Hypergeometric generating functions for values of Dirichlet and other ...

Hypergeometric generating functions for values of Dirichlet and other L functions Jeremy Lovejoy* and Ken Ono†‡ *Centre National de la Recherche Scientifique, Laboratoire Bordelais de Recherche en Informatique, 351 Cours de la Liberation, 33405 Talence Cedex, France; and †Department of Mathematics, University of Wisconsin, Madison, WI 53706 Communicated by Richard A. Askey, University of Wisconsin, Madison, WI, March 24, 2003 (received for review October 7, 2002)

Although there is vast literature on the values of L functions at nonpositive integers, the recent appearance of some of these values as the coefficients of specializations of knot invariants comes as a surprise. Using work of G. E. Andrews [(1981) Adv. Math. 41, 173–185; (1986) q-Series: Their Development and Application in Analysis, Combinatories, Physics, and Computer Algebra, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 66 (Am. Math. Soc, Providence, RI); (1975) Problems and Prospects for Basic Hypergeometric Series: The Theory and Application of Special Functions (Academic, New York); and (1992) Illinois J. Math. 36, 251–274], we revisit this old subject and provide uniform and general results giving such generating functions as specializations of basic hypergeometric functions. For example, we obtain such generating functions for all nontrivial Dirichlet L functions. q series

1. Introduction and Statement of Results uppose that ␹12 is the Dirichlet character with modulus 12 defined by

S

再

1 if n ⬅ 1, 11 共mod 12兲, ␹ 12共n兲 :⫽ ⫺1 if n ⬅ 5, 7 共mod 12兲, 0 otherwise, ⬁ and let L(s, ␹ 12) ⫽ 兺 n⫽1 ␹ 12(n)兾n s be its associated L function. In a recent paper, Zagier (1) obtained the following generating function for the values of L(s, ␹ 12) at negative odd integers

冘 ⬁

⫺2e⫺t/24

冘 ⬁

共1 ⫺ e⫺t兲共1 ⫺ e⫺2t兲· · ·共1 ⫺ e⫺nt兲 ⫽

n⫽0

n⫽0

冉 冊

L共⫺2n ⫺ 1, ␹12兲 ⫺t 䡠 n! 24

n

⫽ ⫺2 ⫺

1681 2 23 t⫺ t ⫺ · · ·. 12 576

[1.1]

The t-series expansion on the left is obtained by using the Taylor series expansion for e ⫺t. In other recent works (see refs. 2–4; K. Hikami, personal communication), similar generating functions have been obtained for the values of certain L functions at nonpositive integers. These examples were derived from q-series identities associated to the ‘‘summation of the tails’’ of a modular form and the combinatorics of q-difference equations. These L-function values are generalized Bernoulli numbers (for example, see ref. 5), and there is vast literature on the subject. These numbers play many important roles in number theory. However, their recent appearance in knot theory, algebraic geometry, and mathematical physics comes as a surprise. In the recent works of Hikami (4) and Zagier (1), they appear as the coefficients of specializations of Kashaev invariants in knot theory. Motivated by these examples, it is natural to seek general results that uniformly provide such L values as the coefficients of generating functions like Eq. 1.1 and the multidimensional series appearing in ref. 4. Here we provide such results. If 0 ⬍ c ⱕ 1 is a rational number, then let ␨ (s, c), for ᑬ(s) ⬎ 1, denote the Hurwitz zeta function

冘 ⬁

␨ 共s, c兲 :⫽

n⫽0

1 . 共n ⫹ c兲s

[1.2]

These functions possess an analytic continuation to ⺓ with the exception of a simple pole at s ⫽ 1 with residue 1. These zeta functions, which generalize Riemann’s zeta function, are the building blocks of many important L functions. All of the L functions in refs. 1 and 4 are finite linear combinations of functions of the form

