Stanley’s Lemma and Multiple Theta Function Identities
arXiv:1312.2172v1 [math.CA] 8 Dec 2013
William Y. C. Chen1 and Lisa H. Sun2 1
Center for Applied Mathematics Tianjin University, Tianjin 300072, P. R. China 2
Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin 300071, P.R. China E-mail: 1
[email protected], 2
[email protected]
Abstract. We present a vector space approach to proving identities on multiple theta functions by verifying a finite number of simpler relations which are often verifiable by using Jacobi’s triple product identity. Consider multiple theta Pm Qnk function identities of the form c θ (a , a , . . . , a ) = 0, where r, m ≥ 2, θ = 1 2 r k k k k=1 i=1 fk,i (a1 , a2 , . . . , ar ), 1 < nk ≤ r and each fk,i is of form (pk,i , q βk,i /pk,i ; q βk,i )∞ with βk,i being a positive integer and pk,i being a monomial in a1 , a2 , . . . , ar and q. For such an identity, θ1 , θ2 , . . . , θm satisfy the same set of linearly independent contiguous relations. Let W be the set of exponent vectors of (a1 , a2 , . . . , ar ) in the contiguous relations. We consider the case when the exponent vectors of (a1 , a2 , . . . , ar ) in pk,1 , pk,2 , . . . , pk,nk are linearly independent for any k. Let VC denote the vector space spanned by multiple theta functions which satisfy the contiguous relations associated with the vectors in W . Using Stanley’s lemma on the fundamental parallelepiped, we find an upper bound of the dimension of VC . This implies that a multiple theta function identity may be reduced to a finite number of simpler relations. Many classical multiple theta function identities fall into this framework, such as Bailey’s generalization of the quintuple product identity, the extended Riemann identity and Riemann’s addition formula. Keywords: theta function, multiple theta function, contiguous relation, Jacobi’s triple product identity, addition formula AMS Classification: 05E45, 14K25
1.
Introduction
In this paper, we provide a vector space approach to proving multiple theta function identities by applying Stanley’s lemma on the fundamental parallelepiped. Let us begin with notation and terminology. Let |q| < 1. Recall that the q-shifted factorial of infinite order is defined by ∞ Y (1 − aq k ). (a; q)∞ = k=0
1
Write (a1 , a2 , . . . , am ; q)∞ = (a1 ; q)∞ (a2 ; q)∞ · · · (am ; q)∞ . The theta function θ(z) is defined by θ(z) = [z; q]∞ = (z, q/z; q)∞ ,
(1.1)
see, for example, Krattenthaler [9]. A multiple theta function is defined by [z1 , z2 , · · · , zn ; q]∞ = [z1 ; q]∞ [z2 ; q]∞ · · · [zn ; q]∞ .
(1.2)
A basic theta function identity is Jacobi’s triple product identity (q, z, q/z; q)∞ =
∞ X
n
(−1)n q ( 2 ) z n
(1.3)
n=−∞
with z 6= 0. For the one variable case, theta function identities can be derived from the theta function solution space of the associated contiguous relation. It is known that theta functions satisfying the contiguous relation 1 (1.4) f (zq α ) = n m f (z) z q form a vector space over C of dimension n, where r, n are positive integers and m is a nonnegative integer. This is a classical result in algebraic geometry, see, for example, [12, p. 212, Theorem 1]. Suppose that f (z) can be written as X f (z) = ak z k . (1.5) k∈Z
Equating coefficients of z k on both sides of the contiguous relation (1.4), we see that the coefficients ak satisfy a recurrence relation of order n. Thus we can prove an identity on theta functions in one variable by showing that both sides satisfy the same contiguous relation in the form of (1.4) with the same initial conditions on the coefficients. This approach applies to many classical theta function identities, such as the quintuple product identity and the septuple product identity, see [1, 2, 5, 7, 16]. As an example, consider the quintuple product identity (q, −z, −q/z; q)∞ (qz 2 , q/z 2 ; q 2 )∞ = (q 3 , qz 3 , q 2 /z 3 ; q 3 )∞ + z(q 3 , q 2 z 3 , q/z 3 ; q 3 )∞ ,
(1.6)
see [8, 11, 15]. Denote the three theta functions in (1.6) by L, R1 and R2 , respectively. It is easily checked that L, R1 and R2 satisfy the contiguous relation f (zq) = −
1 f (z). z3q
(1.7)
Suppose that f (z) has an expansion with coefficients ak as in (1.5). It is clearly from (1.7) that the coefficients ak satisfy a recurrence relation of order three. Therefore, to prove (1.6), it suffices to show that [z −1 ] L = [z −1 ] (R1 + R2 ), 2
[z 0 ] L = [z 0 ] (R1 + R2 ), [z 1 ] L = [z 1 ] (R1 + R2 ), where [z i ] g(z) is the common notation for the coefficient of z i in the power series expansion of g(z). We shall consider multiple theta functions of the following form θ(a1 , a2 , . . . , ar ) =
m Y
fi (a1 , a2 , . . . , ar ),
(1.8)
i=1
where 1 < m ≤ r and γ
γ
γ
fi (a1 , a2 , . . . , ar ) = [(−1)δi a1i,1 a2i,2 · · · ar i,r q zi ; q ti ]∞
(1.9)
with δi ∈ {0, 1}, γi,j ∈ Z, zi ∈ Q and ti ∈ Q+ . Moreover, we assume that for 1 ≤ i ≤ m, the exponent vectors γi = (γi,1 , γi,2 , . . . , γi,r ) in fi are linearly independent. For convenience, γ γ γ we denote the monomial a1i,1 a2i,2 · · · ar i,r by aγi , where a stands for (a1 , a2 , . . . , ar ). Remark that more generally, fi may contains a factor aκ with κ ∈ Zr . For brevity, we only deal with the cases when fi is in the form of (1.9) while all the results presented in this paper still hold for the more general cases. For a multiple theta function θ(a1 , a2 , . . . , ar ), a contiguous relation is meant to be a relation of the following form (−1)ρi θ(a1 q αi,1 , a2 q αi,2 , . . . , ar q αi,r ) = wi,1 wi,2 , w θ(a1 , a2 , . . . , ar ) a1 a2 · · · ar i,r q si
(1.10)
where αi,j , wi,j (1 ≤ j ≤ r) ∈ Z, si ∈ Q and ρi ∈ {0, 1}. Notice that a contiguous relation (1.10) can be uniquely associated with a vector wi = (wi,1 , wi,2 , . . . , wi,r ). Furthermore, we say that two contiguous relations in the form of (1.10) are linearly independent if the corresponding exponent vectors wi are linearly independent. If we restrict our attention to multiple theta functions in the form of (1.8), then the solutions of a system of linearly independent contiguous relations form a finite dimensional vector space. The proof of this fact relies on Stanley’s lemma on the fundamental parallelepiped. This vector space property enables us to verify a multiple theta function identity by reducing it to a finite number of simpler relations. More precisely, consider the following multiple theta function identity m X
ck θk (a1 , a2 , . . . , ar ) = 0,
(1.11)
k=1
where m ≥ 2 and θk (a1 , a2 , . . . , ar ) are given in the form of (1.8). To prove identity (1.11), we first show that for 1 ≤ k ≤ m, θk satisfy the same system C of linearly independent contiguous relations. Let W = {v1 , v2 , . . . , vd } be the set of the exponent vectors of (a1 , a2 , . . . , ar ) in these contiguous relations in C. Let VC denote the vector space spanned by the multiple theta functions as given in (1.8) that satisfy the contiguous relations in C. By Stanley’s lemma on the fundamental parallelepiped, we show that dim VC ≤ |ΠW |, where ΠW = {λ1 v1 + λ2 v2 + · · · + λd vd | 0 ≤ λi < 1, 1 ≤ i ≤ d} ∩ Zr . 3
(1.12)
Hence identity (1.11) can be justified by comparing the coefficients of the terms with exponent vectors in ΠW . For example, let us consider the following identity, (a, −b, q/a, −q/b ; q)∞ + (−a, b, −q/a, q/b ; q)∞ =
2(ab, q 2 /ab, aq/b, bq/a; q 2 )∞ . (q; q 2 )2∞
(1.13)
Berndt [4, P. 45] proved (1.13) based on Ramanujan’s general theta function f (a, b). Chu [6] gave a proof of (1.13) by using q-difference equations. Denote the three multiple theta functions in (1.13) by L1 , L2 and R, respectively. It can be shown that L1 , L2 and R satisfy the following two linearly independent contiguous relations 1 f (aq, bq) =− , f (a, b) ab f (a, bq 2 ) 1 = 2 . f (a, b) b q
(1.14)
Thus W = {(1, 1), (0, 2)}. By Stanley’s lemma, we see that the dimension of VC is bounded by two since ΠW = {(0, 0), (0, 1)}. Hence identity (1.13) can be reduced to the following identities [a0 b0 ] (L1 + L2 ) = [a0 b0 ] R, [a0 b1 ] (L1 + L2 ) = [a0 b1 ] R, which can be certified by Jacobi’s triple product identity (1.3). This paper is organized as follows. In Section 2, we define two vector spaces associated with a multiple theta function of the form (1.8). In Section 3, we employ Stanley’s lemma on the fundamental parallelepiped to derive an upper bound on the dimension of the solution space of a system of linearly independent contiguous relations. In Sections 4 and 5, we apply the vector space approach to several well-known multiple theta function identities and addition formulas such as Bailey’s generalization of the quintuple product identity, the extended Riemann identity and Riemann’s addition formula.
