JMAA, to appear.

A COMBINATORIAL PROOF OF THE FARKAS-KRA THETA FUNCTION IDENTITIES AND THEIR GENERALIZATIONS

Frank G. Garvan Department of Mathematics University of Florida Gainesville, FL 32611 Email: [email protected]fl.edu

(Originally submitted January 6, 1994, revised November 4, 1994)

Abstract. Recently, Farkas and Kra found some cubic theta function identities from their work on automorphic forms. Shortly thereafter, Farkas and Kopeliovich were able to gener- alize these to p-th order theta function identities using the theory of elliptic functions. We give short, elementary proofs of the cubic identities. We show that the p-th order identities follow from more general relations between the coefficients of certain theta functions. Our proof is combinatorial and utilizes certain orthogonal transformations.

The author was supported in part by NSF Grant DMS-9208813. 1980 Mathematics Subject Classification (1991 Revision) Primary 33D10, Secondary 05A19, 11F11, 11F20 Key words and phrases. Theta functions, Dedekind’s eta function, q-series, combinatorial identities, bijective proofs, orthogonal matrices

Typeset by -TEX AMS 1 PROPOSED RUNNING HEAD

THETA FUNCTION IDENTITIES

Mailing Address of Author

Department of Mathematics University of Florida 201 Walker Hall Gainesville, FL 32611

2 1. Introduction The Jacobian theta function ϑ(z, q) is defined by

∞ 2 (1.1) ϑ(z, q) := ( 1)nznqn , X n= − −∞ where q < 1 and z = 0. Recently, H. M. Farkas and I. Kra [8] discovered two cubic theta | | 6 function identities:

3 2 1 3 2 3 1 3 3 1 3 (1.2) ϑ (ω q 2 , q 2 ) + ω ϑ (q 2 , q 2 ) + ωϑ (ωq 2 , q 2 ) = 0 and

3 1 3 3 2 1 3 3 3 3 (1.3) ϑ (ωq 2 , q 2 ) ϑ (ω q 2 , q 2 ) = qϑ (ωq 2 , q 2 ), − where ω = exp(2πi/3). We give short elementary proofs of these identities. In 2 we will § show how (1.2) follows easily from Jacobi’s identity

3 1 3 n n(n+1)/2 (1.4) ϑ (q 2 , q 2 ) = ( 1) (2n + 1)q , X − n 0 ≥ and the fact that n(n + 1)/2 2 (mod 3). The second identity (1.3) depends on the 6≡ following identity

3 1 3 3 2 1 3 2 3 9 27 (1.5) ϑ (ωq 2 , q 2 ) ϑ (ω q 2 , q 2 ) = 3q(ω ω)ϑ (q 2 , q 2 ), − − − and Jacobi’s triple product identity. In 2 we give the details as well as generalizations § of (1.2) and (1.5) related to the work of Newman and Kolberg. It should be noted that (1.3) can also be proved by applying Jacobi’s imaginary transformation to (1.2). This observation is due to H. Farkas (private communication). Farkas and Kopeliovich [6] have found a p-th order generalization of (1.2). It is the case k = 1 of the identity (1.6) below. Let p be odd and suppose k = 1, 3, . . . , p 2 and − gcd (k, 2p) = 1. We have

p 1 − p r` p r ` k (1.6) ζ ϑ (ζ 0 q 2 , q 2 ) = 0, X p− p `=0 where ζp = exp(2πi/p), and r, r0 are any integers that satisfy

2 (1.7) 4r k (mod p), 2r0 k (mod p). ≡ − 3 ≡ When p = 3 and k = 1 (1.6) is precisely (1.2). In [7], Farkas and Kopeliovich apply their method to prove some of Ramanujan’s modular equations. In [12], Kopeliovich shows how the methods of [6], [7] can be extended to multidimensional theta functions and finds a multidimensional analogue of the p-th power identity. We were first led to (1.6) by observing (via MAPLE) that one residue class (mod p) p k p was missing for the exponent of q in the q-series expansion of ϑ (q 2 , q 2 ). The missing 2 residue class was r k (mod p). We also observed other relations among the other ≡ − 4 residue classes. The general result is given in Theorem 3.4. Thus the identity (1.6) follows from a special case of this theorem. Our method of proof is combinatorial and utilizes an orthogonal affine linear transformation. We find two different generalizations of (1.3). We have

p 1 (p 1)/2 − k p − p p r` p r0` (k+p)/2 (p+k)(p k)/8 s` s` p ` (1.8) ζ− ϑ (ζ q 2 , q 2 ) = ( 1) q − (ζ− ζ )ϑ (ζ q 2 , q 2 ), X p p − X p − p p `=0 `=0 where r, r0, s are any integers that satisfy