冉冉 冊 冉

L a,b共s兲 ⫽ 共2a兲 ⫺s ␨ s,

b a⫹b ⫺ ␨ s, 2a 2a

冊冊 冘冉 ⬁

⫽

n⫽0

冊 冘

1 1 ⫺ ⫽ 共2an ⫹ b兲s 共2an ⫹ a ⫹ b兲s

⬁

m⫽0

共⫺1兲m , 共ma ⫹ b兲s

[1.3]

where 0 ⬍ b ⬍ a are integers. We obtain multidimensional basic hypergeometric generating functions for all of these L functions at nonpositive integers. (For a survey of basic hypergeometric series, see refs. 6 or 7.) We shall use the standard notation 共a; q兲 n ⫽ 共1 ⫺ a兲共1 ⫺ aq兲· · ·共1 ⫺ aq n⫺1兲, ‡To

[1.4]

whom correspondence should be addressed. E-mail: [email protected].

6904 – 6909 兩 PNAS 兩 June 10, 2003 兩 vol. 100 兩 no. 12

www.pnas.org兾cgi兾doi兾10.1073兾pnas.1131697100

and throughout we assume that 兩q兩 ⬍ 1 and that the other parameters are restricted to domains that do not contain any singularities of the series or products under consideration. For integers k ⱖ 2, define the series F k(z; q) by F k共z; q兲 ⫽

冘

2

2

2

共q; q兲nk⫺1共z; q兲nk⫺1znk⫺1⫹2nk⫺2⫹· · ·⫹2n1qn1⫹n2⫹· · ·⫹nk⫺2

nk⫺1ⱖ· · ·ⱖn 1ⱖ0

共q; q兲nk⫺1⫺nk⫺2共q; q兲nk⫺2⫺nk⫺3· · ·共q; q兲n2⫺n1共q; q兲n1共⫺z; q兲n1⫹1

.

[1.5]

Theorem 1. If 0 ⬍ b ⬍ a are integers and k ⱖ 2, then

冘 ⬁

2

2

e ⫺共k⫺1兲b tFk共e⫺abt; e⫺a t兲 ⫽

La,b共⫺2n兲 䡠

n⫽0

d 共2k⫺2兲b 共k⫺1兲b2 2 共z q Fk共zaqab; qa 兲兲兩z⫽1,q⫽e⫺t ⫽ 共2k ⫺ 2兲 dz

冘

共共1 ⫺ k兲t兲n , n!

⬁

La,b共⫺2n ⫺ 1兲 䡠

n⫽0

共共1 ⫺ k兲t兲n . n!

冘 ⬁

L共s, ␹ 兲 ⫽

n⫽1

␹共n兲 . ns

[1.6]

Such a ␹ is even (respectively odd) if ␹(⫺1) ⫽ 1 [respectively ␹(⫺1) ⫽ ⫺1]. We give basic hypergeometric generating functions for the values of L(s, ␹ ) at nonpositive integers. Define the series G(z; q) by

冘 ⬁

G共z; q兲 :⫽

n⫽0

共⫺1兲n共q; q兲n共1兾z; q兲n共zq; q兲n⫹1qn共n⫹1兲/2 . 共q; q兲2n⫹1

[1.7]

If ␹ mod f is a Dirichlet character, then define G ␹(z; q) by

冘

关 f/2 兴

G ␹共z; q兲 :⫽

2

2

␹共r兲z⫺rqr G共z fq⫺2 r f; q f 兲.

[1.8]

r⫽1

Theorem 2. Suppose that ␹ mod f is a nontrivial Dirichlet character.

1. If ␹ is odd, then

冘 ⬁

G ␹共1; e ⫺t兲 ⫽

L共⫺2n, ␹兲 䡠

n⫽0

共⫺t兲n . n!

2. If ␹ is even, then d 共G ␹共z; q兲兲兩 z⫽1,q⫽e⫺t ⫽ ⫺ dz

冘 ⬁

L共⫺2n ⫺ 1, ␹兲 䡠

n⫽0

共⫺t兲n . n!