2.
Two vector spaces
In this section, we define two vector spaces over Q associated with a multiple theta function and we show that they are the same vector space. This property will be used in the next section to determine the number of linearly independent contiguous relations needed to verify a multiple theta function identity. Recall that θ(a1 , a2 , . . . , ar ) is a product of m factors γ
γ
γ
fi (a1 , a2 , . . . , ar ) = [(−1)δi a1i,1 a2i,2 · · · ar i,r q zi ; q ti ]∞ .
4
Let Uθ denote the vector space spanned by exponent vectors of (a1 , a2 , . . . , ar ) in fi , that is,
Uθ = {γi = (γi,1 , γi,2 , . . . , γi,r ) | 1 ≤ i ≤ m} .
Let Vθ denote the vector space generated by the exponent vectors wi = (wi,1 , wi,2 , . . . , wi,r ) of contiguous relations (1.10). We have the following assertion. Theorem 2.1. Let θ(a1 , a2 , . . . , ar ) be a multiple theta function as given in (1.8) and Uθ , Vθ defined as above. Then we have U θ = Vθ . The following properties of contiguous relations of theta functions will be needed in the proof of the above theorem. Lemma 2.2. Let f (a) = (av , q t /av ; q t )∞ be a theta function, where v is an integer and t is a positive rational number. Assume that f (a) has a contiguous relation of the following form f (aq α ) =
1 f (a), (−1)ρ aw q s
(2.1)
where α, w ∈ Z, ρ ∈ {0, 1} and s ∈ Q. Then we have v|w and αv is an integer multiple of t. Proof.
By Jacobi’s triple product identity (1.3), f (a) can be written as ∞ X n 1 f (a) = t t (−1)n q t( 2 ) avn . (q ; q )∞ n=−∞
(2.2)
Substituting a with aq α in (2.2) we get an expansion of f (aq α ). Note that the powers of a in the expansions of both sides are all multiples of v, so we deduce that v|w. Plugging the expansions of f (a) and f (aq α ) into the contiguous relation (2.1) and equating coefficients of avn , we obtain that n+ w n w v (−1)n q t( 2 )+αvn = (−1)ρ+n+ v q t( 2 )−s . It follows that ρ + wv is even and the powers of q on both sides are equal for all n ∈ Z, namely, n n + wv t + αvn = t − s. 2 2 Hence
w w t − αv n + t v − s = 0 2 v
for all n ∈ Z, so that t wv = αv. Thus we have αv is an integer multiple of t. Recall that Uθ is the vector space spanned by γ1 , γ2 , . . . , γm . The following lemma shows that any vector v ∈ Uθ gives rise to a contiguous relation of θ(a1 , a2 , . . . , ar ). 5
Lemma 2.3. Let v = (v1 , v2 , . . . , vr ) be a vector in Uθ such that v = k1 γ1 + k2 γ2 + · · · + km γm , where ki ’s are not all zeros, and γi is associated with the factor fi in (1.9). Then there exists a nonzero vector (x1 , x2 , . . . , xr ) and a constant ε 6= 0 such that the following contiguous relation holds Pm (−1) i=1 (δi +1)εki θ(a1 q x1 , a2 q x2 , . . . , ar q xr ) = Pm , (2.3) εki θ(a1 , a2 , . . . , ar ) q i=1 εzi ki +( 2 )ti aεv εvr 1 εv2 where a = (a1 , a2 , . . . , ar ) and aεv = aεv 1 a2 · · · ar .
Proof. To prove the existence of a vector (x1 , x2 , . . . , xr ) and a nonzero constant ε satisfying (2.3), we show that the vector (x1 , x2 , . . . , xr ) and the nonzero constant ε can be chosen as a solution of the following system of linear equations ( xγ i if ki 6= 0, ki ti − ε = 0, (2.4) xγj = 0, if kj = 0, where 1 ≤ i, j ≤ m, x = (x1 , x2 , . . . , xr ), and ε is also considered as a variable. So there are m homogeneous linear equations in r + 1 variables. In other words, the coefficient matrix A of the system of linear equations (2.4) has m rows and r + 1 columns, which can be written as γ1,1 γ1,2 · · · γ1,r ζ1 γ2,1 γ2,2 · · · γ2,r ζ2 A= . , .. .. .. .. .. . . . . γm,1 γm,2 · · ·
γm,r ζm
m×(r+1)
where ζi = −ki ti , if ki 6= 0 and ζi = 0, otherwise. Since 1 < m ≤ r and γ1 , γ2 , . . . , γm are linearly independent, the rank of the matrix A is m. Thus the solution space P of (2.4) in variables x1 , x2 , . . . , xr , ε has dimension r + 1 − m. Consider another system of linear equations xγ1 = 0, xγ2 = 0, .. . xγm = 0.