2 (1.9) 8r k (mod p), 2r0 k (mod p), 4s k (mod p). ≡ − ≡ ≡ − We note that (1.3) follows from putting p = 3, k = 1 in (1.8) and using (1.2) to simplify the result. Our second generalization is

(p 1)/2 − ` `(` 1)/2 p ` 1 p p ` 1 p (1.10) ( 1) q − ϑ (ζpq − 2 , q 2 ) ϑ (ζ q − 2 , q 2 ) X − − p
`=1

p 1 p2 1 p p − − p = ( 1) 2 q 8 ϑ (ζpq 2 , q 2 ). − We note that (1.3) is (1.10) with p = 3. Our proofs of (1.6), (1.8), (1.10) and other related identities are given in 3. For p prime we show that (1.10) follows from our main § Theorem 3.4. For general odd p we show that (1.10) follows from the k = 1 case of (1.6) using Jacobi’s imaginary transformation. Some concluding remarks related to the work of Borwein, Borwein and Garvan [4] are given in 4. § 2. Cubic identities In this section we give short elementary proofs of (1.2), (1.3). Generalizations of (1.2) and other identities related to the work of Newman [13] and Kolberg [11] are given. In the 4 next section, different generalizations of (1.2), (1.3) are given together with elementary proofs. We need Jacobi’s triple product identity [1, p. 21],[2, p. 62],[10, p. 282]

1 1 ∞ n n 1 n 1 (2.1) ϑ(zq 2 , q 2 ) = (1 q )(1 zq )(1 z q ), Y − − n=1 − − −

Jacobi’s identity [2, (3.1.14) p. 65], [10, Theorem 357 p. 285]

∞ (2.2) (1 qn)3 = ( 1)n(2n + 1)qn(n+1)/2, Y − X − n=1 n 0 ≥ and Euler’s pentagonal number theorem [1, Corollary 1.7 p. 11], [2, (3.1.10) p. 54]

∞ n ∞ n n(3n+1)/2 1 3 (2.3) (1 q ) = ( 1) q = ϑ(q 2 , q 2 ). Y X n=1 − n= − −∞ It is well-known that (2.2), (2.3) follow easily from (2.1). We let

∞ n ∞ n 3 3 1 3 (2.4) A(q) := a q := (1 q ) = ϑ (q 2 , q 2 ). X n Y n=0 n=1 −

We observe that n(n + 1)/2 2 (mod 3), so that from (2.2) we have 6≡

∞ ∞ (2.5) 0 = 3a q3n+2 = a (1 + ωn 2 + ω2(n 2))qn X 3n+2 X n − − n=0 n=0 = A(q) + ωA(ωq) + ω2A(ω2q)

3 1 3 3 2 1 3 2 3 1 3 = ϑ (q 2 , q 2 ) + ωϑ (ω q 2 , q 2 ) + ω ϑ (ωq 2 , q 2 ).

Equation (1.2) follows by multiplying both sides of (2.5) by ω2. We now prove (1.3). We define

1 3 ∞ n n(3n+1)/2 ∞ n (2.6) η(q) := ϑ(q 2 , q 2 ) = ( 1) q = (1 q ), X Y n= − n=1 − −∞ by (2.3). Since n(n + 1)/2 2 (mod 3) then from (2.2) we have 6≡ 3 3 3 (2.7) η (q) = 0(q ) + q1(q ), 5 for certain power series 0, 1. Now, (2.8) 3 1 3 3 2 1 3 ϑ (ωq 2 , q 2 ) ϑ (ω q 2 , q 2 ) − = η3(ω2q) η3(ωq) (by (2.6)) − 2 3 = (ω ω)q1(q ) (by (2.7)) − = (ω2 ω) ( 1)n(2n + 1)qn(n+1)/2 − X − n 0 n 1 (mod≥ 3) ≡ = 3q(ω2 ω) ( 1)m(2m + 1)q9m(m+1)/2 (by substituting n = 3m + 1 in − − X − m 0 the sum) ≥ = 3q(ω2 ω)η3(q9). − − From (2.1) we have

1 1 ∞ n n n(n+1)/2 (2.9) ϑ(ωq 2 , q 2 ) = ( 1) ω q X n= − ∞ ∞ = (1 qn)(1 ωqn)(1 ω 1qn 1) Y − − n=1 − − − = (1 ω2) (1 q3n) Y − n=1 − = (1 w2)η(q3). − An easy calculation gives 3(ω2 ω) = (1 ω2)3. Thus combining (2.8), (2.9) we find − − − that