Suppose that ␹ is a nontrivial even (respectively odd) Dirichlet character. It is a classical fact that if n ⱖ 0 is even (respectively odd), then L(⫺n, ␹ ) ⫽ 0. Therefore, Theorem 2 provides the L values at nonpositive integers for all Dirichlet L functions. In section 2, we give formulas for F k(z; q) and G(z; q). To prove these identities, we use important results of Andrews (9–12). In section 3 we recall classical facts about Mellin integral representations of L functions, and then combine them with the identities of section 2 to prove Theorems 1 and 2. In section 4 we give several examples of Theorems 1 and 2. 2. q-Series Identities For convenience, we use the following abbreviation: 共a 1, a 2, . . . , a k兲 n :⫽ 共a1; q兲n共a2; q兲n· · ·共ak; q兲n.

[2.1]

Theorem 2.1. If k ⱖ 2, then the following identity is true

冘 ⬁

F k共z; q兲 ⫽

2

共⫺1兲nz共2k⫺2兲nq共k⫺1兲n .

n⫽0

Lovejoy and Ono

PNAS 兩 June 10, 2003 兩 vol. 100 兩 no. 12 兩 6905

APPLIED MATHEMATICS

The d兾dz series in Theorem 1 is obtained by differentiating summand by summand in z, and then setting z ⫽ 1 and q ⫽ e ⫺t. Although we omit its closed form expression for brevity, we note that it is easily obtained from Eq. 1.5 by using the standard rules for differentiation. Suppose that ␹ mod f is a nontrivial Dirichlet character (i.e. a homomorphism from (⺪兾f⺪)⫻ 3 ⺓⫻) with Dirichlet L function

Theorem 2.2. The following identity is true

冘 ⬁

G共z; q兲 ⫽

冘 ⬁

2

z⫺nqn ⫺

n⫽0

2

zn qn .

n⫽1

Remarks. The series F 2(z; q) is a specialization of the classical Rogers–Fine identity (8)

冘 ⬁

F 2共z; q兲 ⫽

共z兲nzn ⫽ 共⫺z兲n⫹1

n⫽0

冘 ⬁

2

共⫺1兲nz2nqn ,

n⫽0

whereas the series G(z; q) is a specialization of an identity of Andrews (9) related to ‘‘false’’ theta functions in Ramanujan’s lost notebook. Proofs of Theorem 2.1 and 2.2. Two sequences ( ␣ n, ␤ n) are said to form a Bailey pair with respect to a if for every n ⱖ 0,

冘 n

␤n ⫽

k⫽0

␣n . 共q兲n⫺k共aq兲n⫹k

Given such a pair, Andrews (10) has shown that for any natural number k and complex numbers b i, c i we have (subject to convergence conditions)

冉 冉

aq aq , bk ck

aq,

冊 冘 冊 冉

aq b kc k

冉

冊

冘

r 共bk, ck兲nk· · ·共b1, c1兲n1 共b1, c1, . . . , bk, ck兲r ak qk ␣r ⫽ b1c1· · ·bkck aq aq aq aq aq aq aq aq n kⱖn k⫺1ⱖ· · ·ⱖn 1ⱖ0 , ,..., , , ··· , b1 c1 bk ck r bk⫺1 ck⫺1 n b1 c1

⬁

rⱖ0

⬁

冊

⫻

冉

冉

aq bk⫺1ck⫺1

冊

冊 冉 冊 冉 冊 冉 冊 冉 冊 k

··· n k⫺n k⫺1

aq b1 c1

n 2⫺n 1

共q兲nk⫺nk⫺1· · ·共q兲n2⫺n1

aq bk ck

n2

nk

···

aq b1 c1

n1

␤n 1 .