(2.5)
Let Q denote the vector space of solutions (x1 , x2 , . . . , xr ) of the equations in (2.5). Let Q′ be the vector space obtained from Q by substituting every vector (x1 , x2 , . . . , xr ) ∈ Q with (x1 , x2 , . . . , xr , 0). Since γ1 , γ2 , . . . , γm are linearly independent, we see that dim Q′ = dim Q = r − m. It is also clear that any solution (x1 , x2 , . . . , xr ) of (2.5) gives rise to a solution (x1 , x2 , . . . , xr , 0) of (2.4). In other words, Q′ is a subspace of P . Moreover, we have dim Q′ < dim P . Hence there exists a solution (x1 , x2 , . . . , xr , ε) ∈ P \ Q′ such that ε 6= 0. Now we have m
θ(a1 q x1 , a2 q x2 , . . . , ar q xr ) Y fi (a1 q x1 , a2 q x2 , . . . , ar q xr ) = θ(a1 , a2 , . . . , ar ) fi (a1 , a2 , . . . , ar ) i=1
6
=
m γ γ γ Y [(−1)δi a i,1 a i,2 · · · ar i,r q xγi +zi ; q ti ]∞ 1
i=1
=
γ
2
γ
γ
[(−1)δi a1i,1 a2i,2 · · · ar i,r q zi ; q ti ]∞
m γ γ γ Y [(−1)δi a1i,1 a2i,2 · · · ar i,r q εki ti +zi ; q ti ]∞ γ γ γ [(−1)δi a1i,1 a2i,2 · · · ar i,r q zi ; q ti ]∞ i=1
ki 6=0
=
m Y
i=1 ki 6=0
(−1)(δi +1)εki γ γ γ q εzi ki +( )ti (a i,1 a i,2 · · · ar i,r )εki εki 2
1
2
Pm
(−1) i=1 (δi +1)εki = Pm εki q i=1 εzi ki +( 2 )ti aε(k1 γ1 +k2 γ2 +···+km γm ) Pm
(−1)
= q
Pm
i=1
i=1 (δi +1)εki
εzi ki +(εk2 i )ti εv a
,
as desired. It should be noted that our approach does not require an explicit expression of a solution of the system of linear equations in (2.4). In particular, for r = 1, (2.4) reduces to the following equation xγ = ε. kt Since γ, k, t ∈ Q, it is obvious that the above equation has a solution x for any given ε 6= 0. We are now ready to prove Theorem 2.1. Proof of Theorem 2.1. Recall that θ(a1 , a2 , . . . , ar ) =
m Y
fi (a1 , a2 , . . . , ar ),
i=1
γ
γ
γ
with 1 < m ≤ r and the factor fi = [(−1)δi a1i,1 a2i,2 · · · ar i,r q zi ; q ti ]∞ as given by (1.9). For 1 ≤ i ≤ m, let γi = (γi,1 , γi,2 , . . . , γi,r ). We first prove Vθ ⊆ Uθ . By definition, Uθ is spanned by the vectors γ1 , γ2 , . . . , γm and Vθ is the vector space consisting of vectors wi = (wi,1 , wi,2 , . . . , wi,r ) such that θ(a1 q αi,1 , a2 q αi,2 , . . . , ar q αi,r ) (−1)ρi = wi,1 wi,2 , w θ(a1 , a2 , . . . , ar ) a1 a2 · · · ar i,r q si
(2.6)
where αi,j , wi,j ∈ Z, si ∈ Q and ρi ∈ {0, 1}. We proceed to show that any vector in Vθ can be expressed as a linear combination of the vectors γ1 , γ2 , . . . , γm . From (2.6), we find that m
θ(a1 q αi,1 , a2 q αi,2 , . . . , ar q αi,r ) Y fi (a1 q αi,1 , a2 q αi,2 , . . . , ar q αi,r ) = θ(a1 , a2 , . . . , ar ) fi (a1 , a2 , . . . , ar ) i=1
=
m Y [(−1)δi (a1 q αi,1 )γi,1 · · · (ar q αi,r )γi,r q zi ; q ti ]∞ γ
γ
[(−1)δi a1i,1 · · · ar i,r q zi ; q ti ]∞
i=1
7
.
(2.7)
Let µi = αi,1 γi,1 + αi,2 γi,2 + · · · + αi,r γi,r , so that (2.7) can be rewritten as m
γ
γ
γ
θ(a1 q αi,1 , a2 q αi,2 , . . . , ar q αi,r ) Y [(−1)δi a1i,1 a2i,2 · · · ar i,r q zi +µi ; q ti ]∞ . = γ γ γ θ(a1 , a2 , . . . , ar ) [(−1)δi a1i,1 a2i,2 · · · ar i,r q zi ; q ti ]∞
(2.8)
i=1
In view of Lemma 2.2, we see that µi can be written as µi = vi ti with vi being an integer. It follows that Qυi −1 1 m 1 − γi,1 γi,2 γi,r α α α Y i,1 i,2 i,r k=0 θ(a1 q , a2 q , . . . , ar q ) (−1)δi a1 a2 ···ar q zi +kti = Qυi −1 γi,r z +kt δi γi,1 γi,2 i i θ(a1 , a2 , . . . , ar ) k=0 1 − (−1) a1 a2 · · · ar q i=1 =
m Y i=1
(−1)(δi +1)υi . υi γ γ γ υ a1i,1 a2i,2 · · · ar i,r i q zi υi +( 2 )ti
(2.9)
Combining the contiguous relations (2.6) and (2.9), we find that m
Y (−1)ρi (−1)(δi +1)υi = . w w w γi,1 γi,2 γi,r υi zi υi +(υi )ti a1 i,1 a2 i,2 · · · ar i,r q si 2 a a · · · a q r i=1 1 2
Hence we arrive at
wi = υ1 γ1 + υ2 γ2 + · · · + υm γm ,
(2.10)
where wi and γi are defined as before. This leads to the conclusion that Vθ ⊆ Uθ . To prove that Uθ ⊆ Vθ , let v be a vector in Uθ . By Lemma 2.3, we see that there exists a nonzero vector (x1 , x2 , . . . , xr ) and a constant ε 6= 0 such that εv is the exponent vector of a in the denominator of the contiguous relation θ(a1 q x1 , a2 q x2 , . . . , ar q xr ) θ(a1 , a2 , . . . , ar ) as given in (2.3). Therefore εv belongs to the vector space Vθ , that is, v ∈ Vθ . This yields Uθ ⊆ Vθ , and hence the proof is complete.
3.
Stanley’s lemma and an upper bound of dim VC
In this section, we consider the solution space of a system of linearly independent contiguous relations. Let C be a system of linearly independent contiguous relations in the form of (1.10). Denote by W = {w1 , w2 , . . . , wd } the exponent vectors associated with the contiguous relations in C. Let VC denote the vector space spanned by the multiple theta functions as given in (1.8) that satisfy the contiguous relations in C. Based on Stanley’s lemma [14, Lemma 4.5.7 (i)] on the fundamental parallelepiped, we obtain an upper bound of the dimension of VC . Stanley’s lemma is stated as follows.
8
Lemma 3.1. Let v1 , v2 , . . . , vd be linearly independent integer vectors over Q and let F be the set of linear combinations of v1 , v2 , . . . , vd with rational coefficients, namely, F = {γ ∈ Zr | γ = c1 v1 + c2 v2 + · · · + cd vd , ci ∈ Q for 1 ≤ i ≤ d}. Suppose that Π = {λ1 v1 + λ2 v2 + · · · + λd vd | 0 ≤ λi < 1 for 1 ≤ i ≤ d} is the fundamental parallelepiped generated by v1 , v2 , . . . , vd . Then every element γ ∈ F can be expressed uniquely in the following form γ = β + b1 v1 + b2 v2 + · · · + bd vd ,
(3.1)
where β ∈ Π ∩ Zr and b1 , b2 , . . . , bd are integers. Conversely, any vector γ in the form (3.1) belongs to F . The following theorem gives an upper bound of the dimension of VC . Theorem 3.2. The dimension of VC is bounded by |ΠW |, where ΠW = {λ1 w1 + λ2 w2 + · · · + λd wd | 0 ≤ λi < 1, 1 ≤ i ≤ d} ∩ Zr .