3 1 3 3 2 1 3 ϑ (ωq 2 , q 2 ) ϑ (ω q 2 , q 2 )(2.10) − = 3q (ω2 ω)η3(q9) = q (1 ω2)3η3(q9) − − − 3 3 3 = q ϑ (wq 2 , q 2 ), which is (1.3). We now consider other cubic identities related to the work of Kolberg [11]. Let p > 2 be a prime. We define

n 1 n(n+1) h = ( 1) (2n + 1)q 2 (s = 0, 1, . . . , p 1); s X 1 − − 2 n(n+1) s (mod p) n≡ 0 ≥ 6 so that p 1 − h = η3(q). X s s=0

The proof of the following lemma is analogous to that of (2.7), (2.8).

Lemma 2.11. (Kolberg) If 8s + 1 is a quadratic non-residue (mod p), then hs = 0. If 8s + 1 0 (mod p), then ≡

1 (p 1) 1 (p2 1) 3 p2 (2.12) hs = ( 1) 2 − p q 8 − η (q ). −

We may cast this result in terms of theta functions.

Corollary 2.13. Let p be an odd prime, let 8s + 1 be a quadratic non-residue (mod p), 2 let t = (p 1)/8, and let ζp = exp(2πi/p). Then −

p 1 − ks 3 k(p+1)/2 1 3k(p+1)/2 3 (2.14) ζ ϑ (ζ q 2 , ζ q 2 ) = 0, X p− p p k=0 and

p 1 − p2 3p2 kt 3 k(p+1)/2 1 3k(p+1)/2 3 (p 1)/2 2 t 3 (2.15) ζ− ϑ (ζ q 2 , ζ q 2 ) = ( 1) − p q ϑ (q 2 , q 2 ). X p p p − k=0

Newman [13] has found results analogous to (2.12) or (2.15) for higher powers of η(q).

To describe Newman’s results we need some notation. For an integer r we define pr(n) by

∞ ∞ (2.16) p (n)qn = (1 qn)r, X r Y n=0 n=1 − so that p3(n) = an, which was defined in (2.4). We note that pr(n) is defined to be zero when n is not a non-negative integer.

Theorem 2.17. (Newman) Suppose r is one of the numbers 2, 4, 6, 8, 10, 14, 26. Let p be a prime > 3 such that r(p + 1) 0 (mod 24) and let ∆ = r(p2 1)/24. Then ≡ −

(r/2) 1 (2.18) pr(np + ∆) = ( p) − pr(n/p). −7 3. The p-th power generalizations In this section we prove (1.6), (1.8) and (1.10), the generalizations of (1.2) and (1.3). We also derive other analogous identities. The methods of 2 don’t suffice so we use a § different approach. Throughout this section k, p are odd integers with p 3 and gcd(2p, k) = 1. We define ≥ n p k p T (k, p, n) as the coefficient of q in the series expansion of ϑ (q 2 , q 2 ); ie.

n ~n ~1 p ~n 2+ k ~n ~1 p k p (3.1) T (k, p, n)q = ( 1) · q 2 || || 2 · = ϑ (q 2 , q 2 ), X Xp − n n0 ~n Z ≥ ∈ where ~1 = (1, 1,..., 1). By considering each of the transformations

(i) ~n ~n, 7→ − and

(ii) ~n ~n + ~1, 7→ we have

(3.2) T ( k, p, n) = T (k, p, n), − and

p(p+k) p(k p) (3.3) T (k, p, n) = T (k + 2p, p, n ) = T (2p k, p, n + − ). − − 2 − − 2 Hence it is enough to consider k = 1, 3, . . . , p 1 for fixed p. The main result is − Theorem 3.4. Let p be an odd integer p 3. Suppose gcd(2p, k) = 1 and k = 1, 3, . . . , p ≥ − 2. Then

` `(` k) (3.5) T (k, p, pn + r) = ( 1) T (2` k, p, pn + r − ), − − − 2 k+1 k+p 2 k` where ` is any integer that satisfies ` − , gcd (2` k, p) = 1 and r 2 ≤ ≤ 2 − ≡ − 4 (mod p); and

(3.6) T (k, p, pn + s) = ( 1)`T (2` + k, p, pn + s `(`+k) ), − − 2 k+1 k+p 2 k` where ` is any integer that satisfies − ` − − , gcd (2` + k, p) = 1 and s 2 ≤ ≤ 2 ≡ 4 (mod p).

