Rank-Crank type PDEs and generalized Lambert series identities

0.25 MB·English

by  Song Heng Chan

RANK-CRANK TYPE PDES AND
GENERALIZED LAMBERT SERIES IDENTITIES
SONG HENG CHAN, ATUL DIXIT AND FRANK G. GARVAN
2
Dedicated to our friends, Mourad Ismail and Dennis Stanton
1
0
2
Abstract. We show how Rank-Cranktype PDEs for higher orderAppell functions

n due to Zwegers may be obtained from a generalized Lambert series identity due to
a
the first author. Special cases are the Rank-Crank PDE due to Atkin and the third
J
authorandaPDEforalevel5Appellfunctionalsofoundbythethirdauthor. These
8
two specialPDEs are relatedto generalizedLambert series identities due to Watson,

] and Jackson respectively. The first author’s Lambert series identities are common
T
generalizations. We also show how Atkin and Swinnerton-Dyer’s proof using elliptic
N functions can be extended to prove these generalized Lambert series identities.
.
h
t
a
m
[

1
1. Introduction
v
3
F. J. Dyson [9], [10, p. 52] defined the rank of a partition as the largest part minus
6
6 the number of parts. Dyson conjectured that the residue of the rank mod 5 divides
1
the partitions of 5n + 4 into 5 equal clases thereby providing a combinatorial inter-
.
1
pretation of Ramanujan’s famous partition congruences p(5n + 4) 0(mod5). He
0 ≡
2 also conjectured that the rank mod 7 likewise gives Ramanujan’s partition congru-
1 ence p(7n+5) 0(mod7). Dyson’s rank conjectures were proved by A. O. L. Atkin
: ≡
v and H. P. F. Swinnerton-Dyer [3]. The following was the crucial identity that Atkin
i
X and Swinnerton-Dyer needed for the proof of the Dyson rank conjectures. It was first
r proved by G.N. Watson [18].
a
[ζ2] ∞ ( 1)nq3n(n+1)/2 [ζ] [ζ2] (q)2
ζ ∞ − + ∞ ∞ ∞ (1.1)
[ζ] 1 zqn [z/ζ] [z] [ζz]
∞ nX=−∞ − ∞ ∞ ∞
= ∞ ( 1)nq3n(n+1)/2 ζ−3n + ζ3n+3 .
− 1 zqn/ζ 1 zζqn
n= (cid:18) − − (cid:19)
X−∞
Date: January 8, 2012.
2010 Mathematics Subject Classification. 11F11, 11P82,11P83, 33D15.
Key words and phrases. Rank-CrankPDE,higher levelAppell function, Lambertseries,partition,
q-series, basic hypergeometric function, quasimodular form.
The third author was supported in part by NSA Grant H98230-09-1-0051.
1
2 SONGHENGCHAN,ATUL DIXITANDFRANKG. GARVAN
Throughout we use the standard q-notation
(x) := (x;q) := 1,
0 0
n 1
−
(x) := (x;q) := (1 xqm),
n n
−
m=0
Y
(x , ,x ) := (x , ,x ;q) := (x ;q) (x ;q) ,
1 m n 1 m n 1 n m n
··· ··· ···
[x , ,x ] := [x , ,x ;q] := (x ,q/x , ,x ,q/x ;q) .
1 m n 1 m n 1 1 m m n
··· ··· ···
when n is a nonnegative integer. Assuming q < 1 we also use this notation when
| |
n = by interpreting its meaning as the limit as n . Later M. Jackson [14]
∞ → ∞
proved an analogue of the above identity,
ζ2[ζ2] [xζ] [x/ζ] ∞ ( 1)nq5n(n+1)/2 [ζ] [ζ2] [xζ] [ζ/x] (q)2
∞ ∞ ∞ − + ∞ ∞ ∞ ∞ ∞
[ζ] [x]2 1 zqn [z/x] [z/ζ] [z] [zζ] [zx]
∞ ∞ nX=−∞ − ∞ ∞ ∞ ∞ ∞
+ ζ [ζ]∞[ζ2]∞ ∞ ( 1)nq5n(n+1)/2 x−5n + x5n+5 (1.2)
x[x] [x2] − 1 zqn/x 1 zxqn
∞ ∞ nX=−∞ (cid:18) − − (cid:19)
= ∞ ( 1)nq5n(n+1)/2 ζ−5n + ζ5n+5 .