(3.2)
Proof. Write wi = (wi,1 , wi,2 , . . . , wi,r ) for 1 ≤ i ≤ d. Note that Vθ stands for the vector space spanned by w1 , w2 , . . . , wd . Assume that θ(a1 , a2 , . . . , ar ) ∈ VC . Let X θ(a1 , a2 , . . . , ar ) = hη1 ,η2 ,...,ηr aη11 aη22 · · · aηr r . (3.3) (η1 ,η2 ,...,ηr )∈Zr
By Jacobi’s triple product identity (1.3), we get γ
γ
γ
fi (a1 , a2 , . . . , ar ) = [(−1)δi a1i,1 a2i,2 · · · ar i,r q zi ; q ti ]∞ =
∞ X ni 1 γ γ γ n (−1)(1+δi )ni q ti ( 2 )+zi ni a1i,1 a2i,2 · · · ar i,r i . t t (q i ; q i )∞ n =−∞
(3.4)
i
Hence θ(a1 , a2 , . . . , ar ) =
m Y i=1
=
∞ X ni 1 γ γ γ n (−1)(1+δi )ni q ti ( 2 )+zi ni a1i,1 a2i,2 · · · ar i,r i t t i i (q ; q )∞ n =−∞
X
mi ∈Z
i
X
(n1 ,n2 ,...,nm )
! ni (−1)(1+δi )ni q ti ( 2 )+zi ni mr 1 m2 am 1 a2 · · · ar , (q ti ; q ti )∞
(3.5)
where the inner sum ranges over (n1 , n2 , . . . , nm ) ∈ Zm such that n1 γ1 + n2 γ2 + · · · + nm γm = (m1 , m2 , . . . , mr ). Comparing (3.3) and (3.5), we get hη1 ,η2 ,...,ηr =
X
(n1 ,n2 ,...,nm )
ni (−1)(1+δi )ni q ti ( 2 )+zi ni , (q ti ; q ti )∞
9
where the sum ranges over (n1 , n2 , . . . , nm ) ∈ Zm such that η = (η1 , η2 , . . . , ηr ) = n1 γ1 + n2 γ2 + · · · + nm γm .
(3.6)
Since γ1 , γ2 , . . . , γm are linearly independent, we see that for a given exponent vector η of a nonzero term in the expansion (3.3) there is a unique vector (n1 , n2 , . . . , nm ) in Zm that satisfies (3.6). It follows that η ∈ Uθ . By Theorem 2.1, that is, Uθ = Vθ , we have η ∈ Vθ . Since w1 , w2 , . . . , wd form a basis of Vθ , η as given in (3.6) can be expressed as η = g1 w1 + g2 w2 + · · · + gd wd ,
(3.7)
where gi ∈ Q. According to Lemma 3.1, η can be uniquely written as η = β + b1 w1 + b2 w2 + · · · + bd wd ,
(3.8)
where b1 , b2 , . . . , bd are integers and β ∈ ΠW . Next we show that VC is isomorphic to a vector space whose dimension does not exceed |ΠW |. Let dw = |ΠW | and β1 , β2 , . . . , βdw be the elements of ΠW listed in any order. Let Hθ = (hβ1 , hβ2 , . . . , hβdw ),
(3.9)
where hβi is the coefficient of aβi in the expansion (3.3) of θ(a). Let H = {Hθ | θ ∈ VC }, and let VH be the vector space spanned by H. It is clear that dim VH ≤ dw . Define a linear map φ : VC → VH by φ(θ) = Hθ . We proceed to show that φ is a bijection so that dim VC = dim VH ≤ dw . To prove that φ is a bijection, it suffices to show that there is a unique θ ∈ VC such that φ(θ) = Hθ . Assume to the contrary that φ(θ1 ) = φ(θ2 ) = Hθ , where θ1 , θ2 ∈ VC and θ1 6= θ2 . Let θ = θ1 − θ2 . Then θ is in VC and φ(θ) = 0. Assume that θ is expanded as (3.3) with hη being the coefficient of aη . Since φ(θ) = (hβ1 , hβ2 , . . . , hβdw ) = 0, we have hβi = 0 for βi ∈ ΠW . It remains to show that θ = 0. Recall that for any vector wi = (wi,1 , wi,2 , . . . , wi,r ) in W , it corresponds to a contiguous relation in C, namely, (−1)δi θ(a1 q αi,1 , a2 q αi,2 , . . . , ar q αi,r ) = wi,1 wi,2 . w θ(a1 , a2 , . . . , ar ) a1 a2 · · · ar i,r q si
(3.10)
Rewriting the above contiguous relation as w
w
w
θ(a1 , a2 , . . . , ar ) = (−1)δi a1 i,1 a2 i,2 · · · ar i,r q si θ(a1 q αi,1 , a2 q αi,2 , . . . , ar q αi,r ) and equating the coefficients of the terms with exponent vector η = (η1 , η2 , . . . , ηr ) on both sides, we obtain the following relation hη = (−1)δi q si +αi (η−wi ) hη−wi ,
10
where αi = (αi,1 , αi,2 , . . . , αi,r ). Iterating the above relation bi times, we deduce that hη = (−1)bi δi q bi si +bi αi η−(
bi +1 2
)αi wi h
η−bi wi .
(3.11)
Iterating (3.11) for i = 1, 2, . . . , d, we find that Pd
hη = (−1)
i=1 bi δi
q
Pd
i=1
bi si −(bi2+1)αi wi +bi (αi η−
Pi−1
Substituting (3.8) into (3.12), we get Pd
hη = (−1)
i=1 bi δi
Pd
= (−1)
i=1 bi δi
q q
j=1 bj αi wj )
hη−b1 w1 −b2 w2 −···−bd wd .
Pd
bi si −(bi2+1)αi wi +bi (αi β+bi αi wi +
Pd
P αi wi 2 αi wi bi − 2 bi +bi si +bi (αi β+ dj=i+1 bj αi wj ) 2
i=1
i=1
Pd
j=i+1 bj αi wj )
(3.12)
hβ
hβ ,
(3.13)
where β belongs to ΠW . But hβ = 0 for any β ∈ ΠW , so we deduce that hη = 0 for any exponent vector η of a nonzero term of θ in the expansion (3.3). Thus we obtain θ = 0, namely, θ1 = θ2 , a contradiction to the assumption, and so the proof is complete.
4.
