q m1 . n ⭈⭈⭈ Ž bp q m p . n z n n! Ž b1 . n ⭈⭈⭈ Ž bp . 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 a1 , . . . , a p Ž/ 0, y1, y2, . . . . are real and b1 , . . . , bp ) 0, then F, R, and I are equi¨ alent. Proof. We have just shown that if b1 , . . . , bp ) 0, then I implies R. 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 a1 , . . . , a p / 0, y1, y2, . . . in Theorem 1; otherwise the function p Fp Ž a1 , . . . , a p ; b1 , . . . , bp ; z . reduces to a polynomial, so that the implication F « I does not hold. Theorem 2 states that if p Fp Ž z . is a real entire function and a1 , . . . , a p are real, then p Fp Ž a1 , . . . , a p ; b1 , . . . , bp ; z . has only a finite number of real zeros. The following example, however, shows that this is not the case in general. EXAMPLE 1. The real entire function f Ž z . s2 F2 Ž i, yi; 1, 1; z . has infinitely many real zeros. This may be shown as follows. According to Ž8. of
259
ZEROS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS
Section 3 or the asymptotic equality given in w1, Sect. 9x, we have 2
y
e ye
f Ž yt . s
⌫ Ž i . ⌫ Ž y2 i . ⌫Ž1 y i. q
2
tyi Ž 1 q O Ž ty1 . .
⌫ Ž yi . ⌫ Ž 2 i . ⌫Ž1 q i.
2
t i Ž 1 q O Ž ty1 . . ,
as t ª ⬁ with t ) 0. Therefore there exist real constants A and B, with A ) 0, such that f Ž yt . s A cos Ž B q log t . q O Ž ty1 .
Ž t ª ⬁, t ) 0 . ,
and this proves our assertion. Next, we consider the assumptions of Theorem 3, namely a1 , . . . , a p / 0, y1, y2, . . . and b1 , . . . , bp ) 0. The following example is about the condition a1 , . . . , a p / 0, y1, y2, . . . . EXAMPLE 2. Let b ) 0. Then
Ž x. s
⌫ Ž b. ⌫Ž b q x.
is a real entire function of order 1 and has negative real zeros only. Hence Ž b .y1 4⬁ Ž .4⬁ w n ns0 s n ns0 is a complex zero decreasing sequence by 2, Theorem 1.4x. In particular, it is a multiplier sequence. Let m be a positive 4⬁ integer. If we apply the multiplier sequence Ž b .y1 n ns0 to the polynomial m Ž1 y z . , then we obtain 1 F1Žym; b; z ., because m 1 F1 Ž ym; b; z . s
Ž ym . n z n
Ý Ž b. n ns0
n!
m
s
1
Ý ns0 Ž b . n
ž mn / Ž yz . . n
Therefore 1 F1Žym; b; z . has real zeros only for each m s 1, 2, . . . and b ) 0. Consequently, we must assume a1 , . . . , a p / 0, y1, y2, . . . in order to obtain the implication R « I in Theorem 3. Our last example is concerned with the condition b1 , . . . , bp ) 0. EXAMPLE 3. Let b - 0 and b / y1, y2, . . . . If m is a positive integer greater than yb, then 1 F1Ž b q m; b; z . has at least 2wŽ1 y b .r2x nonreal zeros. Proof. Let m be a positive integer greater than yb. Then Ž b q 4⬁ m.y1 n ns0 is a complex zero decreasing sequence. On the other hand, the proof of Theorem 1 shows that 1 F1Ž b q m; b; z . s P Ž z . e z for some real
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KI AND KIM
polynomial P Ž z . of degree m. Hence the number of nonreal zeros of 1 F1
Ž b q m; b; z . s
⬁
Ý ns0
Ž b q m. n z n Ž b . n n!
is at least that of 0 F1 Ž b; z . s
⬁
zn
1
Ý Ž b. n
ns0
n!
s
⬁
Ý ns0
Ž b q m. n z n . Ž b q m . n Ž b . n n! 1
Now our assertion follows from a well-known theorem of Hurwitz w5, 15.27x which states that if b - 0 and b / y1, y2, . . . , then 0 F1Ž b; z . has exactly 2wŽ1 y b .r2x 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 k Ž z . / 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 Ž1998᎐2000.. The second author was supported by the Korea Science and Engineering Foundation ŽKOSEF. through the Global Analysis Research Center ŽGARC. 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 Ž1907., 59᎐116. 2. T. Craven and G. Csordas, Complex zero decreasing sequences, Methods Appl. Anal. 2 Ž1995., 420᎐441. 3. E. Hille, Note on some hypergeometric series of higher order, J. London Math. Soc. 4 Ž1929., 50᎐54. ¨ 4. G. Polya zwei Arten von Faktorenfolgen in der Theorie der algebrais´ and J. Schur, Uber chen Gleichungen, J. Reine Angew. Math. 144 Ž1914., 89᎐113. 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 Ž1935., 286᎐293.