− 1 zζ 1qn 1 zζqn
n= (cid:18) − − − (cid:19)
X−∞
Recently, the first author [8, p.603] found a generalization of the above two identities,
namely,
xm[x /x , ,x /x ,x x , ,x x ,x2] ∞ ( 1)nq(2m+1)n(n+1)/2
1 2 1 ··· m 1 1 m ··· 1 2 1 ∞ −
[x ] [x , ,x ]2 1 zqn
1 2 m
∞ ··· ∞ nX=−∞ −
[x /x , ,x /x ,x ,x x , ,x x ,x2] (q)2
+ 1 2 ··· 1 m 1 1 m ··· 1 2 1 ∞ ∞
[z/x ,z/x , ,z/x ,z,zx , ,zx ]
1 2 m m 1
··· ··· ∞
x [x /x , ,x /x ,x ,x x , ,x x ,x2]
+ 1 1 3 ··· 1 m 1 1 m ··· 1 3 1 ∞ (1.3)
x [x /x , ,x /x ,x ,x x , ,x2]
(cid:26) 2 2 3 ··· 2 m 2 2 m ··· 2 ∞
(2m+1)n (2m+1)(n+1)
∞ x− x
( 1)nq(2m+1)n(n+1)/2 2 + 2 +idem(x ;x , ,x )
× − 1 zqn/x 1 zx qn 2 3 ··· m
n= − 2 − 2 ! (cid:27)
X−∞
(2m+1)n (2m+1)(n+1)
∞ x− x
= ( 1)nq(2m+1)n(n+1)/2 1 + 1 ,
− 1 zqn/x 1 zx qn
1 1 !
n= − −
X−∞
where g(a ,a , ,a )+ idem(a ;a , ,a ) denotes the sum
1 2 m 1 2 n
··· ···
n
g(a ,a , ,a ,a ,a , ,a ),
i 2 i 1 1 i+1 m
··· − ···
i=1
X
in which the i-th term of the sum is obtained from the first by interchanging a and a .
1 i
Equation (1.3) was proved using partial fractions. Indeed, the m = 1 case of (1.3)
is equivalent to (1.1), while the m = 2 case is equivalent to (1.2). The fact that the
right-hand side of (1.2) is independent of x, and that the right-hand side of (1.3) is
RANK-CRANK TYPE PDES 3
independent of x ,x , ,x seems to be intriguing at first. Indeed, one purpose of
2 3 m
···
this article is to show that the left-hand sides of (1.2) and (1.3) are really elliptic
functions of order less than 2, in fact entire functions as we show, in the respective
variables (x for (1.2) and x for (1.3) while holding x , ,x fixed) and therefore
2 3 m
···
that they must be constants which are nothing but the right-hand sides of (1.2) and
(1.3) respectively. Since (1.2) follows from (1.3), we show this only for (1.3). This is
done in Section 2.
Let N(m,n) denote the number of partitions of n with rank m. Then the rank
generating function R(z,q) is given by
∞ n ∞ qn2
R(z,q) = N(m,n)zmqn = . (1.4)
(zq) (z 1q)
n − n
n=0m= n n=0
X X− X
In [1], G. E. Andrews and the third author defined the crank of a partition, a partition
statistic hypothesized by Dyson in [9]. It is the largest part if the partition contains
no ones, and otherwise is the number of parts larger than the number of ones minus
the number of ones. For n > 1, we let M(m,n) denote the number of partitions of n
with crank m. If we amend the definition of M(m,n) for n = 1, then the generating
function can be given as an infinite product. Accordingly, we assume
M(0,1) = 1, M( 1,1) = M(1,1) = 1, and M(m,1) = 0 otherwise.