Multiple theta function identities
In this section, we demonstrate how to apply the vector space approach to prove multiple theta function identities. In general, we consider a multiple theta function identity in the form of (1.11), that is, m X ck θk (a1 , a2 , . . . , ar ) = 0, (4.1) k=1
where 1 < m ≤ r. Recall that
θ(a1 , a2 , . . . , ar ) =
d Y
fi (a1 , a2 , . . . , ar ),
i=1
where
γ
γ
γ
fi (a1 , a2 , . . . , ar ) = [(−1)δi a1i,1 a2i,2 · · · ar i,r q zi ; q ti ]∞ . The first example is Bailey’s generalization of the quintuple product identity, (b/a,aq/b, df /a, aq/df, ef /a, aq/ef, bde/a, aq/bde; q)∞ = (f /a, aq/f, bd/a, aq/bd, be/a, aq/be, def /a, aq/def ; q)∞ b − (d, q/d, e, q/e, f /b, bq/f, bdef /a2 , a2 q/bdef ; q)∞ , a
(4.2)
see [3]. Clearly, (4.2) is of the form (4.1), that is, m = 3, d = 4, and θ1 − θ2 + θ3 = 0, where θ1 = (b/a, aq/b, df /a, aq/df, ef /a, aq/ef, bde/a, aq/bde; q)∞ , 11
(4.3)
θ2 = (f /a, aq/f, bd/a, aq/bd, be/a, aq/be, def /a, aq/def ; q)∞ , θ3 =
b (d, q/d, e, q/e, f /b, bq/f, bdef /a2 , a2 q/bdef ; q)∞ . a
An identity of the form (4.1) can be verified via three steps. Step 1. For any 1 ≤ k ≤ m, let Uk be the vector space spanned by exponent vectors (γi,1 , γi,2 , . . . , γi,r ) of (a1 , a2 , . . . , ar ) in the factors of θk and let Vk be the vector space consisting of exponent vectors (wi,1 , wi,2 , . . . , wi,r ) of all contiguous relations of θk in the form (1.10), namely, (−1)ρi θ(a1 q αi,1 , a2 q αi,2 , . . . , ar q αi,r ) = wi,1 wi,2 . w θ(a1 , a2 , . . . , ar ) a1 a2 · · · ar i,r q si
(4.4)
Since the exponent vectors (γi,1 , γi,2 , . . . , γi,r ) in the factors of θ are supposed to be linearly independent, the dimension of the vector space Uk is d. Moreover, one can verify that U1 = U2 = · · · = Um . By Theorem 2.1, we have Uk = Vk , which implies that dim Vk = d. We begin with d linearly independent contiguous relations satisfied by any θk , which can be found by using the procedure given in Lemma 2.3. For the above identity (4.2), it can be easily checked that U1 = U2 = U3 . The exponent vectors in the factors of θ1 are linearly independent, so that dim U1 = 4. By Theorem 2.1, we have dim V1 = 4. Following the procedure given in Lemma 2.3, we find that θ1 satisfies the following four linearly independent contiguous relations b2 de θ(aq, b, d, e, f q) = 2 2, θ(a, b, d, e, f ) a q f θ(aq, bq, dq, e, f ) = , θ(a, b, d, e, f ) bdq 1 θ(aq, b, dq, eq, f ) = , θ(a, b, d, e, f ) deq b θ(aq, q, dq, e, f q) = . θ(a, b, d, e, f ) df q
(4.5)
In fact, since U1 = U2 = U3 , we see that θ2 and θ3 also satisfy the above contiguous relations. Step 2. Let W denote the set of exponent vectors in the d linearly independent contiguous relations obtained in Step 1. Applying Stanley’s lemma, it can be shown that all the multiple theta functions in the form of (1.8) that satisfy the contiguous relations associated to W form a vector space, denoted by VC . By Theorem 3.2, the dimension of VC is bounded by |ΠW |, where ΠW is the set of integer vectors in the fundamental parallelepiped associated with W , as given by (3.2). Let ΠW = {β1 , β2 , . . . , βdw }. For example, for the contiguous relations in (4.5), we have W = {(2, −2, −1, −1, 0), (0, 1, 1, 0, −1), (0, 0, 1, 1, 0), (0, −1, 1, 0, 1)}, and ΠW = {λ1 (2, −2, −1, −1, 0) + λ2 (0, 1, 1, 0, −1) + λ3 (0, 0, 1, 1, 0) 12
+ λ4 (0, −1, 1, 0, 1) | 0 ≤ λi < 1, i = 1, 2, 3, 4} ∩ Z5 = (0, 0, 0, 0, 0), (0, 0, 1, 0, 0), (1, −1, 0, 0, 0), (1, −1, 1, 0, 0) .
Let β1 = (0, 0, 0, 0, 0), β2 = (0, 0, 1, 0, 0), β3 = (1, −1, 0, 0, 0) and β4 = (1, −1, 1, 0, 0). Since dw = 4, by Theorem 3.2, we have dim VC ≤ 4. Step 3. We use the upper bound of dim VC to transform identity (4.1) into a finite number of relations satisfied by the coefficients c1 , c2 , . . . , cm . More precisely, for 1 ≤ k ≤ m, let Hθk = (hk,β1 , hk,β2 , . . . , hk,βdw ), where hk,βi is the coefficient of the term with exponent vector βi ∈ ΠW in the expansion (3.3) of θk . Since θ1 , θ2 , . . . , θm satisfy the same linearly independent contiguous relations associated with W , the coefficients of each θk satisfy the same recurrence relation (3.13). Following the procedure to prove Theorem 3.2, any coefficient in the expansion of θk can be iteratively computed from Hθk . This implies that identity (4.1) can be proved by verifying the following relation m X ck Hθk = 0, k=1
or equivalently, for any βi ∈ ΠW
m X
ck hk,βi = 0.
k=1
For example, (4.2) can be reduced to the relation Hθ1 − Hθ2 + Hθ3 = 0, where Hθk = (hk,β1 , hk,β2 , hk,β3 , hk,β4 ) for 1 ≤ k ≤ 3. That is to say that for any βi ∈ ΠW h1,βi − h2,βi + h3,βi = 0.
(4.6)
In summary, identity (4.2) holds if the above relations can be verified for 1 ≤ i ≤ 4. Let us consider the case k = 1 and i = 1. In this case, h1,β1 = [a0 b0 d0 e0 f 0 ] θ1 . By Jacobi’s triple product identity (1.3), we have [a0 b0 d0 e0 f 0 ] θ1 = [a0 b0 d0 e0 f 0 ] (b/a, aq/b, df /a, aq/df, ef /a, aq/ef, bde/a, aq/bde; q)∞ = [a0 b0 d0 e0 f 0 ]
∞ ∞ X X n 1 n (n n 2 ) (b/a) (−1)n q ( 2 ) (df /a)n (−1) q 4 (q; q)∞ n=−∞ n=−∞
×
∞ X
n (−1) q ( 2 ) (ef /a)n
n
=
1 (q; q)4∞
n
(−1)n q ( 2 ) (bde/a)n
n=−∞
n=−∞
X
∞ X
n1
n2
n3
n4
(−1)n1 +n2 +n3 +n4 q ( 2 )+( 2 )+( 2 )+( 2 ) ,
(n1 ,n2 ,n3 ,n4 )∈Z4
13
(4.7)
where the summation in (4.7) ranges over all (n1 , n2 , n3 , n4 ) ∈ Z4 such that n1 + n2 + n3 + n4 = 0, n1 + n4 = 0, n2 + n4 = 0, n3 + n4 = 0, n + n = 0. 2 3
Since the above system of equations has a unique solution n1 = n2 = n3 = n4 = 0, we get [a0 b0 d0 e0 f 0 ] θ1 =
1 . (q; q)4∞
Following the same procedure, we get [a0 b0 d0 e0 f 0 ] θ2 =
1 , (q; q)4∞
[a0 b0 d0 e0 f 0 ] θ3 = 0,
and hence relation (4.6) holds for i = 1. Similarly, for i = 2, 3, 4, we obtain [a0 b0 d1 e0 f 0 ] θ1 = 0
[a0 b0 d1 e0 f 0 ] θ2 = 0
[a0 b0 d1 e0 f 0 ] θ3 = 0
q [a1 b−1 d0 e0 f 0 ] θ2 = 0, [a1 b−1 d0 e0 f 0 ] θ1 = − (q;q) 4 ,
[a1 b−1 d0 e0 f 0 ] θ3 =
∞
[a1 b−1 d1 e0 f 0 ] θ1 = 0,
q , (q;q)4∞
2
2
q q [a1 b−1 d1 e0 f 0 ] θ2 = − (q;q) [a1 b−1 d1 e0 f 0 ] θ3 = − (q;q) 4 , 4 . ∞
∞
Thus (4.6) is verified for any i. This proves identity (4.2). The next example is concerned with the extended Riemann identity on theta functions proved by Malekar and Bhate [10, Theorem 3.1] by using eigenvectors corresponding to the discrete Fourier transform Φ(2), which can be reformulated as follows. Example 4.1 (Extended Riemann Identity). We have 1
4qxyuv[−q 2 x2 , −q 2 y 2 , −q 2 u2 , −q 2 v 2 ; q 2 ]∞ + 4q 2 xy[−q 2 x2 , −q 2 y 2 , −qu2 , −qv 2 ; q 2 ]∞ 1
+ 4q 2 uv[−qx2 , −qy 2 , −q 2 u2 , −q 2 v 2 ; q 2 ]∞ + 4[−qx2 , −qy 2 , −qu2 , −qv 2 ; q 2 ]∞ 1
=
1
1 1 1 1 1 1 1 1 1 1 (q 2 ; q 2 )4∞ [−q 4 x, −q 4 y, −q 4 u, −q 4 v; q 2 ]∞ + [q 4 x, q 4 y, −q 4 u, −q 4 v; q 2 ]∞ 2 2 4 (q ; q )∞ 1 1 1 1 1 1 1 1 1 1 + [−q 4 x, −q 4 y, q 4 u, q 4 v; q 2 ]∞ + [q 4 x, q 4 y, q 4 u, q 4 v; q 2 ]∞ .