[2.2]

The theorems follow by inserting the right Bailey pairs into the above equation and making appropriate specializations of the parameters. Substituting the Bailey pair with respect to a (9),

␣j ⫽

共a兲 j共1 ⫺ aq 2j兲共⫺1兲jqj共j⫺1兲/2 共q兲j共1 ⫺ a兲

再

1, j ⫽ 0 ␤j ⫽ 0, otherwise,

and

yields a limiting case of Andrews’ multidimensional generalization of Watson’s transformation (11),

冉 冉

aq aq , bk ck

aq,

冊 冘 冊 冉 ⬁

aq b kc k

⬁

冘

⫽

冉

冊

r 共a, q 冑a, ⫺q 冑a, b1, c1, . . . , bk, ck兲r ak qk 共⫺1兲rqr共r⫺1兲/2 b1c1· · ·bkck aq aq aq aq rⱖ0 q, 冑a, ⫺ 冑a, , ,..., , b1 c1 bk ck r

冊

n k⫺1ⱖ· · ·ⱖn 1ⱖ0

冉

冉

共bk, ck兲nk⫺1· · ·共b2, c2兲n1 aq aq , bk⫺1 ck⫺1

冊 冉 冊 ···

n k⫺1

aq aq , b1 c1

aq bk⫺1ck⫺1

冊

··· n k⫺1⫺n k⫺2

冉 冊 aq b1 c1

共q兲nk⫺1⫺nk⫺2· · ·共q兲n1

n1

冉 冊 冉 冊 aq bk ck

n k⫺1

···

aq b2 c2

n1

.

[2.3]

n1

Keeping in mind that (x) r兾x r 3 (⫺1) rq r(r⫺1)/2 and (y兾x) r 3 1 as x 3 ⬁, let a ⫽ z 2, b 1 ⫽ ⫺z, c k ⫽ z, b k ⫽ q, and all remaining b i, c i 3 ⬁ in Eq. 2.3. After a bit of simplification, this is Theorem 2.1. Next insert in Eq. 2.2 the Bailey pair with respect to q (12),

␣ j ⫽ 共⫺z兲⫺jqj共j⫹1兲/2共1 ⫺ z2j⫹1兲 and ␤j ⫽ The k ⫽ 1 case is Andrews’ generalized false theta identity (9),

冘 ⬁

n⫽0

共z兲n⫹1共q兾z, b, c兲n

冉冊 q2 bc

共q兲2n⫹1

6906 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.1131697100

n 2

⫽

2

共q 兾b, q 兾c兲⬁ 共q, q2兾bc兲⬁

冘 ⬁

n⫽0

共z兲j⫹1共q兾z兲j . 共q2兲2j

冉冊

q2 n bc qn共n⫹1兲/2共共⫺z兲n⫹1 ⫹ 共⫺z兲⫺n兲. 共q2兾b, q2兾c兲n 共b, c兲n

Lovejoy and Ono

Letting b ⫽ q, z ⫽ zq, and c 3 ⬁ gives Theorem 2.2. 3. Proofs of Theorems 1 and 2 To prove Theorems 1 and 2, we require the following classical results regarding the Mellin integral representations of L functions. Proposition 3.1. Suppose that ⌿ is a periodic function with modulus f with mean value zero, and let

冘 ⬁

L共s兲 ⫽

n⫽1

⌿共n兲 ⫽ f ⫺s ns

冘 冉 冊 f

r . f

⌿共r兲␨ s,

r⫽1

As t n 0 we have

冘 ⬁

冘 ⬁

2

⌿共n兲e⫺n t ⬃

n⫽1

冘 ⬁

共⫺t兲n , n!

L共⫺2n兲 䡠

n⫽0

冘 ⬁

2

⌿共n兲ne⫺n t ⬃

n⫽1

L共⫺2n ⫺ 1兲 䡠

n⫽0

共⫺t兲n . n!

Proof. By the hypothesis on ⌿, L(s) has an analytic continuation to ⺓. Suppose that H(t) is the asymptotic expansion as t n 0

given by

冘 ⬁

冘 ⬁

b共n兲tn ⬃

n⫽0

2

⌿共n兲e⫺n t.