− −
Then the crank generating function C(z,q) is given by
n
∞ (q)
C(z,q) = M(m,n)zmqn = ∞ . (1.5)
(zq) (z 1q)
−
Xn=0mX=−n ∞ ∞
Atkin and the third author [2] found the so-called Rank-Crank PDE, a partial differen-
tial equation (PDE) which relates R(z,q) and C(z,q). To state this PDE in its original
form, we first define the differential operators
∂ ∂
δ = z , δ = q . (1.6)
z q
∂z ∂q
Then the Rank-Crank PDE can be written as
1 1
z(q)2 [C (z,q)]3 = 3δ + δ + δ2 R (z,q), (1.7)
∗ q 2 z 2 z ∗
∞
(cid:18) (cid:19)
where
R(z,q)
R (z,q) := ,
∗
1 z
−
C(z,q)
C (z,q) := . (1.8)
∗
1 z
−
In [2], it was shown how the Rank-Crank PDE and certain results for the derivatives
of Eisenstein series lead to exact relations between rank and crank moments. As in
[13], define N (m,n) by
k
1 ∞
N (m,n)qn = ( 1)n 1qn((2k 1)n 1)/2+mn(1 qn), (1.9)
k − − − | |
(q) − −
Xn≥0 ∞ Xn=1
4 SONGHENGCHAN,ATUL DIXITANDFRANKG. GARVAN
of any positive integer k. When k = 1 this is the generating function for the crank, and
when k = 2 it is the generating function for the rank. When k 2, N (m,n) can be
k
≥
interpreted combinatorially asthe number ofpartitionsof ninto k 1 successive Durfee
−
squares with k-rank equal to m. See [13, Eq.(1.11)] for a definition of the k-rank. We
define
n
R (z,q) := N (m,n)zmqn. (1.10)
k k
n 0m= n
X≥ X−
From [13, Eq.(4.5)], this generating function can be written as
qn21+n22+···+n2k−1
R (z,q) = , (1.11)
k
(q) (q) (zq) (z 1q)
nk−1≥nkX−2≥···≥n1≥1 nk−1−nk−2··· n2−n1 n1 − n1
when k 2. In Section 3, we show that R (z,q) is related to the level 2k 1 Appell
k
≥ −
function
Σ(2k 1)(z,q) := ∞ (−1)nq(2k−1)n(n+1)/2. (1.12)
−
1 zqn
n= −
X−∞
We obtain the following
Theorem 1.1. For k 1,
≥
R (z,q)
k
k 3
1 −
= zk 1(1 z)Σ(2k 1)(z,q) zθ (q)+z(1 z) zmθ (q) ,
− − 1,2k 1 2m+3,2k 1
(q) − − − − − !
∞ mX=0
(1.13)
where
∞
θ (q) = ( 1)nqn((2k 1)n+j)/2, (1.14)
j,2k 1 −
− −
n=
X−∞
for j = 1, 3, ..., 2k 3.
−
This theorem generalizes Lemma 7.9 in [12] which gives a relation between the rank
generatingfunctionR(z,q)andalevel3Appellfunction. Thek = 1caseofthetheorem
gives the familiar partial fraction expansion for the reciprocal of Jacobi’s theta product
(z)A∞(fezw−1yqe)a∞rs[1a6g,opt.h1e],th[1ir7d, pa.ut1h3o6r],fosiunncde a 4∞nt=h−o∞rd(e−r1P)nDqEn(,nw+1h)/ic2h=is0a.n analogue of the
P
Rank-Crank PDE and is related to the 3-rank [13]. To state this PDE we define
1
G(5)(z,q) := Σ(5)(z,q). (1.15)
(q)3
∞
Then
24(q)2 [C (z,q)]5
∗
∞
= 24(1 10Φ (q))G(5)(z,q)
3
−
+ 100δ +50δ +100δ δ +35δ2 +20δ δ2 +100δ2 +10δ3 +δ4 G(5)(z,q), (1.16)
q z q z z q z q z z
(cid:0) (cid:1)
RANK-CRANK TYPE PDES 5
where
∞ n3qn
Φ (q) := . (1.17)
3 1 qn
n=1 −
X
This PDE can be written more compactly as
24(q)2 [C (z,q)]5 = (H2 E )G(5)(z,q), (1.18)
∗ 4
−
∞ ∗
where H is the operator
∗
H := 5+10δ +5δ +δ2,
q z z
∗
and
E := E (q) := 1+240Φ (q),
4 4 3
is the usual Eisenstein series of weight 4. The PDE (1.16) was first conjectured by the
third author and then subsequently proved and generalized by Zwegers [21]. It was also
Zwegers who first observed that (1.16) could be written in a more compact form. We
now describe Zwegers’s generalization. Define for l Z , the level l Appell function
>0
∈
as
∞ ( 1)lnqln(n+1)/2wn
A (u,v) := A (u,v;τ) := zl/2 − , (1.19)
l l 1 zqn
n= −
X−∞
wherez = e2πiu,w = e2πiv,q = e2πiτ, anddefinethemodifiedrankandcrankgenerating
functions as follows.
z1/2q 1/24
−
:= (u;τ) := R(z,q), (1.20)
R R 1 z
−
z1/2q 1/24
−
:= (u;τ) := C(z,q). (1.21)
C C 1 z
−
Here and throughout we assume Imτ > 0 so that q < 1. Then the following theorem
| |
due to Zwegers gives for odd l, the (l 1)th order analogue of the Rank-Crank PDE.