(4.8)
Proof. It is easy to check that each multiple theta function in (4.8) satisfies the following four linearly independent contiguous relations 1 f (xq, y, u, v) = 2 , f (x, y, u, v) x q 1 f (x, yq, u, v) = 2 , f (x, y, u, v) y q 14
f (x, y, uq, v) 1 = 2 , f (x, y, u, v) u q 1 f (x, y, u, vq) = 2 . f (x, y, u, v) v q Then we have W = {(2, 0, 0, 0), (0, 2, 0, 0), (0, 0, 2, 0), (0, 0, 0, 2)} and ΠW = {λ1 (2, 0, 0, 0) + λ2 (0, 2, 0, 0) + λ3 (0, 0, 2, 0) + λ4 (0, 0, 0, 2) | 0 ≤ λi < 1, 1 ≤ i ≤ 4} ∩ Z4 = {(a1 , a2 , a3 , a4 ) | ai = 0 or 1, 1 ≤ i ≤ 4}. By Theorem 3.2, (4.8) can be verified by showing that it holds only for the the terms xη1 y η2 uη3 v η4 with (η1 , η2 , η3 , η4 ) ∈ ΠW . In view of the symmetries of x, y and u, v, it is sufficient to show that it holds for the terms with exponent vectors in Π′W = {(0, 0, 0, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 0, 1, 0), (1, 1, 1, 0), (1, 1, 1, 1)} In fact, in each case, the required equality can be justified by Jacobi’s triple product identity (1.3). For example, let (η1 , η2 , η3 , η4 ) = (0, 0, 0, 0). Denote the multiple theta functions on both sides by L1 , L2 , L3 , L4 and R1 , R2 , R3 , R4 , respectively. For the left hand side, it is clear that [x0 y 0 u0 v 0 ] 4 (L1 + L2 + L3 + L4 ) = [x0 y 0 u0 v 0 ] 4 [−qx2 , −qy 2 , −qu2 , −qv 2 ; q 2 ]∞ = [x0 y 0 u0 v 0 ]
=
4 (q 2 ; q 2 )4∞
4
X
(q 2 ; q 2 )4∞ n∈Z
n
q 2( 2 ) qx2
n X
n
q 2( 2 ) qy 2
n X
n
q 2( 2 ) qu2
n
q 2( 2 ) qv 2
n∈Z
n∈Z
n∈Z
n X
n
.
Similarly, we have 1
[x0 y 0 u0 v 0 ]
1
1
1
4 (q 2 ; q 2 )4∞ (q 2 ; q 2 )4∞ 4 = 2 2 4 . (R + R + R + R ) = 1 2 3 4 (q 2 ; q 2 )4∞ (q 2 ; q 2 )4∞ (q 12 ; q 12 )4 (q ; q )∞ ∞
Hence identity (4.8) holds with respect to the coefficients of the term x0 y 0 u0 v 0 . The other cases can be verified in the same way. The following identity was derived by Chu [6], which can be verified by our approach. Example 4.2. We have [d/b, c/b, cd/α, α; q]∞ − [c, d, α/b, cd/bα; q]∞ = c[cd/b, 1/b, d/α, α/c; q]∞ .
15
(4.9)
Proof. Denote the three terms on both sides of the identity by L1 , L2 and R, respectively. It is easy to check that L1 , L2 and R satisfy the following four linearly independent contiguous relations α f (bq, cq, d, α) = , f (b, c, d, α) bcq α f (bq, c, dq, α) = , f (b, c, d, α) bdq f (b, cq, d, αq) b = , f (b, c, d, α) cα cd f (bq, c, d, α) = 2 . f (b, c, d, α) b q Thus we have W = {(1, 1, 0, −1), (1, 0, 1, −1), (−1, 1, 0, 1), (2, −1, −1, 0)} and ΠW = {λ1 (1, 1, 0, −1) + λ2 (1, 0, 1, −1) + λ3 (−1, 1, 0, 1) + λ4 (2, −1, −1, 0) | 0 ≤ λi < 1, 1 ≤ i ≤ 4} ∩ Z4 = (0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), (2, 0, 0, −1) . Using Jacobi’s triple product identity (1.3), we have [b0 c0 d0 α0 ] L1 =
1 [b0 c0 d0 α0 ] L2 = − (q;q) [b0 c0 d0 α0 ] R = 0, 4 ,
1 , (q;q)4∞
∞
[b1 c0 d0 α0 ] L1 = 0
[b1 c0 d0 α0 ] L2 = 0
[b0 c1 d0 α0 ] L1 = 0,
[b0 c1 d0 α0 ] L2 = 2
1 , (q;q)4∞
q [b2 c0 d0 α−1 ] L2 = 0, [b2 c0 d0 α−1 ] L1 = − (q;q) 4 , ∞
[b1 c0 d0 α0 ] R = 0 [b0 c1 d0 α0 ] R =
1 , (q;q)4∞ 2
q [b2 c0 d0 α−1 ] R = − (q;q) 4 . ∞
Using Theorem 3.2, we obtain (4.9).
5.