[3.1]

n⫽1

Using the Mellin integral representation for L(2s) (for ᑬ(s) ⬎ 1), and by letting T ⫽ n 2t, it turns out that

冕 冉冘 ⬁

0

⬁

n⫽1

2

冊

冘 ⬁

⌿共n兲e⫺n t ts⫺1dt ⫽

⌿共n兲

n⫽1

冕

⬁

冕 冉冘 ⬁

0

⌿共n兲 n2s

n⫽1

0

For any N ⬎ 0, this combined with Eq. 3.1 implies that ⌫共s兲L共2s兲 ⫽

冘 ⬁

2

e⫺n tts⫺1dt ⫽

冊

N⫺1

⬁

e⫺TT s⫺1dT ⫽ ⌫共s兲L共2s兲.

[3.2]

0

冘

N⫺1

b共n兲tn ⫹ O共tN兲 t s⫺1dt ⫽

n⫽0

冕

n⫽0

b共n兲 ⫹ F共s兲, s⫹n

where F(s) is analytic for ᑬ(s) ⬎ ⫺N. Therefore, b(n) is the residue at s ⫽ ⫺n of ⌫(s)L(2s), and so b共n兲 ⫽

共⫺1兲n 䡠 L共⫺2n兲. n!

⬁ The same argument applies to the asymptotic expansion of 兺 n⫽1 ⌿(n)ne ⫺n t. We now prove Theorems 1 and 2 by using Theorems 2.1 and 2.2 and Proposition 3.1. Proof of Theorem 1. By Theorem 2.1, we have 2

冘 ⬁

2

2

z 共2k⫺2兲bq共k⫺1兲b Fk共zaqab; qa 兲 ⫽

2

共⫺1兲nz共2k⫺2兲共an⫹b兲q共k⫺1兲共an⫹b兲 .

[3.3]

n⫽0

By letting z ⫽ 1 and q ⫽ e ⫺t in Eq. 3.3, we obtain a power series in t. To see that the t series is well defined, we use the fact that the constant term in the Taylor expansion of 1 ⫺ e ⫺mt is zero for every positive integer m. Each summand in Eq. 1.5 therefore has the property that the numerator contains 2n k⫺1 such factors whereas the denominator has n k⫺1 many [note: the (⫺z; q) n1⫹1 does not contribute any]. Therefore, Theorem 1 for L a,b(⫺2n) follows from Proposition 3.1. To obtain Theorem 1 for L a,b(⫺2n ⫺ 1), we apply the argument above to the series that is obtained by differentiating Eq. 3.3 in z summand by summand before letting z ⫽ 1 and q ⫽ e ⫺t. As above, the form of Eq. 1.5 implies that the resulting t series is well defined. Proof of Theorem 2. By Theorem 2.2, if 0 ⬍ r ⬍ f, then

冘

z⫺共 fn⫹r兲q共 fn⫹r兲 ⫺

冘

␹共r兲z⫺rq r G共z fq⫺2 r f; q f 兲.

⬁

2

2

z ⫺rqr G共z fq⫺2 r f; q f 兲 ⫽

冘 ⬁

2

n⫽0

2

z共 fn⫺r兲q共 fn⫺r兲 .