By taking ` = k in the theorem we obtain the following corollary which is equivalent to (1.6). 8 Corollary 3.7. Let p be an odd integer p 3. Suppose gcd(2p, k) = 1 and k = 1, 3, . . . , p ≥ − 2. Then

k2 (3.8) T (k, p, pn + r) = 0, where r (mod p). ≡ − 4

We illustrate the theorem and its corollary for p = 3, 5 and 7. For p = 3, there is only one relation

(3.9) T (1, 3, 3n + 2) = 0, which is equivalent to (1.2). For p = 5, there are four relations:

(3.10) T (1, 5, 5n + 1) = 0, (3.11) T (1, 5, 5n + 2) = T (3, 5, 5n + 1), (3.12) T (1, 5, 5n + 4) = T (3, 5, 5n + 3), − (3.13) T (3, 5, 5n + 4) = 0.

For p = 7, there are 9 relations:

(3.14) T (1, 7, 7n + 1) = T (5, 7, 7n 2), − − (3.15) T (1, 7, 7n + 2) = T (3, 7, 7n + 1), − (3.16) T (1, 7, 7n + 3) = T (3, 7, 7n + 2), (3.17) T (1, 7, 7n + 4) = T (5, 7, 7n + 1), (3.18) T (1, 7, 7n + 5) = 0, (3.19) T (3, 7, 7n + 3) = 0, (3.20) T (3, 7, 7n + 4) = T (5, 7, 7n + 2), (3.21) T (3, 7, 7n + 6) = T (5, 7, 7n + 4), − (3.22) T (5, 7, 7n + 6) = 0.

Our proof of Theorem 3.4 is combinatorial and utilizes an orthogonal affine linear trans- formation L` given below in (3.30). For fixed odd p 3 we define the p p matrix Q as ≥ × the matrix whose columns are

2 (3.23) ~vi = ~ei ~1, (1 i p), − p ≤ ≤ 9 where the ~ei are the columns of the identity matrix so that

2 (3.24) Q = I J, − p where J is the p p matrix all of whose entries are one. The following properties of Q are × easily proved:

(3.25) QT = Q, (3.26) QT Q = I (Q is an orthogonal matrix),

(3.27) Q~1 = ~1. − It should be noted that Q is an example of a Householder matrix. The use of Householder matrices is important in numerical linear algebra [L, p. 358]. For ~n Zp, we define ∈ p k (3.28) F (~n) := F (~n,k) = ~n 2 + ~n ~1, 2|| || 2 · so that

~n ~1 F (~n) p k p (3.29) ( 1) · q = ϑ (q 2 , q 2 ). Xp − ~n Z ∈ For any integer ` we define the affine linear transformation

p p ` (3.30) L` : R R by L`(~n) = Q~n ~1. −→ − p

Then the following properties of L` are easily proved: `(` k) (3.31) F (L`(~n), k) = F (~n, 2` k) + − , − 2 1 (3.32) L`(~n) = ~n (2(~n ~1) + `)~1, − p ·

(3.33) L`(~n) ~1 = (~n ~1) `, · − · − 2 (3.34) L` = I.

For  = +1 or 1, we define −

 p p 2 k ~n ~1 (3.35) (N) = ~n Z : N = ~n + ~n ~1 and ( 1) · =  . Sk,p { ∈ 2|| || 2 · − } 10 We turn to the proof of Theorem 3.4. First we prove (3.5) by showing that

 0 `(` k) (3.36) L` : (N) (N + − ) Sk,p −→ Sk0,p 2 is a bijection if N k` (mod p), gcd (k, 2p) = 1 and gcd (2` k, p) = 1, and where ≡ − 4 − ` k`  0 = ( 1)  and k0 = 2` k. Suppose N (mod p), gcd (k, 2p) = 1 and ~n (N). − − ≡ − 4 ∈ Sk,p Then k k` ~n ~1 (mod p), 2 · ≡ − 4 ` and ~n ~1 (mod p) since gcd (k, p) = 1. · ≡ −2

From (3.32) it follows that p L`(~n) Z . ∈

If we let ~m := L`(~n) then

~m ~1 ~n ~1 ` ( 1) · = ( 1)− · − by (3.33) − − = ( 1)` − = 0, and

p k0 ~m ~m + ~m ~1 = F (L`(~n), 2` k) 2 · 2 · − `(k `) = F (~n,k) + − by (3.31) 2 `(k `) = N + − . 2

0 `(k `) It follows that L`(~n) (N + − ), and the map in (3.36) is well-defined. If in ∈ Sk0,p 2 addition, gcd (2` k, p) = 1, then analogously we find that −  L`(~m) (N) ∈ Sk,p