−
Theorem 1.2 (Zwegers[21]). Let l 3 be an odd integer. Define
≥
l ∂ 1 ∂2 l(2k 1)
:= + − E ,
Hk πi∂τ (2πi)2∂u2 − 12 2
[k] := , (1.22)
2k 1 2k 3 3 1
H H − H − ···H H
where E2(τ) = 1 − 24 ∞n=1σ1(n)qn is the usual Eisenstein series of weight 2 with
σ (n) = dα. Then there exist holomorphic modular forms f (which can be con-
stαructed expdl|incitly), withPj = 4,6,8, ,l 1, on SL (Z) of weighjt j, such that
2
P ··· −
(l 5)/2
−
[(l 1)/2] + f [k] A (u,0) = (l 1)!ηl l, (1.23)
− l 2k 1 l
H − − H  − C
k=0
X
 
where η is the Dedekind η-function, given by η(τ) = q1/24(q) .
∞
6 SONGHENGCHAN,ATUL DIXITANDFRANKG. GARVAN
Zwegers proved this theorem using the formulas and methods motivated from the
theory of Jacobi forms. In contrast to this, the proof of the Rank-Crank PDE by Atkin
and the third author, which corresponds to the l = 3 case of Zwegers’s PDE, depends
upon simply taking the second derivative with respect to ζ of both sides of (1.1). The
maingoal of this paper is to show how a generalized Rank-Crank PDE ofany oddorder
follows from the Lambert series identity (1.3) in a similar fashion. We will obtain
Zwegers’s result in a different form. In our form the coefficients are quasimodular
forms rather than holomorphic modular forms, but in contrast, our coefficients are
given recursively. See Theorem 4.4 and Corollary 4.5.
This paper is organized as follows. In Section 2, we prove (1.3) using the theory
of elliptic functions. Then in Section 3 we prove Theorem 1.1, which is the theorem
that relates R (z,q) with the level 2k 1 Appell function Σ(2k 1)(z,q). In Section 4
k −
−
we prove our main result that shows how (1.3) can be used to derive the higher order
Rank-Crank-type PDEs of Zwegers.
Inthelightof(1.19),itshouldbeobservedthattheidentities(1.1)and(1.2)arereally
the identities involving certain combinations of level 3 and level 5 Appell functions
respectively while (1.3) is an identity involving a combination of level (2m+1) Appell
functions. However, the analogue for level 1 Appell function which cannot be derived
from (1.3) was found by R. Lewis [15, Equation 11] and is as follows.
[z]∞[ζ2]∞(q)2∞ = ∞ ( 1)nqn(n+1)/2 ζ−n + ζn+1 . (1.24)
[zζ] [ζ] [zζ 1] − 1 zqn/ζ 1 zζqn
∞ ∞ − ∞ nX=−∞ (cid:18) − − (cid:19)
2. General Lambert series identity through elliptic function theory
Atkin and Swinnerton-Dyer’s proof of (1.1) depends in essence on the theory of
elliptic functions. In this section, we show how this method of proof can be used to
prove (1.3). Let
z = e2πiu,x = e2πiv,x = e2πiw, (2.1)
1 2
and let
x = e2πiaj, j = 3, ,m, (2.2)
j
···
where u,v,w,a ,...,a are all complex numbers. Also recall that q = e2πiτ, where Im
3 m
τ > 0.
Let
[b] =: J(a,q) =: J(a), where b = e2πia,a C. (2.3)
∞ ∈
Then using the Jacobi triple product identity [4, p. 10, Theorem 1.3.3], we easily find
that,
ieπiaq 1/8
−
J(a,q) = θ(a), (2.4)
(q)
∞
where
∞ πi n+1 2τ+2πi n+1 z+1
θ(z) = θ(z;τ) := e (cid:16) 2(cid:17) (cid:16) 2(cid:17)(cid:16) 2(cid:17). (2.5)
n=
X−∞
RANK-CRANK TYPE PDES 7
Comparingthiswiththeclassical definitionofθ (z) [11, p.355, Section13.19, Equation
1
10], we find that upon replacing q by q1/2 in this classical definition, θ(z) = θ (z).