[3.4]

n⫽1

Recall the definition of G ␹(z; q) (see Eq. 1.8) 关 f/2 兴

G ␹共z; q兲 :⫽

2

2

[3.5]

r⫽1

By Eq. 3.4, this implies that Lovejoy and Ono

PNAS 兩 June 10, 2003 兩 vol. 100 兩 no. 12 兩 6907

APPLIED MATHEMATICS

H共t兲 ⫽

冘冘

关 f/2 兴

G ␹共z; q兲 ⫽

⬁

2

2

␹共r兲共z⫺共fn⫹r兲q共fn⫹r兲 ⫺ z共fn⫹f⫺r兲q共fn⫹f⫺r兲 兲.

r⫽1 n⫽0

Therefore, we have

冘冘

关 f/2 兴

G ␹共1; e ⫺t兲 ⫽

⬁

2

2

␹共r兲共e⫺共fn⫹r兲 t ⫺ e⫺共fn⫹f⫺r兲 t兲,

r⫽1 n⫽0

d 共G ␹共z; q兲兲兩 z⫽1,q⫽e⫺t ⫽ ⫺ dz

冘冘

关 f/2 兴

⬁

2

2

␹共r兲共共fn ⫹ r兲e⫺共fn⫹r兲 t ⫹ 共fn ⫹ f ⫺ r兲e⫺共fn⫹f⫺r兲 t兲.

r⫽1 n⫽0

An inspection of Eq. 1.7 shows that both t series are well defined. Since ␹ (f兾2) ⫽ 0 for even f, Proposition 3.1 implies that if ␹ is odd, then

冘 ⬁

G ␹共1; e ⫺t兲 ⫽

L共⫺2n, ␹兲 䡠

n⫽0

共⫺t兲n . n!

Similarly if ␹ is even, then Proposition 3.1 implies that d 共G ␹共z; q兲兲 z⫽1,q⫽e⫺t ⫽ ⫺ dz

冘 ⬁

L共⫺2n ⫺ 1, ␹兲 䡠

n⫽0

共⫺t兲n . n!

This completes the proof of Theorem 2. 4. Examples Here we give examples of Theorems 1 and 2. Example 1. Here we give the k ⫽ 2 example of Theorem 1 for the function

冘 ⬁

L 2,1共s兲 ⫽

n⫽1

␹⫺1共n兲 , ns

where ␹⫺1 is the unique nontrivial Dirichlet character modulo 4. Theorem 1 implies that

冘 ⬁

e ⫺tF2共e⫺2t; e⫺4t兲 ⫽ e⫺t

n⫽0

共1 ⫺ e⫺2t兲共1 ⫺ e⫺6t兲· · ·共1 ⫺ e⫺2t⫺4共n⫺1兲t兲e⫺2nt 共1 ⫹ e⫺2t兲共1 ⫹ e⫺6t兲· · ·共1 ⫹ e⫺2t⫺4nt兲

5 1 1 61 1385 4 t ⫹··· ⫽ ⫹ t ⫹ t2 ⫹ t3 ⫹ 2 2 4 12 48

冘 ⬁

⫽

n⫽0

L共⫺2n, ␹⫺1兲 䡠

共⫺t兲n . n!

Theorem 1 provides a generating function for the values L(⫺2n ⫺ 1, ␹ ⫺1), where n ⱖ 0. Since ␹⫺1 is odd, these values are zero. Theorem 1 implies that

冘 ⬁

0⫽2

n⫽0

L共⫺2n ⫺ 1, ␹⫺1兲 䡠

d 共⫺t兲n ⫽ 共z2qF2共z2q2; q4兲兲兩z⫽1,q⫽e⫺t ⫽ n! dz ⫽

冉

冊 冉

冘 冉 ⬁

n⫽0

冊

d 共z2q2; q4兲nz2n⫹2q2n⫹1 兩z⫽1,q⫽e⫺t dz 共⫺z2q2; q4兲n⫹1

冊 冉

冊

1 17 1 3 15 1 1 ⫹ t ⫺ t2 ⫺ · · · ⫹ ⫺ ⫹ t ⫹ t2 ⫺ · · · ⫹ ⫺2t ⫹ t2 ⫹ · · · ⫹ · · · . 2 2 4 2 2 4 2