 `(k `) 2 where ~m is any element of 0 (N + − ). Since L = I, we see that the transformation Sk0,p 2 ` L` in (3.36) defines a bijection and (3.5) follows. Equation (3.6) follows by replacing k by k in (3.5) and using (3.2). This completes the proof of Theorem 3.4. − 11 k+p We now turn to (1.8) and consider L` where ` = . In this case we have 2` k = p 2 − so that Theorem 3.4 does not apply. However we are still able to identify the image of  k2 (N) under L` when N (mod p). For  = +1 or 1 we define Sk,p ≡ − 8 − (3.37)

 p p 2 p ~m ~1 k (M) = ~m Z : M = ~m + ~m ~1, ( 1) · = , and ~m ~1 (mod p) . Rk,p { ∈ 2|| || 2 · − · ≡ − 4 }

2 For gcd (k, 2p) = 1 and N k (mod p) we will prove that ≡ − 8

 0 (k+p)(k p) (3.38) L k+p : k,p(N) k,p(N + 8 − ) 2 S −→ R

k+p 2 2 k is a bijection, where 0 = ( 1) . We suppose N 8 (mod p), gcd (k, 2p) = 1 and  − ≡ − ~n k,p(N). If we let ~m = L k+p (~n) then, as in the proof of (3.36), we find that ∈ S 2

p ~m ~1 p p (k + p)(k p) ~m = L k+p (~n) Z , ( 1) · = 0, ~m ~m + ~m ~1 = N + − . 2 ∈ − 2 · 2 · 8

In addition we find that

k + p k ~m ~1 = L k+p (~n) ~1 = (~n ~1) (mod p), · 2 · − · − 2 ≡ − 4

2 since N k (mod p) and gcd (k, p) = 1 imply ~n ~1 k (mod p). Hence, ≡ − 8 · ≡ − 4

0 (k + p)(k p) ~m = L k+p (~n) k,p(N + − ). 2 ∈ R 8

k2  (k+p)(k p) Now suppose N (mod p), gcd (k, 2p) = 1 and ~m 0 (N + − ). If we ≡ − 8 ∈ Rk,p 8 k+p let ~n := L k+p (~m) and  = ( 1) 2 0 then we find that 2 −

p ~n ~1 p k ~n = L k+p (~m) Z , ( 1) · = , ~n ~n + ~n ~1 = N. 2 ∈ − 2 · 2 ·

 2 Hence ~n k,p(N). Since L k+p = I, it follows that the map in (3.38) is a bijection. ∈ S 2 We now show how the bijection in (3.38) may be interpreted in terms of theta-functions and hence prove (1.8). If ζp = exp(2πi/p) then

p 1 − p if k 0 (mod p), ζjk =  ≡ X p 0 otherwise. j=0 12 For 0 j p 1, we define ≤ ≤ −

p ∞ (3.39) G := G (q) = ( 1)mζjmqpm(m+1)/2! j j X p m= − −∞ p p p j 2 2 = ϑ (ζpq , q ), and

~m ~1 p ~m 2+ p ~m ~1 Fj := Fj(q) = ( 1) · q 2 || || 2 · . Xp − ~m Z ~m ~1 j∈(mod p) · ≡

We observe that m(m + 1) is invariant under m 1 m. We have 7→ − −

( ~1 ~m) ~1 = (~m ~1) p − − · − · − so that by considering ~m ~1 ~m we find that 7→ − −

Fj = Fp j. − −

Hence

p 1 − (3.40) G = ζkjF j X p k k=1

(p 1)/2 − kj kj = (ζ ζ− )Fk. X p − p k=1

We claim that

(p 1)/2 − 1 kj kj (3.41) Fj = (ζ− ζ )Gk. p X p − p k=1 13 Now

(p 1)/2 − kj kj (ζ− ζ )Gk X p − p k=1

(p 1)/2 (p 1)/2 − − kj kj `k `k = (ζ− ζ ) (ζ ζ− )F` X p − p X p − p k=1 `=1

(p 1)/2 (p 1)/2 − −  k(` j) k(j+`) k(j+`) k(j `) = ζ − ζ ζ− + ζ − F` X X p − p − p p `=1  k=1  p 1 − 2kj = p 1 ζ ! Fj (since the inner sum above is zero when ` = j) − − X p 6 k=1