1
−
From [20, p. 8], we see that
θ(z +1) = θ(z), (2.6)
−
θ(z +τ) = e πiτ 2πizθ(z), (2.7)
− −
−
θ( z) = θ(z), (2.8)
− −
θ′(0;τ) = 2πq1/8(q)3 . (2.9)
− ∞
Using (2.6) and (2.7), we have
J(a+1,q) = J(a,q), (2.10)
J(a+τ,q) = e 2πiaJ(a,q), (2.11)
−
−
J(a τ,q) = q 1e2πiaJ(a,q), (2.12)
−
− −
J( a,q) = e 2πiaJ(a,q). (2.13)
−
− −
Using (2.3), we rephrase (1.3) as follows:
e2πimvJ(w v)J(w +v)J(2v) J(a v)J(a +v) ∞ ( 1)nq(2m+1)n(n+1)/2
j j
− − −
J(v)(J(w))2 (J(a ))2 1 e2πiuqn
j
3 j m n= −
≤Y≤ X−∞
J(v w)J(v)J(v+w)J(2v)(q)2 J(v a )J(v+a )
j j
+ − −
∞
J(u v)J(u w)J(u)J(u+w)J(u+v) J(u a )J(u+a )
j j
− − 3 j m −
≤Y≤
J(v)J(2v) J(v a )J(v +a )
+ e2πi(v w) − j j
−
J(w)J(2w) J(w a )J(w +a )
(cid:26) 3 j m − j j
≤Y≤
∞ ( 1)nq(2m+1)n(n+1)/2 e−2πiw(2m+1)n + e2πiw(2m+1)(n+1) +idem(w;a , ,a )
× − 1 e2πi(u w)qn 1 e2πi(u+w)qn 3 ··· m
n= (cid:18) − − − (cid:19) (cid:27)
X−∞
= ∞ ( 1)nq(2m+1)n(n+1)/2 e−2πiv(2m+1)n + e2πiv(2m+1)(n+1) . (2.14)
− 1 e2πi(u v)qn 1 e2πi(u+v)qn
n= (cid:18) − − − (cid:19)
X−∞
Fix a , ,a , consider the left-hand side of (2.14) as a function of w only and denote
3 m
···
it by g(w). Let f (w) denote the expression in line 1 of (2.14), f (w) the expression in
1 2
line 2 of (2.14) and f (w) the expression in lines 3 and 4 of (2.14). Then, using (2.10),
3
(2.11) and (2.12), we see
f (w+1) = f (w) = f (w+τ),
1 1 1
f (w+1) = f (w) = f (w+τ),
2 2 2
f (w+1) = f (w). (2.15)
3 3
8 SONGHENGCHAN,ATUL DIXITANDFRANKG. GARVAN
Another application of (2.11) and (2.12) gives
f (w +τ)
3
e2πi(v w τ)J(v)J(2v) J(v a )J(v +a )
− − j j
= −
J(w +τ)J(2w +2τ) J(w +τ a )J(w +τ +a )
j j
3 j m −
≤Y≤
∞ ( 1)nq(2m+1)n(n+1)/2 e−2πi(w+τ)(2m+1)n + e2πi(w+τ)(2m+1)(n+1)
× − 1 e2πi(u w τ)qn 1 e2πi(u+w+τ)qn
n= (cid:18) − − − − (cid:19)
X−∞
m
J(v)J(2v)J(v w τ)J(v +w +τ) J(v a )J(v +a )
+ e2πi(v−ak) − − − j j
J(a )J(2a )J(a w τ)J(a +w +τ) J(a a )J(a +a )
k k k k k j k j
k=3 − − 2<j<m+1 −
X Yj6=k
∞ ( 1)nq(2m+1)n(n+1)/2 e−2πiak(2m+1)n + e2πiak(2m+1)(n+1)
×n= − (cid:18)1−e2πi(u−ak)qn 1−e2πi(u+ak)qn(cid:19)
X−∞
J(v)J(2v) J(v a )J(v+a ) ∞
= e2πiv+4πimw − j j ( 1)nq(2m+1)(n+1)(n+2)/2
