Zeta functions, topology and quantum physics

ZETA FUNCTIONS, TOPOLOGY
AND QUANTUM PHYSICS


Developments in Mathematics
VOLUME 14
Series Editor:
Krishnaswami Alladi, University of Florida, U.S.A.

Aims and Scope
Developments in Mathematics is a book series publishing
(i)

Proceedings of conferences dealing with the latest research
advances,

(ii)

Research monographs, and

(iii)

Contributed volumes focusing on certain areas of special
interest.

Editors of conference proceedings are urged to include a few survey
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ZETA FUNCTIONS, TOPOLOGY
AND QUANTUM PHYSICS

Edited by
TAKASHI AOKI
Kinki University, Japan
SHIGERU KANEMITSU
Kinki University, Japan
MIKIO NAKAHARA
Kinki University, Japan
YASUO OHNO
Kinki University, Japan

a- springer

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Zeta functions, topology, and quantum physics 1 edited by Takashi Aoki ... [et al.].
p. cm. - (Developments in mathematics ; v. 14)
Includes bibliographicalreferences.
ISBN 0-387-24972-9 (acid-freepaper) - ISBN 0-387-24981-8 (e-book)
1. Functions, Zeta-Congresses. 2. Mathematical physics-Congresses. 3.
Differential geometry-Congresses. I. Aoki, Takashi, 1953- 11.Series.

AMS Subiect Classifications: 1 1Mxx. 35Qxx. 34Mxx. 14Gxx. 51PO5
ISBN-10: 0-387-24972-9
e-ISBN-10: 0-387-24981-8

ISBN-13: 978-0387-24972-8
e-ISBN-13: 978-0387-24981-0

Printed on acid-free paper.
O 2005 Springer Science+Business Media, Inc.

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Contents

xi
xii
xiv

Preface
Conference schedule
List of participants
Gollnitz-Gordon partitions with weights and parity conditions
Krishnaswami Alladi and Alexander Berlcovich
1
Introduction
2
A new weighted partition theorem
3
Series representations
4
A new infinite hierarchy
Acknowledgments
References
Partition Identities for the Multiple Zeta Function
David M. Bradley
1
Introduction
2
Definitions
3

Rational Functions
4
Stuffles and Partition Identities
References

A perturbative theory of the evolution of the center of typhoons
Sergey Dobrolchotov, Evgeny Semenov, Brunello Tirozzi
Introduction
D namics of vortex square-root type singularities and Hugonibtd s l o v chains
Equation for the smooth and singular part of the solutions
3
Cauchy-Riemann conditions
Derivation of the Hugonibt-Maslov chain using complex vari4
ables and its integrals
Acknowledgments
1
2

References

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1


viii

ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS

Algebraic Aspects of Multiple Zeta Values

Michael E. Hoffman
1
Introduction
2
The Shuffle Algebra
3
The Harmonic Algebra and Quasi-Symmetric Functions
4
Derivations and an Action by Quasi-Symmetric Functions
5
Cyclic Derivations
6
Finite Multiple Sums and Mod p Results

51

References

71

On the local factor of the zeta function of quadratic orders
Masanobu Kaneko
Acknowledgments

79

References

79

51
54
56
60
63
64

Sums involving the Hurwitz zeta-function values
S. Kanemitsu, A. Schinzel, Y. Tanigawa
1
Introduction and statement of results
2
Proof of results
References

89

Crystal Symmetry Viewed as Zeta Symmetry
91
Shigeru Kanemitsu, Yoshio Tanigawa, Haruo Tsukada, Masami Yoshimoto
1
Introduction
92
2
Lattice zeta-functions and Epstein zeta-functions
103
3
Abel means and screened Coulomb potential
120
References

128

Sum relations for multiple zeta values
Yasuo Ohno
1
Introduction
2
Generalizations of the sum formula
3
Identities associated with Arakawa-Kaneko zeta functions
4
Multiple zeta-star values and restriction on weight, depth, and
height
Acknowledgment

131

142
143

References

143

The Sum Formula for Multiple Zeta Values
OKUDA Jun-ichi and UENO Kimio
1
Introduction