= pFj, and (3.41) follows. This means we may now interpret the bijection (3.38) as a theta- 2 function identity. If r k (mod p) then we have ≡ − 8

k+p (k+p)(k p) pN+r+ − (3.42) T (k, p, pN + r) = ( 1) 2 Coefficient of q 8 in Fs(q), − where s k (mod p). The equation (1.8) follows easily via (3.41). ≡ − 4 Our proof of (3.41) leads one to consider the matrix

(3.43) S = sin(2πij/p) p 1 . 1 i,j −
≤ ≤ 2

Our result (3.41) is equivalent to

p (3.44) S2 = I. 4

For

(3.45) C = cos(2πij/p) p 1 1 i,j −
≤ ≤ 2 we find

p 1 (3.46) C2 = I J. 4 − 2 14 We note that the matrices S and C occurred in a problem in the American Mathematical Monthly proposed by Ron Evans and Jerrold Griggs [5]. See page 64 of the January 1991 issue for the solution. We now turn to the proof of (1.10). First we show how (1.10) follows from Theorem 3.4 when p is a prime. Then we show how, for general odd p, (1.10) follows from the k = 1 case of (1.6) using Jacobi’s imaginary transformation. We assume p is an odd prime. First we write (1.10) in terms of the T (k, p, n) defined in (3.1). For k 0 (mod p) we define k0 (mod p) by 6≡ 2 (3.47) k0 (mod p). ≡ k

k+1 By replacing ` by 2 and using (3.1), (3.39) we find that (1.10) may be written as

p 2 − k+1 k2 1 k n p 1 p2 1 − k0n 0 n − − (3.48) ( 1) 2 q 8 T (k, p, n) ζ ζ q = ( 1) 2 q 8 G1, X − X p − p
− k=1 n k odd or (3.49)

p 2 p 1 2 2 2 − − k+1 k 1 k (r ( k 1 )) k0(r ( 8− )) 0 8− k 1 pn+r ( 1) 2 ζp − ζ − T (k, p, pn + r − )q X X X − − p
− 8 n k=1 r=0 k odd

p 1 p2 1 − − = ( 1) 2 q 8 G1. − p Since G1 is a function of q we must first show that

p 2 2 2 2 − k+1 k 1 k (r ( k 1 )) k0(r ( 8− )) 0 8− k 1 (3.50) ( 1) 2 ζp − ζ − T (k, p, pn + r − ) = 0, X − − p
− 8 k=1 k odd for 0 r p 1 with r 1 (mod p). We prove (3.50) by showing that each term on ≤ ≤ − 6≡ − 8 the left side is either zero or can be paired with another term of opposite sign. Let 0 r p 1 be fixed and suppose r 1 (mod p). We consider each of the ≤ ≤ − 6≡ − 8 integers 1 k p 2 with k odd. Each such integer k is one of two types: Type(I) or ≤ ≤ − Type(II). The types correspond to the two parts of Theorem 3.4 namely, (3.5) and (3.6). More explicitly the types are defined as follows. First we consider such an integer k. The sequence of consecutive integers

k (p 2) k + (p 2) (3.51) − − ,..., − 2 2 15 k represent all residue classes (mod p) except the class congruent to 2 (mod p). Since 1 r (mod p) we may choose an integer `1 from the sequence (3.51) such that 6≡ − 8

1 2 (3.52) `1 (k (8r + 1)) (mod p), ≡ 2k − so that

2 k 1 k`1 (3.53) r  −  (mod p). − 8 ≡ − 4

k+1 k+(p 2) If `1 − we say k is of Type(I) and we let ` = `1. On the other hand if 2 ≤ ≤ 2 k (p 2) k 1 − − `1 − we say k is of Type(II) and we let ` = `1. 2 ≤ ≤ 2 − We now fix n. We define the set

:= 1, 3, . . . , p 2 . S { − } For k we define the weight ω(k) as ∈ S

2 2 2 k+1 k 1 k (r ( k 1 )) k0(r ( 8− )) 0 8− k 1 ω(k) := ( 1) 2 ζp − ζ − T (k, p, pn + r − ), − − p
− 8 so that (3.50) may be written as

(3.54) ω(k) = 0. X k ∈S

Equation (3.54) and hence (3.50) will follow by constructing a sign-reversing involution Φ