Addition formulas for multiple theta functions
Addition formulas for theta functions play an important role in the theory of elliptic functions, see, for example, [4, 9, 13, 16]. In this section, we show that some classical addition formulas for theta functions can be verified by using our vector space approach, such as Riemann’s addition formula and the addition formulas for the Jacobi theta functions. First, let us consider the Riemann’s addition formula, see, for example, Krattenthaler [9]. Weierstrass showed that it is equivalent to an identity on sigma functions, see Whittaker and Watson [16, p. 451]. 16
Example 5.1 (Riemann’s addition formula). Let θ(z) = (z, q/z; q)∞ . We have θ(xy)θ(x/y)θ(uv)θ(u/v) − θ(xv)θ(x/v)θ(uy)θ(u/y) =
u θ(yv)θ(y/v)θ(xu)θ(x/u). y
(5.1)
Proof. Denote the three multiple theta functions in (5.1) by L1 , L2 and R, respectively. It can be checked that L1 , L2 and R satisfy the following four linearly independent contiguous relations 1 f (xq, y, u, v) = 2, f (x, y, u, v) x 1 f (x, yq, u, v) = 2 , f (x, y, u, v) y q 1 f (x, y, uq, v) = 2, f (x, y, u, v) u 1 f (x, y, u, vq) = 2. f (x, y, u, v) v Then we have W = {(2, 0, 0, 0), (0, 2, 0, 0), (0, 0, 2, 0), (0, 0, 0, 2)} and ΠW = {λ1 (2, 0, 0, 0) + λ2 (0, 2, 0, 0) + λ3 (0, 0, 2, 0) + λ4 (0, 0, 0, 2) | 0 ≤ λi < 1, 1 ≤ i ≤ 4} ∩ Z4 = {(a1 , a2 , a3 , a4 ) | ai = 0 or 1, 1 ≤ i ≤ 4}. Owing to the symmetries of the parameters x, y, u and v, it is suffices to show that identity (5.1) holds for the terms with exponent vectors in Π′W = {(0, 0, 0, 0), (1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 0), (1, 1, 1, 1)}. Using Jacobi’s triple product identity (1.3), we have [x0 y 0 u0 v 0 ] L1 =
1 , (q; q)4∞
[x1 y 0 u0 v 0 ] L1 = 0, [x1 y 1 u0 v 0 ] L1 =
−1 , (q; q)4∞
[x1 y 1 u1 v 0 ] L1 = 0, [x1 y 1 u1 v 1 ] L1 =
1 , (q; q)4∞
[x0 y 0 u0 v 0 ] L2 =
1 , (q; q)4∞
[x0 y 0 u0 v 0 ] R = 0,
[x1 y 0 u0 v 0 ] L2 = 0,
[x1 y 0 u0 v 0 ] R = 0,
[x1 y 1 u0 v 0 ] L2 = 0,
[x1 y 1 u0 v 0 ] R =
[x1 y 1 u1 v 0 ] L2 = 0,
[x1 y 1 u1 v 0 ] R = 0,
[x1 y 1 u1 v 1 ] L2 =
Using Theorem 3.2, we complete the proof. 17
1 , (q; q)4∞
−1 , (q; q)4∞
[x1 y 1 u1 v 1 ] R = 0.
Our approach is also applicable to addition formulas on Jacobi theta functions (5.2) due to Whittaker and Watson [16, Chapter XXI]. Recall that the four Jacobi theta functions are given by ϑ1 (z, q) =
∞ X
1
1 2
(−1)n− 2 q (n+ 2 ) e(2n+1)iz ,
n=−∞
ϑ2 (z, q) =
∞ X
1 2
q (n+ 2 ) e(2n+1)iz ,
n=−∞
ϑ3 (z, q) =
∞ X
2
q n e2niz ,
n=−∞
ϑ4 (z, q) =
∞ X
2
(−1)n q n e2niz .
(5.2)
n=−∞
Let 2w′ = −w + x + y + z, 2x′ = w − x + y + z, 2y ′ = w + x − y + z, 2z ′ = w + x + y − z, see Whittaker and Watson [16, Chapter XXI]. For example, we consider two addition formulas given by Whittaker and Watson [16]. Example 5.2 (Whittaker and Watson [16, p. 468]). We have ϑ1 (w)ϑ1 (x)ϑ1 (y)ϑ1 (z) + ϑ2 (w)ϑ2 (x)ϑ2 (y)ϑ2 (z) = ϑ1 (w′ )ϑ1 (x′ )ϑ1 (y ′ )ϑ1 (z ′ ) + ϑ2 (w′ )ϑ2 (x′ )ϑ2 (y ′ )ϑ2 (z ′ ).
(5.3)
Proof. Making the substitutions eiw → w, eix → x, eiy → y, eiz → z and q 2 → q, (5.3) becomes θ(w2 q; q)θ(x2 q; q)θ(y 2 q; q)θ(z 2 q; q) + θ(−w2 q; q)θ(−x2 q; q)θ(−y 2 q; q)θ(−z 2 q; q) = θ(qxyz/w; q)θ(qwyz/x; q)θ(qwxz/y; q)θ(qwxy/z; q) + θ(−qxyz/w; q)θ(−qwyz/x; q)θ(−qwxz/y; q)θ(−qwxy/z; q),
(5.4)
where θ(z) = (z, q/z; q)∞ . Denote the multiple theta functions in (5.4) by L1 , L2 and R1 , R2 , respectively. It is easy to check that L1 , L2 and R1 , R2 satisfy the following four linearly independent contiguous relations f (wq, xq, y, z) 1 = 2 2 2, f (w, x, y, z) w x q 1 f (wq, x, yq, z) = 2 2 2, f (w, x, y, z) w y q 18
f (wq, x, yq, zq) 1 = 2 2 2, f (w, x, y, z) w z q 1 f (w, xq, yq, z) = 2 2 2. f (w, x, y, z) x y q Hence W = {(2, 2, 0, 0), (2, 0, 2, 0), (2, 0, 0, 2), (0, 2, 2, 0)} and ΠW = {λ1 (2, 2, 0, 0) + λ2 (2, 0, 2, 0) + λ3 (2, 0, 0, 2) + λ4 (0, 2, 2, 0) | 0 ≤ λi < 1, i = 1, 2, 3, 4} ∩ Z4
= (0, 0, 0, 0), (1, 1, 0, 0), (1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 1, 0), (2, 1, 1, 0), (2, 1, 0, 1), (1, 2, 1, 0), (2, 0, 1, 1), (1, 1, 2, 0), (1, 1, 1, 1), (3, 1, 1, 1), (2, 2, 1, 1), (2, 2, 2, 0), (2, 1, 2, 1), (3, 2, 2, 1), (1, 1, 1, 0), (2, 2, 1, 0), (2, 1, 2, 0), (2, 1, 1, 1), (1, 2, 2, 0), (3, 2, 2, 0), (3, 2, 1, 1), (2, 3, 2, 0), (3, 1, 2, 1), (2, 2, 3, 0), (2, 2, 2, 1), (4, 2, 2, 1), (3, 3, 2, 1), (3, 3, 3, 0), (3, 2, 3, 1), (4, 3, 3, 1) .
Notice that any nonzero term on both sides of identity (5.4) has even powers in w, x, y and z. Hence it is sufficient to show that identity (5.4) holds only for the terms with exponent vectors belonging to Π′W = {(0, 0, 0, 0), (2, 2, 2, 0)}. By Jacobi’s triple product identity (1.3), we obtain that [w0 z 0 y 0 z 0 ] L1 = 1/(q; q)4∞ ,
[w0 z 0 y 0 z 0 ] L2 = 1/(q; q)4∞ ,
[w0 z 0 y 0 z 0 ] R1 = 1/(q; q)4∞ ,
[w0 z 0 y 0 z 0 ] R2 = 1/(q; q)4∞ ,
[w2 z 2 y 2 z 0 ] L1 = −q 3 /(q; q)4∞ ,
[w2 z 2 y 2 z 0 ] L2 = q 3 /(q; q)4∞ ,
[w2 z 2 y 2 z 0 ] R1 = 0,
[w2 z 2 y 2 z 0 ]R2 = 0.
This completes the proof. Example 5.3 (Whittaker and Watson [16, p. 468]). We have 2ϑ3 (w)ϑ3 (x)ϑ3 (y)ϑ3 (z) = − ϑ1 (w′ )ϑ1 (x′ )ϑ1 (y ′ )ϑ1 (z ′ ) + ϑ2 (w′ )ϑ2 (x′ )ϑ2 (y ′ )ϑ2 (z ′ ) + ϑ3 (w′ )ϑ3 (x′ )ϑ3 (y ′ )ϑ3 (z ′ ) + ϑ4 (w′ )ϑ4 (x′ )ϑ4 (y ′ )ϑ4 (z ′ ).