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131
133
140


ix

Contents
Acknowledgment
2
Shuffle Algebra
Multiple Polylogarithms and the formal KZ equation
3
Mellin transforms of polylogarithms and the sum formula for
4
MZVs
Knizhnik-Zamolodchikov
equation over the configuration space
5
x3 (@)

References
Zeta functions over zeros of general zeta and L-functions
Andre' Voros
1
Generalities
2
The first family { T ( s , x))

3

The second family ( 2(a,
v))

4

The third family {3(a,
y))
Concrete examples

5

References
Hopf Algebras and Transcendental Numbers
Michel Waldschmidt
Transcendence, exponential polynomials and commutative lin1
ear algebraic groups
2
Bicommutative Hopf algebras
Hopf algebras and multiple zeta values
3
References

218

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Preface

This volume contains papers by invited speakers of the symposium
"Zeta Functions, Topology and Quantum Physics" held at Kinki University in Osaka, Japan, during the period of March 3-6, 2003. The
aims of this symposium were to establish mutual understanding and to
exchange ideas among researchers working in various fields which have
relation to zeta functions and zeta values.
We are very happy to add this volume to the series Developments
in Mathematics from Springer. In this respect, Professor Krishnaswami
Alladi helped us a lot by showing his keen and enthusiastic interest in
publishing this volume and by contributing his paper with Alexander
Berkovich.
We gratefully acknowledge financial support from Kinki University.
We would like to thank Professor Megumu Munakata, Vice-Rector of
Kinki University, and Professor Nobuki Kawashima, Director of School
of Interdisciplinary Studies of Science and Engineering, Kinki University, for their interest and support. We also thank John Martindale of
Springer for his excellent editorial work.
Osaka, October 2004
Takashi Aoki
Shigeru Kanemitsu
Mikio Nakahara
Yasuo Ohno

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xii

ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS

Z e t a Functions, Topology,

and
Q u a n t u m Physics
Kinki University, Osaka, Japan
3 - 6 March

2003

3 March
M. Waldschmidt (Paris VI)
How to prove relations between polyzeta values using automata
H. Tsukada (Kinki Univ.)
Crystal symmetry viewed as zeta symmetry
(cowork with S. Kanemitsu, Y. Tanigawa and M. Yoshimoto)
S. Akiyama (Niigata Univ.)
Quasi-crystals and Pisot dual tiling
K. Alladi (Florida)
Insights into the structure of Rogers-Ramanujan type identities, some
from physics
A. Voros (Saclay)
Zeta functions for the Riemann zeros
4 March
Y. Ohno (Kinki Univ.)
Sum relations for multiple zeta values
M. Hoffman (U. S. Naval Acad.)
Algebraic aspects of multiple zeta values
B. Tirozzi (Rome)
Application of shallow water equation to typhoons
J. Okuda (Waseda Univ.)
Multiple zeta values and Mellin transforms of multiple polylogarithms
(cowork with K. Ueno)

D. Broadhurst (The Open Univ.)
Polylogarithms in quantum field theory
High School Session (Two lectures for younger generation)
(i) K. Alladi (Univ. Florida)

Prime numbers and primality testing
(ii) M. Waldschmidt (Univ. Paris VI)
Error correcting codes

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xiii

Conference schedule

5 March
M. Kaneko (Kyushu Univ.)
On a new q-analogue of the Riemann zeta function
K. F'ukaya (Kyoto Univ.)
Theta function and its potential generalization which appear in Mirror
symmetry

6 March
T . Ibukiyama (Osaka Univ.)
Graded rings of Siege1 modular forms and differential operators
D. Bradley (Maine)
Multiple polylogarithms and multiple zeta values: Some results and
conjectures
J. Murakami (Waseda Univ.)