Φ: ; ω(Φ(k)) = ω(k). S −→ S − Our map Φ is defined simply as Φ(k) = 2` k where ` = `(k) is defined above. If k is of − Type(I) then an easy calculation shows that

k2 1 (2` k)2 1 (3.55) k0 r  −  (2` k)0 r  − −  (mod p), − 8
≡ − − 8
k2 1 `(` k) (2` k)2 1 (3.56) − + − = − − , 8 2 8 k+1 ` 2` k+1 (3.57) ( 1) 2 ( 1) = ( 1) −2 . − − − − 16 Also we have

k2 1 k` (3.58) r  −  (mod p), − 8 ≡ − 4 by (3.52). Hence by (3.5) we have

(3.59) ω(Φ(k)) = ω(2` k) = ω(k). − −

We find that

(3.60) 1 2` k p 2, ≤ − ≤ − 2` k (2` k) + p 2 (3.61) − ` − − , 2 ≤ ≤ 2 and

(2` k)2 1 ` (3.62) r  − −  (2` k) (mod p). − 8 ≡ −4 −

It follows that Φ(k) , Φ(k) is also Type(I) with the same value of ` as k, so that ∈ S

Φ(Φ(k)) = 2` (2` k) = k. − −

Similarly, when k is of Type(II) we find that

ω(Φ(k)) = ω(k) − using (3.6), Φ(k) , Φ(k) is of Type(II) and Φ(Φ(k)) = k. Hence Φ is a sign-reversing ∈ S involution on as required. S From (3.49) we now see that the proof of (1.10) is reduced to proving

(3.63)

p 2 2 k2 1 k 1 2 − k+1 k0(r ( − )) k 1 k0(r0 ( 8− )) 0 8 pn+r0 ( 1) 2 ζp − ζ − T (k, p, pn + r0 − )q X X − − p
− 8 n k=1 k odd

p 1 p2 1 − − = ( 1) 2 q 8 G1, − 17 p2 1 where 1 r0 p 1 and r0 − (mod p). From (3.42) we find that the left side of ≤ ≤ − ≡ 8 (3.63) may be written as

(3.64) p 2 p 1 p2 1 − −2 8− k/4 k/4 ( 1) q ζp ζp− F( k/4)(q) (where each k/4 is reduced modulo p), − X −
− − k=1 k odd p 1 (since k (k odd, k = 1, 3, . . . , p 2) forms p 1 p2 1 − − − k ± − = ( 1) 2 q 8 ζ Fk(q) a complete non-zero residue system mod- − X p k=1 ulo p ),

p 1 p2 1 − − = ( 1) 2 q 8 G1 (by (3.40)), − and this gives the result (3.63) and this completes the proof of (1.10) for p prime. We now show how, for general odd p, (1.10) follows from the k = 1 case of (1.6) using Jacobi’s imaginary transformation formula [2, Eq. (2.2.5) p.38]

∞ 1 (n+x)2π ∞ 2πikx πk2s (3.65) e s = √s e e , X − X − n= k= −∞ −∞ where re(s) > 0. If we let x = m + `τ and s = iτ then (3.65) becomes 2p 2p −

2 ∞ πikm 1 (k2+ `k )+ ` (3.66) √ iτ e p q 2 p 8p2 − X k= −∞ 2 πim` ∞ πin` 1 (n2+ nm )+ m = e− 2p2 e p q 2 p 8p2 , X − 0 n= −∞

πi where q = exp(πiτ) q0 = q( 1/τ) = exp( ) and im(τ) > 0. The k = 1 case of (1.6) − − τ may be written as

p 1 p `(1 p2) n`(1+p) − ∞ n −4 n 2 (pn+1) (3.67) ζp ( 1) ζp q 2 ! = 0. X X − `=0 n= −∞

We observe that

πin` n`(1+p) exp( p ), ` odd, n 2 ` ( 1) ζp = exp(nπi ` 1 + ) = ( − { − p } πin(` p) exp( p− ), ` even. 18 1 If we multiply both sides of (3.67) by √ iτq 8p2 , apply the transformation (3.66) with p − ` = 1, and replace q0 by q we obtain (3.68) p 2 p p 2 p 2 − πi`p `2 ∞ πin n πi − πi`p (` p) ∞ πin n (pn+`) − (pn+` p) e − 2 q 8 e p q 2 ! + e 2 e − 2 q 8 e p q 2 ! X X − X X − − `=1 n= `=1 n= ` odd −∞ ` even −∞ p 2 πi p ∞ πin pn (n+1) + e 2 q 8 e p q 2 ! = 0. − X − n= −∞ On replacing ` by p ` in the outer sum in the second term, then simplifying and reversing − the order of summation in the inner sum we have (3.69) p 2 p p − πi`p `2 ∞ πin n ∞ πin n − (pn+`) (pn+`) e 2 q 8 ( e− p q 2 ! e p q 2 ! ) X X − X `=1 n= n= ` odd −∞ −∞ p 2 πi p ∞ πin pn (n+1) = e 2 q 8 e p q 2 ! . − X − − n= −∞

p+1 πi/2 1 On multiplying both sides by ( 1) 2 e q− 8 and simplifying we have − p 2 p p − `+1 `2 1 ∞ πin n ∞ πin n − (pn+`) (pn+`) ( 1) 2 q 8 ( e− p q 2 ! e p q 2 ! ) X − X − X `=1 n= n= ` odd −∞ −∞ p 2 p 1 p 1 ∞ πin pn (n+1) = ( 1) −2 q 8− e p q 2 ! , X − − n= −∞