Example 2. Here we compute the values of L(s, ␹ ⫺1) at even nonpositive integers again by using Theorem 2. By Theorem 2, we find

that

6908 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.1131697100

Lovejoy and Ono

冘 ⬁

G ␹⫺1共1; e⫺t兲 ⫽ e⫺tG共e8t; e⫺16t兲 ⫽ e⫺t

n⫽0

冘 ⬁

⫽

L共⫺2n, ␹⫺1兲 䡠

n⫽0

共⫺1兲n共e⫺16t; e⫺16t兲n共e⫺8t; e⫺16t兲n共e⫺8t; e⫺16t兲n⫹1e⫺8n共n⫹1兲t 共e⫺16t; e⫺16t兲2n⫹1

5 61 1385 4 共⫺t兲n 1 1 ⫽ ⫹ t ⫹ t2 ⫹ t3 ⫹ t ⫹ · · ·. n! 2 2 4 12 48

Example 3. Suppose that ␹5 is the Dirichlet character modulo 5 given by the Legendre symbol modulo 5. By definition, we have

that

G ␹5共z; q兲 ⫽ z ⫺1qG共z5q⫺10; q25兲 ⫺ z⫺2q4G共z5q⫺20; q25兲, and so we have G ␹5共z; q兲 ⫽

⫺

冘

n⫽0

⬁

n⫽0

25n 共 n⫹1 兲 ⫹1 2

共⫺1兲n共q25; q25兲n共z⫺5q20; q25兲n共z5q5; q25兲n⫹1z⫺2q 共q25; q25兲2n⫹1

25n 共 n⫹1 兲 ⫹4 2

.

By differentiating in z summand by summand, then letting z ⫽ 1 and q ⫽ e ⫺t, Theorem 2 gives 67 2 361 3 d 共G ␹5共z; q兲兲兩 z⫽1,q⫽e⫺t ⫽ ⫹ 2 t ⫹ t2 ⫹ t ⫹···⫹ dz 5 5 3

冘 ⬁

⫽⫺

n⫽0

L共⫺2n ⫺ 1, ␹5兲 䡠

共⫺t兲n . n!

K.O. is supported by National Science Foundation Grant DMS-9874947, an Alfred P. Sloan Foundation Research Fellowship, a David and Lucile Packard Research Fellowship, and an H. I. Romnes Fellowship. Zagier, D. (2001) Topology 40, 945–960. Andrews, G. E., Jime´nez-Urroz, J. & Ono, K. (2001) Duke Math. J. 108, 395–419. Coogan, G. H. & Ono, K. (2003) Proc. Am. Math. Soc. 131, 719–724. Hikami, K. (2003) Volume Conjecture and Asymptotic Expansion of q Series, preprint. 5. Apostol, T. (1984) Introduction to Analytic Number Theory (Springer, New York). 6. Andrews, G. E., Askey, R. & Roy, R. (1998) Special Functions (Cambridge Univ. Press, Cambridge, U.K.). 7. Gasper, G. & Rahman, M. (1990) Basic Hypergeometric Series (Cambridge Univ. Press, Cambridge, U.K.).

1. 2. 3. 4.

Lovejoy and Ono

8. Fine, N. (1988) Basic Hypergeometric Series and Applications (Am. Math. Soc, Providence, RI). 9. Andrews, G. E. (1981) Adv. Math. 41, 173–185. 10. Andrews, G. E. (1986) q-Series: Their Development and Application in Analysis, Combinatories, Physics, and Computer Algebra, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 66 (Am. Math. Soc, Providence, RI). 11. Andrews, G. E. (1975) Problems and Prospects for Basic Hypergeometric Series: The Theory and Application of Special Functions (Academic, New York). 12. Andrews, G. E. (1992) Illinois J. Math. 36, 251–274.

PNAS 兩 June 10, 2003 兩 vol. 100 兩 no. 12 兩 6909

APPLIED MATHEMATICS

冘

共⫺1兲n共q25; q25兲n共z⫺5q10; q25兲n共z5q15; q25兲n⫹1z⫺1q 共q25; q25兲2n⫹1

⬁