Multiple zeta values and quantum invariants of knots
A. Schinzel (Warsaw)
An extension of some formulae of Lerch
G. Lachaud (CNRS)
Eisenstein series and the Riemann hypothesis

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xiv

ZETA FUNCTIONS, T O P O L O G Y AND QUANTUM PHYSICS

List of participants
Akiyama, Shigeki
Alladi, Krishnaswami
Aoki, Takashi
Arakawa, Tsuneo
Asada, Akira
Asai, Tsunenobu
Bradley, David M.
Broadhurst, David J .
Chinen, Koji
Fujita, Keiko
Fujiwara, Hidenori
Fukaya, Kenji
Hasegawa, Hiroyasu
Hata, Kazuya
Hirabayashi, Mikihito
Hironaka, Yumiko

Hoffman, Michael E.
Ibukiyama, Tomoyoshi
Ishii, Tadamasa
Izumi, Shuzo
Kaneko, Masanobu
Kanemitsu, Shigeru
Kawashima, Nobuki
Kimura, Daiji
Kogiso, Takeyoshi
Komatsu, Takao
Kondo, Yasushi
Kubota, Yoshihiro
Kumagai, Hiroshi
Kuribayashi, Masanori
Lachaud, Gilles
Maruyama, Fumitsuna
Mima, Yuki
Mizuno, Yoshinori
Munakata, Megumu
Munemoto, Tomoyuki
Murakami, Jun
Nagaoka, Shoyu
Nakagawa, Koichi
Nakagawa, Nobuo
Nakahara, Mikio
Nishihara, Hideaki

Niigata University, Japan
University of Florida, Gainesville, USA
Kinki University, Japan

Rikkyo University, Japan
Hyogo, Japan
Kinki University, Japan
University of Maine, USA
Open University, UK
Osaka Institute of Technology, Japan
Saga University, Japan
Kinki University, Japan
Kyoto University, Japan
Kinki University, Japan
Kinki University, Japan
Kanazawa Institute of Technology, Japan
Waseda University, Japan
U. S. Naval Academy, USA
Osaka University, Japan
Kinki University, Japan
Kinki University, Japan
Kyushu University, Japan
Kinki University, Japan
Kinki University, Japan
Hiroshima University, Japan
Josai University, Japan
Mie University, Japan
Kinki University, Japan
The University of the Air, Japan
Kagoshima National College of Technology, Japan
Osaka University, Japan
Institut Mathhmatiques, Luminy, France
Toyo University, Japan
Kinki University, Japan

Osaka University, Japan
Kinki University, Japan
Kinki University, Japan
Waseda University, Japan
Kinki University, Japan
Hoshi University, Japan
Kinki University, Japan
Kinki University, Japan
Osaka University, Japan

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List of participants
Ochiai, Hiroyuki
Ohishi, Ryoko
Ohno, Yasuo
Ohyama, Yousuke
Okazaki, Ryotaro
Okuda, Jun-ichi
Owa, Shigeyoshi
Sakuma, Kazuhiro
Sato, Fumihiro
Schinzel, Andrzej
Sugiyama, Kazunari
Suzuki, Masatoshi
Takahashi, Hiroaki
Takahashi, Koichi
Takei, Yoshitsugu
Tanaka, Satoshi

Tanaka, Tatsushi
Tanigawa, Yoshio
Tanimura, Shogo
Tazawa, Shinsei
Terajima, Hitomi
Tirozzi, Brunello
Toda, Masayuki
Tohyama, Masaki
Tsukada, Haruo
Uchiyama, Tadashi
Ueno, Kimio
Ushio, Kazuhiko
Voros, Andre
Waldschmidt, Michel
Watanabe, Masashi
Yoshimoto, Masami
Yuasa, Manabu

Nagoya University, Japan
University of Tokyo, Japan
Kinki University, Japan
Osaka University, Japan
Doshisha University, Japan
Waseda University, Japan
Kinki University, Japan
Kinki University, Japan
Rikkyo University, Japan
Polish Academy of Science, Institute of Mathematics, Poland
Tsukuba University, Japan
Nagoya University, Japan

Takamatsu National College of Technology, Japan
Kinki University, Japan
RIMS, Kyoto University, Japan
Kinki University, Japan
Kyushu University, Japan
Nagoya University, Japan
Kyoto University, Japan
Kinki University, Japan
Kobe University, Japan
University of Rome, La Sapienza, Italy
Kinki University, Japan
Tokyo University of Science, Japan
Kinki University, Japan
Kinki University, Japan
Waseda University, Japan
Kinki University, Japan
CEA, Saclay, France
Institut Mathhmatiques, Paris, France
Kyushu University, Japan
Nagoya University, Japan
Kinki University, Japan

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Zeta Functions, Topology and Quantum Physics, pp. 1-18
T. Aoki, S. Kanemitsu, M. Nakahara and Y. Ohno, eds.
© 2005 Springer Science + Business Media, Inc.