J(w)J(2w) J(w a )J(w+a ) −
3 j m − j j (cid:18)n=
≤Y≤ X−∞
e−2πiw(2m+1)(n+1)q−(2m+1)(n+1) + ∞ ( 1)nq(2m+1)n(n 1)/2e2πiw(2m+1)nq(2m+1)n
−
× 1 e2πi(u w)qn − 1 e2πi(u+w)qn
− − n= − (cid:19)
X−∞
m
J(v)J(2v)J(v w)J(v+w) J(v a )J(v +a )
+ e2πi(v−ak) − − j j
J(a )J(2a )J(a w)J(a +w) J(a a )J(a +a )
k k k k k j k j
k=3 − 2<j<m+1 −
X Yj6=k
∞ ( 1)nq(2m+1)n(n+1)/2 e−2πiak(2m+1)n + e2πiak(2m+1)(n+1)
×n= − (cid:18)1−e2πi(u−ak)qn 1−e2πi(u+ak)qn(cid:19)
X−∞
= f (w). (2.16)
3
Thus from (2.15) and (2.16), we deduce that g is a doubly periodic function in w with
periods 1 and τ. Our next task is to show that g is an entire function of w and hence
a constant (with respect to w). We show that the poles of g at w = u and w = u are
−
actually removable singularities by proving that lim (w u)(f (w)+f (w)) = 0
w u 2 3
→± ∓
which readily implies that lim (w u)g(w) = 0. Let
w u
→± ∓
A := A(v,a , ,a ;q) := J(v)J(2v) J(v a )J(v +a ). (2.17)
3 m j j
··· −
3 j m
≤Y≤
Next, applying (2.4), (2.13) and (2.9), we see that
lim(w u)(f (w)+f (w))
2 3
w u −
→
J(v w)J(v+w)(q)2 1
= A lim(w u) − ∞
w→u − (cid:26)J(u−w)J(u+w)J(u−v)J(u)J(u+v) 3 j m J(u−aj)J(u+aj)
≤Y≤
+ e2πi(v−w) 1 1 + ∞ ( 1)nq(2m+1)n(n+1)/2
J(w)J(2w) J(w a )J(w +a ) 1 e2πi(u w) −
3 j m − j j (cid:18) − − n=−∞
≤Y≤ Xn6=0
RANK-CRANK TYPE PDES 9
e−2πiw(2m+1)n + ∞ ( 1)nq(2m+1)n(n+1)/2e2πiw(2m+1)(n+1)
× 1 e2πi(u w)qn − 1 e2πi(u+w)qn
− − n= − (cid:19)(cid:27)
X−∞
e2πi(v u) 1 iq1/8(q)3 1
−
= A − ∞ +
J(u)J(2u) J(u a )J(u+a ) θ′(0) 2πi
3 j m − j j (cid:18) (cid:19)
≤Y≤
= 0. (2.18)
Similarly, lim (w+u)(f (w)+f (w)) = 0. Now the only other possibility of a pole
w u 2 3
→−
of g is at 0, which arises from f and f each having a pole at 0. Again, to show that
1 3
this is a removable singularity, it suffices to show that lim w(f (w)+f (w)) = 0.
w 0 1 3
→
To show this, we need Jacobi’s duplication formula for theta functions [19, p. 488, Ex.
5]
θ(2w)θ θ θ = 2θ(w)θ (w)θ (w)θ (w). (2.19)
2 3 4 2 3 4
Let
J(2v) J(a v)J(a +v) ∞ ( 1)nq(2m+1)n(n+1)/2
B := B(u,v,a , ,a ;q) := e2πimv j − j − .