(5.5)
Proof. By the substitutions eiw → w, eix → x, eiy → y, eiz → z, we can rewrite (5.5) as follows 2θ(−qw2 ; q 2 )θ(−qx2 ; q 2 )θ(−qy 2 ; q 2 )θ(−qz 2 ; q 2 ) = −qwxyzθ(q 2 xyz/w; q 2 )θ(q 2 wyz/x; q 2 )θ(q 2 wxz/y; q 2 )θ(q 2 wxy/z; q 2 ) 19
+ qwxyzθ(−q 2 xyz/w; q 2 )θ(−q 2 wyz/x; q 2 )θ(−q 2 wxz/y; q 2 )θ(−q 2 wxy/z; q 2 ) + θ(−qxyz/w; q 2 )θ(−qwyz/x; q 2 )θ(−qwxz/y; q 2 )θ(−qwxy/z; q 2 ) + θ(qxyz/w; q 2 )θ(qwyz/x; q 2 )θ(qwxz/y; q 2 )θ(qwxy/z; q 2 ),
(5.6)
where θ(z) = (z, q/z; q)∞ . Denote the terms in the above identity by L, R1 , R2 , R3 , R4 , respectively. It is easy to check that L, R1 , R2 , R3 , R4 satisfy the following four linearly independent contiguous relations 1 f (wq 2 , xq 2 , y, z) = 2 2 2, f (w, x, y, z) w x q f (wq 2 , x, yq 2 , z) 1 = 2 2 2, f (w, x, y, z) w y q f (wq 2 , x, y, zq 2 ) 1 = 2 2 2, f (w, x, y, z) w z q 1 f (w, x, yq 2 , zq 2 ) = 2 2 2. f (w, x, y, z) y z q Then we have W = {(2, 2, 0, 0), (2, 0, 2, 0), (2, 0, 0, 2), (0, 0, 2, 2)} and ΠW = {λ1 (2, 2, 0, 0) + λ2 (2, 0, 2, 0) + λ3 (2, 0, 0, 2) + λ4 (0, 0, 2, 2) | 0 ≤ λi < 1} ∩ Z4 = (0, 0, 0, 0), (1, 1, 0, 0), (1, 0, 1, 0), (1, 0, 0, 1), (0, 0, 1, 1), (2, 1, 1, 0), (2, 1, 0, 1), (1, 1, 1, 1), (2, 0, 1, 1), (1, 0, 2, 1), (1, 0, 1, 2), (3, 1, 1, 1),
(2, 1, 1, 2), (2, 1, 2, 1), (2, 0, 2, 2), (3, 1, 2, 2), (1, 0, 1, 1), (2, 1, 1, 1), (2, 0, 2, 1), (2, 0, 1, 2), (1, 0, 2, 2), (3, 1, 2, 1), (3, 1, 1, 2), (2, 1, 2, 2) (3, 0, 2, 2), (2, 0, 3, 2), (2, 0, 2, 3), (4, 1, 2, 2), (3, 1, 2, 3), (3, 1, 3, 2), (3, 0, 3, 3), (4, 1, 3, 3) .
Since (5.6) only contains terms with even powers in w, x, y, z, (5.4) can be justified by showing that it holds only for the terms with exponent vectors in Π′W = {(0, 0, 0, 0), (2, 0, 2, 2)}. By Jacobi’s triple product identity (1.3), we get [w0 z 0 y 0 z 0 ] L = 2/(q 2 ; q 2 )4∞ ,
[w2 x0 y 2 z 2 ] L = 2q 3 /(q 2 ; q 2 )4∞ ,
[w0 x0 y 0 z 0 ] R1 = 0,
[w0 x0 y 0 z 0 ] R2 = 0,
[w0 x0 y 0 z 0 ] R3 = 1/(q 2 ; q 2 )4∞ ,
[w0 x0 y 0 z 0 ] R4 = 1/(q 2 ; q 2 )4∞ ,
[w2 x0 y 2 z 2 ] R1 = q 3 /(q 2 ; q 2 )4∞ ,
[w2 x0 y 2 z 2 ] R2 = q 3 /(q 2 ; q 2 )4∞ , 20
[w2 x0 y 2 z 2 ] R3 = 0,
[w2 x0 y 2 z 2 ] R4 = 0.
Hence the proof is complete. To conclude this paper, we remark that our vector space approach can be used to discover multiple theta function identities. For example, consider the following three linearly independent contiguous relations 1 f (aq, bq, cq) = , f (a, b, c) abc f (aq 2 , b, c) 1 = 2 , f (a, b, c) a q 1 f (a, bq 2 , c) = 2 . f (a, b, c) b q
(5.7)
Then we have W = {(1, 1, 1), (2, 0, 0), (0, 2, 0)} and ΠW = {(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0)}. Thus the dimension of VC is bounded by four. By comparing the coefficients of (a, b, c) with exponent vectors in ΠW , we find the following identity which might be new (q; q 2 )2∞ [a, b, −c; q]∞ + [a, −b, c; q]∞ + [−a, b, c; q]∞ = (a, q/a; q)∞ [bc, bq/c; q 2 ]∞ + (b, q/b; q)∞ [ac, aq/c; q 2 ]∞ + (c, q/c; q)∞ [ab, aq/b; q 2 ]∞ , (5.8)
which reduces to (1.13) by setting c = −1. Acknowledgments. This work was supported by the 973 Project and the National Science Foundation of China.
References [1] C. Adiga, B.C. Berndt, S. Bhargava, and G. Watson, Chapter 16 of Ramanujan’s Second Notebook: Theta-functions and q-Series, Mem. Amer. Math. Soc. (1985), no. 315. [2] G.E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999. [3] W.N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J. Math. 7 (1936), 105–115. [4] B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1985. [5] B.C. Berndt, Number Theory in the Spirit of Ramanujan, AMS Press, Providence, 2006. [6] W.C. Chu, Theta function identities and Ramanujan’s congruences on the partition function, Quart. J. Math. 56 (2005), 491–506. [7] J.A. Ewell, A note on a Jacobian identity, Proc. Amer. Math. Soc. 126 (2) (1998), 421–423.
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[8] H.M. Farkars and I. Kra, On the quintuple product identity, Proc. Amer. Math. Soc. 27 (1999), 771–778. [9] C. Krattenthaler, Advanced determinant calculus: A complement, Linear Algebra Appl. 411 (2005), 68–166. [10] R.A. Malekar and H. Bhate, Discrete Fourier transform and Riemann identities for θ functions, Appl. Math. Lett. 25 (2012), 1415–1419. [11] L.J. Rogers, On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc. 16 (1917), 315–336. [12] I.R. Shafarevich, Basic Algebraic Geometry 2, 2nd ed., Springer-Verlag, New York, 1996. [13] L.-C. Shen, On the additive formulae of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5, Trans. Amer. Math. Soc. 345 (1994), 323–345. [14] R.P. Stanley, Enumerative Combinatorics, Vol. 1, 2nd ed., Cambridge University Press, Cambridge, 2011. [15] G.N. Watson, Theorems stated by Ramanujan, VII: Theorems on continued fractions, J. London. Math. Soc. 4 (1929), 39–48. [16] E.T. Whittaker and G.N. Watson, A Course of Morden Analysis, 4th. ed., Cambridge University Press, Cambridge, 1996.
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