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2

ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS

The analytic representation of Theorem 1 is

when i = 1, and

when i = 3. In (1.1)) (1.2), and in what follows, we have used the
standard notation

for any complex number a, and
03

for 141 < 1. The products on the right in (1.1)) (1.2) are also equal to

and

respectively, which have obvious interpretations as generating functions
of partitions into parts in certain residue classes (mod 8)) repetition
allowed. The equally well known Gollnitz-Gordon partition theorem is

-

Theorem 2. For i = 1,3, the number of partitions into parts
fi, 4
(mod 8) equals the number ofpartitions into parts differing by 2 2, where
the inequality is strict i f a part is even, and the smallest part is 2 i.

The analytic representation of Theorem 2 is

when i = 1, and

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Gollnitz-Gordon partitions with weights and parity conditions

3

when i = 3. Actually (1.3) and (1.4) are equations (36) and (37) in
Slater's famous list [9], but it was Gollnitz [6] and Gordon [7] who independently realized their combinatorial interpretation.
By a reformulation of the (Big) Theorem of Gollnitz [6] (not Theorem 1) using certain quartic transformations, Alladi [l]provided a uniform treatment of all four partition functions Qi(n), i = 0 , 1 , 2 , 3 in
terms of partitions into parts differing by 1 4, and with certain powers
of 2 as weights attached. As a consequence, it was noticed in [I] that
Q2(n) and Qo(n) possess certain more interesting properties than their
well known counterparts Ql (n) and Qs(n). In particular, Q2(n) alone
among the four functions satisfies the property that for every positive
integer k, Q2(n) is a multiple of 2k for almost all n which was proved by
Gordon in an Appendix to [I].
Our goal is to prove Theorem 3 in 52 which shows that by attaching
weights which are powers of 2 to the Gollnitz-Gordon partitions of n, and
by imposing certain parity conditions, this is made equal to Q2(n). Here
by a Gollnitx-Gordon partition we mean a partition into parts differing
by 2 2, where the inequality is strict if a part is even. There is a
similar result for Qo(n), and this is stated as Theorem 4 at the end of
52. Theorems 3 and 4 are nice complements to Theorem 1 and to results
of Alladi [I].
A combinatorial proof of Theorem 3 is given in full in the next section.

Theorem 4 is only stated, and its proof which is similar, is omitted.
In proving Theorem 3 we are able to cast it as an analytic identity (see
(3.2) in 53) which equates a double series with the product which is the
generating function of Q2(n). It turns out that there is a two parameter
refinement of (3.2) (see (3.3) of $3) which leads to similar double series
representations for all four products

m>O,m$i (mod 4)

for i = 0,1,2,3. It will be shown in 53 that only in the cases i = 1 , 3 do
these double series reduce to the single series in (1.1) and (1.2).
Actually, the double series identity (3.2) is the case k = 2 of a new
infinite hierarchy of identities valid for every k 1 1. In $4 we use a
limiting case of Bailey's lemma to derive this hierarchy. We give a partition theoretic interpretation of the case k = 1 and state without proof a
doubly bounded polynomial identity which yields our new hierarchy as
a limiting case. This polynomial identity will be investigated in detail
elsewhere.

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4

2.

ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS

A new weighted partition theorem

Normally, by the parity of an integer we mean its residue class

(mod 2). Here by the parity of an odd (or even) integer we mean its
residue class (mod 4).
Next, given a partition n into parts differing by 2, by a chain x in
n we mean a maximal string of parts differing by exactly 2. Thus every
partition into parts differing by 2 2 can be decomposed into chains.
Note that if one part of a chain is odd (resp. even), then all parts of the
chain are odd (resp. even). Hence we may refer to a chain as an odd
chain or an even chain. Also let X(X) denote the least part of a chain x
and X(n) the least part of n.
Note that in a Gollnitz-Gordon partition, since the gap between even
parts is > 2, this is the same as saying that every even chain is of length
1, that is, it has only one element.
Finally, given part b of partition T, by t(b; n ) = t(b) we denote the
number of odd parts of n that are < b. With this new statistic t we now
have

>

Theorem 3. Let S denote the set of all special Gollnitz-Gordon partitions, namely, Gollnitx-Gordon partitions n satisfying the parity condition that for every even part b of n
b = 2t(b)

Decompose each IT E S into chains

4x1 =

2,

if

(mod 4).

(2.1)

x and define the weight w(x) as

-

x is an odd chain, X(X) > 5,

and X(X) 1
1, otherwise.

+ 2t(X(x))

(mod 4),

(2.2)

The weight w(n) of the partition n is defined multiplicatively as

the product over all chains

x of T.

W e then have

where a ( n ) is the sum of the parts of T.

+ + +

Proof: Consider the partition n : bl b2 - - . bN, n E S, where
contrary to the standard practice of writing parts in descending order, we
< bN. Subtract 0 from bl, 2 from b2, . . . , 2 N - 2
now have bl < b2 <

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5

Gollnitz-Gordon partitions with weights and parity conditions

from bN, to get a partition n*. We call this process the Euler subtraction.
Note that in n* the even parts cannot repeat, but the odd parts can.
Let the parts of n* be bT 5 ba 5 . . . 5 b&.
Now identify the parts of n which are odd, and which are the smallest
parts of chains and satisfy both the parity and low bound conditions in
(2.2). Mark such parts with a tilde at the top. That is, if bk is such a
part, we write bk = bk for purposes of identification. Let bk yield bi = b i
after the Euler subtraction.
Next, split the parts of n* into two piles nT and ng, with nT consisting
only of certain odd parts, and n; containing the remaining parts. In this
decomposition we adopt the following rule:
(a) the odd parts of n* which are not identified as above are put in
7rT.
(b) the odd parts of n* which have been identified could be put in
either nT or ng.
Thus we have two choices for each identified part.
Let us say, in a certain given situation, after making the choices, we
have n l parts in nT and n2 parts in ng. We now add 0 to the smallest

part of n;, 2 to the second smallest part of ng, ..., 2n2 - 2 to the largest
part of n;, 2n2 to the smallest part of nT, 2n2 2 to the second smallest
part of nT, ..., 2(nl n2) - 2 = 2 N - 2 to the largest part of nT. We
call this the Bressoud redistribution process. As a consequence of this
redistribution, we have created two partitions
(out of nT) and 7r2 (out
of n;) satisfying the following conditions:
(i) n1 consists only of distinct odd parts, with each odd part being
greater than twice the number of parts of n2.
(ii) Since both the even and odd parts of n; are distinct, the parts
of n2 differ by 2 4. Also since the odd parts of n; are chosen from the
smallest of parts of certain chains in n, the odd parts of .rm actually differ
by 2 6, and each such odd part is 2 5.
In transforming the original partition n into the pair (nl, n2), we need
to see how the parity conditions of n given by (2.1) and (2.2) transform
to parity conditions in nl and ~ 2 .
First observe that since the parity conditions on n are imposed only
.
the identified odd parts of n , the transformed
on the even parts of 7 ~ and
parity conditions (to be determined below) will be imposed only on 7r2
and not on nl. Thus nl will satisfy only condition (i) above.
Suppose bk is an even part of n and that t(bk;n ) = t, that is there are
t odd parts of n which are less than bk. Now bk becomes

-

-

+

+

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-


6

ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS

after the Euler subtraction. Notice that t(b$;n*) = t(bk;n) = t. Now
suppose that from among the t odd parts of n* less than b$, r of them
are put in n; and the remaining t - r odd parts are put in n$. Then
b$ becomes the ( k - r ) - t h smallest part in n$. SO in the Bressoud
redistribution process, 2(k - r ) - 2 is added to b; making it a new even
part ek-r in n2. Thus