(p 1)/2 which is (1.10) with ζp replaced by ζp− − which is a primitive p-th root of unity, so that (1.10) clearly follows.

4. Concluding remarks The following classical identity follows easily from Theorem 4 and Corollary 1 in [8]

(4.1) ϑ4( 1, q) = ϑ4(1, q) + qϑ4( q, q). − − It is of interest that the two theta functions ϑ2( 1, q) and ϑ2(1, q) parametrize the − arithmetic-geometric mean iteration of Gauss [2, Chapter 1] whose limit is identified with 19 1 1 the hypergeometric function 2F1( , ; 1; ). Recently, Borwein and Borwein [3] were able 2 2 · to parametrize a cubic mean iteration whose limit is identified with the the hypergeo- 1 2 metric function 2F1( , ; 1; ). For a survey of recent related results see [9]. Some of the 3 3 · properties of the Borweins’ cubic iteration bear some striking similarities to the classical case of Gauss. This time the parametrization is in terms of

∞ 2 2 (4.2) a(q) := qm +mn+n , X m,n= −∞ and

∞ 2 2 (4.3) b(q) := ωn mqm +mn+n where ω = exp(2πi/3). X − m,n= −∞ The analogue of (4.1) is

(4.4) a3(q) = b3(q) + c3(q), where

∞ (m+ 1 )2+(m+ 1 )(n+ 1 )+(n+ 1 )2 (4.5) c(q) = q 3 3 3 3 . X m,n= −∞ Equation (4.4) bears a striking resemblance to (1.2). It is however a different identity. It would be interesting to investigate whether the theta-functions in (1.2) occurred in the parametrization of some two-term iteration whose limit could be explicitly identified. There are connections with (1.2), (1.3). From (2.7), (2.8) we have

3 3 3 2 3 3 3 9 (4.6) η (q) = 0(q ) + q1(q ) + q 2(q ) = 0(q ) 3 qη (q ), − for certain power series i(q). The fact that 2(q) = 0 led to identity (1.2). The identity 3 3 1(q) = 3 η (q ) led to identity (1.3). Below we identify 0(q) which leads analogously to − a third identity. We find that

(4.7) 0(q) = η(q) a(q), where a(q) is defined in (4.2). From [4] we have η3(q) (4.8) b(q) = , η(q3) 3 3 1 η (q ) (4.9) c(q) = 3 q 3 , η(q) (4.10) b(q) = a(q3) c(q3). 20 − If we multiply both sides of (4.10) by η(q3) we obtain

(4.11) η3(q) = η(q3) a(q3) 3 qη3(q9), − which gives (4.7). In [4] we showed how the cubic identity (4.4) follows easily from (4.8) and (4.10). Using (1.2) and (2.9) we find that (4.7) may be written as the identity

3 2 1 3 2 3 1 3 1 1 3 (4.12) ϑ (ω q 2 , q 2 ) ω ϑ (ωq 2 , q 2 ) = ϑ(ωq 2 , q 2 ) a(q ). − It should be pointed out that Kopeliovich [12] has found a multidimensional general- ization of the p-th power identity (1.6). It would interesting to see whether the methods of this paper can be extended to the multidimensional case.

Acknowledgments I would first like to thank Bruce Berndt and Jon Borwein for communicating the Farkas- Kra cubic identities to me. When I wrote the original version of this paper I was unaware of the work of Farkas and Kopeliovich. I had independently discovered and proved the p-th power generalization (1.6) of (1.3). I was also unaware that Kopeliovich had found a multidimensional analogue. I would like to thank the referee and others for pointing out these oversights. I would also like to thank Farkas, Kra and Kopeliovich for sending preprints of their papers and for useful discussions. In the original version I had found only one generalization of (1.3), namely (1.8). I tried to find a more straightforward generalization after H. Farkas pointed out to me that (1.3) follows from (1.2) via a modular transformation. Thus I found (1.10) by applying a similar transformation to (1.6). But my goal was then to find a more combinatorial proof.

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