3 m
··· J(v) (J(a ))2 1 e2πiuqn
j
3 j m n= −
≤Y≤ X−∞
(2.20)
Then from (2.14) and (2.20),
lim w(f (w)+f (w))
1 3
w 0
→
J(w v)J(w +v) e2πi(v w) 1
−
= lim w B − +A
w→0 (cid:26) (J(w))2 J(w)J(2w) 3 j m J(w −aj)J(w +aj)
≤Y≤
∞ ( 1)nq(2m+1)n(n+1)/2 e−2πiw(2m+1)n + e2πiw(2m+1)(n+1)
× − 1 e2πi(u w)qn 1 e2πi(u+w)qn
n= (cid:18) − − − (cid:19)(cid:27)
X−∞
m
J(v)J(2v)J(v w)J(v+w) J(v a )J(v +a )
+ lim w e2πi(v−ak) − − j j
w→0 k=3 J(ak)J(2ak)J(ak −w)J(ak +w) 2<j<m+1 J(ak −aj)J(ak +aj)
X Yj6=k
w θ(w v)θ(w +v) e 5πiw 1
= lim B − Ae2πivq1/4(q)2 −
w→0 θ(w)(cid:26) θ(w) − ∞θ(2w) 3 j m J(w −aj)J(w +aj)
≤Y≤
∞ ( 1)nq(2m+1)n(n+1)/2 e−2πiw(2m+1)n + e2πiw(2m+1)(n+1)
× − 1 e2πi(u w)qn 1 e2πi(u+w)qn
n= (cid:18) − − − (cid:19)(cid:27)
X−∞
1 D(w)
= lim , (2.21)
θ′(0) w 0 θ(w)
→
10 SONGHENGCHAN,ATUL DIXITANDFRANKG. GARVAN
where
D(w) := D(w,v,a , ,a ;q) := Bθ(w v)θ(w +v) Ae2πivq1/4(q)2 E(w)
3 m
··· − −
∞
e 5πiwθ(w) 1
−
E(w) := E(w;u,a ,a ;q) :=
3 m
··· θ(2w) J(w a )J(w +a )
j j
3 j m −
≤Y≤
∞ ( 1)nq(2m+1)n(n+1)/2 e−2πiw(2m+1)n + e2πiw(2m+1)(n+1) .
× − 1 e2πi(u w)qn 1 e2πi(u+w)qn
n= (cid:18) − − − (cid:19)
X−∞
(2.22)
Now using (2.19), (2.4) and (2.13), we find that as w 0,
→
e2πivAq1/4(q)2 ∞ ( 1)nq(2m+1)n(n+1)/2
D(w) Bθ2(v) −
∞
→ − − ( 1)m 2e 2πi(a3+ +am)(J(a ) J(a ))2 1 e2πiuqn
− − ··· 3 m
− ··· n= −
X−∞
= 0, (2.23)
which is observed by putting the expressions for B and C back in the first expression
on the right side in (2.23). Thus,
D′(0)
lim w(f (w)+f (w)) = . (2.24)
w 0 1 3 θ′(0)2
→
Now we calculate D′(0).
D′(w) = B θ′(w v)θ(w+v)+θ(w v)θ′(w +v) e2πivAq1/4(q)2 E′(w). (2.25)
− − −
∞
(cid:16) (cid:17)
Using (2.4) and (2.19), we have
e πiw(2m+1)θ θ θ 1
E(w) = ( 1)m−2qm4−2(q)2(m−2) − 2 3 4
− ∞ 2θ2(w)θ3(w)θ4(w) θ(w aj)θ(w +aj)
3 j m −
≤Y≤
∞ ( 1)nq(2m+1)n(n+1)/2 e−2πiw(2m+1)n + e2πiw(2m+1)(n+1) . (2.26)
× − 1 e2πi(u w)qn 1 e2πi(u+w)qn
n= (cid:18) − − − (cid:19)
X−∞
Differentiating both sides with respect to w and simplifying, we obtain
1
E′(w) = ( 1)m−2qm4−2(q)2(m−2)e−πiw(2m+1)θ2θ3θ4
2 −
∞
θ′(w) θ′(w) θ′(w) θ′(w a ) θ′(w +a )
πi(2m+1)+ 2 + 3 + 3 + − j + j
θ (w) θ (w) θ (w) θ(w a ) θ(w +a )
2 3 3 3 j m(cid:18) − j j (cid:19)
≤X≤
× − θ (w)θ (w)θ (w) θ(w a )θ(w +a )
(cid:26) 2 3 4 j j
−
3 j m
≤Y≤
∞ ( 1)nq(2m+1)n(n+1)/2 e−2πiw(2m+1)n + e2πiw(2m+1)(n+1)
× − 1 e2πi(u w)qn 1 e2πi(u+w)qn
n= (cid:18) − − − (cid:19)
X−∞
1 1
′
+ F (w) , (2.27)
θ (w)θ (w)θ (w) θ(w a )θ(w+a )
2 3 4 3 j m − j j (cid:27)
≤Y≤