We see from (2.1) and (2.3) that

ek+

E 2t

- 2r = 2(t - r ) = 2t(ek-,;

n2)

(mod 4 )

(2.4)

and so the parity condition (2.1) on the even parts does not change when
.
we may write (2.4) in short as
going to ~ 2 Thus

e

= 2t(e)

(2.5)

(mod 4 )

for any even part in n2.
Now we need to determine the parity conditions on the odd parts in
7r2 which are derived from some of the identified odd parts of T . To
-this end
- suppose that is an identified odd part of n which becomes
is
b; = bk - (2k - 2 ) in n* due to the Euler subtraction, and that
placed in ~r; Let t $ k ; T ) = t. Notice that

zk

&,

Suppose that from among the t odd parts of n* which are

r of them
are placed in a; and the remaining t - r are placed in T;. Then
becomes the ( k - r ) - t h smallest part in n$. Thus under the Bressoud
redistribution, 2(k - r ) - 2 is added to it to yield the part fk given by

as in (2.3). Therefore the parity condition (2.2) yields

fk

E

1 +2t

- 2r

= 1 +2(t - r )

(mod 4 ) .

But t ( f k n2)
; = t - r . So this could be expressed in short as

for any odd part of ~ 2 Thus
.
the pair of partitions ( n l ,n2)is determined
by condition (i) on nl, and conditions (ii) and the parity conditions (2.5)
and (2.6) on n2.

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Gollnitz-Gordon partitions with weights and parity conditions

7

In going from 7r to the pair (7r1, 7r2) we had a choice of deciding whether
an identified part of 7r would end up in 7rl or 7r2. This choice is precisely
the weight w ( x ) = 2 associated with certain chains X. The weight of
the partition 7r is computed multiplicatively because these choices are
independent. So what we have established up to now is:
Lemma 1. The weighted count of the special Gollnitz-Gordon partitions
of n equals the number of bipartitions (7rl, 7r2) of n satisfying conditions
(i), (ii), (2.5) and (2.6).
Next, we discuss a bijective map

where 7r3 is a partition into distinct multiples of 4 and
into distinct odd parts such that

71-4 is

a partition

Here by U(T) we mean the number of parts of a partition 7r and by A(7r)
the largest part of 7r.
To describe the map (2.7) we represent .rra as a Ferrers graph with
weights 1 , 2 or 4, at each node. We construct the graph as follows:
1) With each odd (resp. even) part f (resp. e) of ./ra we associate a
row of 3+f:2t(f)
(resp. T
e+2t(e)

) nodes.
2) We place a 1 at end of any row that represents an odd part of 7r2.
3) Every node in the column directly above each 1 is given weight 2.
4) Each remaining node is given weight 4.
Every part of 7r2 is given by the sum of weights in an associated row.
It is clear from these weights, that the partition represented by this
weighted Ferrers graph satisfies precisely the conditions (ii), (2.5) and
(2.6) that characterize 7r2.

Next we extract from this weighted Ferrers graph all columns with a 1
at the bottom, and assemble these columns as rows to form a 2-modular
Ferrers graph as shown below.

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8

ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS

Clearly this 2-modular graph represents a partition 7r4 that satisfies
condition (2.9).
After this extraction, the decorated graph of 7r2 becomes a 4-modular
graph (in this case a graph with weight 4 at every node). This graph 7r3
clearly satisfies (2.8).

It is easy to check that (2.7) is a bijection. Thus Lemma 1 can be
recasted in the form

Lemma 2. The weighted count of the special Gollnitz-Gordon partitions

of n as in Theorem 3 is equal to the number of partitions of n i n the
)
form (TI, 7r3, ~ 4 where
(iii) 7r3 consists only of distinct multiples of 4,
(iv) 7r4 has distinct odd parts and A(7r4) < 2v(r3),
(v) 7rl has distinct odd parts and X(7rl) > 2v(r3),
Finally, observe that conditions (iv) and (v) above yield partitions
into distinct odd parts (without any other conditions). This together
with (iii) yields partitions counted by Q2(n), thereby completing the
combinatorial proof of Theorem 3.
In a similar fashion, we can obtain the following representation for
Qo(n) with weights and parity conditions imposed on the Gollnitz-Gordon
partitions:

Theorem 4. Let S* denote the set of all special Gollnitz-Gordon partitions, namely, Gollnitz-Gordon partitions 7r satisfying the parity condition that for every even part b of IT

Decompose each T E S* into chains

W(X>

=

x

and define the weight w(x) as

2, if x is an odd chain, X(X) 2 3,
and X(X) = 2t(X(x)) - 1 (mod 4),
1, otherwise.

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(2.11)


Gollnitz-Gordon partitions with weights and parity conditions
The weight w(n-) of the partition n- is defined multiplicatively as

the product over all chains

x of n-.

We then have

where a ( n ) is the s u m of the parts ofn-.

3.

Series representations

If we let u(nl) = nl and u(n-2) = n2, then (2.7) and conditions (iii),
(iv), and (v) of Lemma 2 imply that the generating function of all such
triples of partitions (nl, n-3, n4) is

If the expression in (3.1) is summed over all non-negative integers n l
and n2, it yields

By just following the above steps we can actually get a two parameter
refinement of (3.2), namely,

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10

ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS

One may view (3.2) as the analytic version of Theorem 3. In reality,
the correct way to view (3.2) is that, if the summand on the left is
decomposed into three factors as (3.1), then (3.2) is the analytic version
of the statement that the number of partitions of an integer n into the
triple of partitions (7r1, 7r3, 7r4) is equal to Q2(n). This is of course only
the final step of the proof given above. and (3.2), which is quite simple,
is equivalent to it.
The advantage in the two parameter refinement (3.3) is that by suitable choice of the parameters we get similar representations involving
Qi(n) for i = 0,1,3. For example, if we replace w by wq-2 in (3.3) we
get

which is the analytic representation of Theorem 4 above.
Next, replacing z by zq and w by wq-l in (3.3) we get

Now choose z = 1 in (3.5). Then the double series on the left becomes

If we now put n = n l
the form

+ n2 and j

= n2, then (3.6) could be rewritten in

which is the single series identity (1.1) in a refined form.

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Gollnitz-Gordon partitions with weights and parity conditions

Similarly, replacing w by wqW3and z by zq in (3.2) we get

Now the choice z = 1 makes the double series in (3.8) as

which is a refinement of the single series identity (1.2). Thus precisely
in the cases i = 1'3, can the double series be reduced to single series by
setting one of the parameters z = 1.

4.

A new infinite hierarchy

Identity (3.2) given above is just the case k = 2 of a new infinite
hierarchy of multiple series identities (4.12) given below.
To derive this hierarchy, we will need the definition of a Bailey pair,
and a special case of Bailey's lemma which produces a new Bailey pair
from a given Bailey pair [2].

Definition: A pair of sequences an(q), Pn(q) is called a Bailey pair
(relative to 1) if for all n 2 0

By setting a = 1,pl = -q3, and letting p2 + oo in the formulas (3.29)
and (3.30) of [2], we obtain the following limiting case of Bailey's lemma:

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12

ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS

Lemma 3. Suppose (an(q),,&(q)) is a Bailey pair.
pn(1)(4)) is another Bailey pair, where

Then (aL1)(q),

From (a,(1)(q), Pn(1)(9)) one can produce next Bailey pair (a,(2) (q),
,&(2) (4)) simply using (a,(1)(q), ,&(1)(9)) as the initial Bailey pair. It is
easy to check that the k-fold iteration of (the limiting case of) Bailey's
Lemma yields

+

+

+

where 7? = ( n l , n z , . . . , n k ) and Ni = ni ni+l ... nk, with i =
1 , 2 , . . . , k . In [8], [9] Slater derived A-M families of Bailey pairs to
produce the celebrated list of 130 identities of the Rogers-Ramanujan
type. We shall need her E(4) pair:

It follows from (4.1) and (4.4)-(4.6) that

where q-binomial coefficients are defined as

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