DETERMINANT IDENTITIES FOR THETA FUNCTIONS

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1 DETERMINANT IDENTITIES FOR THETA FUNCTIONS SHAUN COOPER AND PEE CHOON TOH Abstract. Two proofs of a theta function identity of R. W. Gosper and R. Schroeppel are given. A cubic analogue is presented, and several interesting special cases are noted. 1. Introduction Let τ be a complex number with positive imaginary part, and let q = expiπτ so that q < 1. The Jacobian theta functions may be defined by z τ = i ϑ z τ = ϑ 3 z τ = and ϑ 4 z τ = n= n= n= n= 1 n q n+ 1 e in+1z, q n+ 1 e in+1z, q n e inz, 1 n q n e inz. The following result was stated without proof by R. W. Gosper and R. Schroeppel [7]. Theorem 1.1 Gosper and Schroeppel. Let w 1, w, w 3, z 1, z and z 3 be complex variables, and consider the 3 3 matrix whose j, k entry is ϑ r w j z k τϑ s w j + z k τ, where r, s {1,, 3, 4}. Then 1 ϑ r w j z k τϑ s w j + z k τ 1 j,k 3 Gosper and Schroeppel observed that many of the identities in the treatise by Whittaker and Watson [10, Chapter 1] are special cases of the identity 1. For example, [7, 4vars], cf. [10, Ex. 5, p. 451]: Date: November 7, Mathematics Subject Classification. Primary 33E05; Secondary 05A19, 11F7, 15A15. 1

2 SHAUN COOPER AND PEE CHOON TOH Corollary 1.. Let a 1, a, b 1 and b be complex variables. Then, for s = 1,, 3 or 4, we have a j b k τϑ s a j + b k τ 1 j,k = a 1 a τϑ s a 1 + a τ b 1 b τϑ s b 1 + b τ. Proof. Setting w 1 = z 1 and w = z in 1 and using the fact that z τ is an odd function of z and so 0 τ = 0, we get z 1 z τϑ s z 1 + z τ z z 3 τϑ s z + z 3 τ w 3 z 1 τϑ s w 3 + z 1 τ + z 1 z 3 τϑ s z 1 + z 3 τ z z 1 τϑ s z + z 1 τ w 3 z τϑ s w 3 + z τ w 3 z 3 τϑ s w 3 + z 3 τ z z 1 τϑ s z + z 1 τ z 1 z τϑ s z 1 + z τ Now cancel the common factor z 1 z τϑ s z 1 +z τ and replace z 1, z, z 3, w 3 with a, a 1, b 1, b to complete the proof. The remainder of this work is arranged as follows. We give two proofs of Theorem 1.1 in Sections and 3. In Section 4, we give an analogous result for cubic theta functions. Finally, in Section 5 we show that a special case of a erminant of cubic theta functions is equivalent to an identity of M. Hirschhorn et al [8] and how this identity leads to a fundamental result from the theory of elliptic functions in signature 3.. First proof of the Gosper-Schroeppel identity We begin by observing that by the translational properties of theta functions, we have ϑ r w j z k τϑ s w j + z k τ = γ W j Z k τ W j + Z k τ, where the values of W j, Z k and γ, which depend on w j, z k, q, r and s, can be worked out from [10, p. 464, Ex. ]. For example, with r = 4 and s = 3, we have W j = w j + π 4 + πτ, Z k = z k + π 4 and γ = iq1/ e iw j. Therefore ϑ 4 w j z k τϑ 3 w j + z k τ 1 j,k 3 = iq 1/ e iw j W j Z k τ W j + Z k τ 1 j,k 3 = iq 3/ e iw 1+w +w 3 W j Z k τ W j + Z k τ 1 j,k 3. Consequently, it suffices to prove Theorem 1.1 in the case r = s = 1. By [10, p. 465], the function satisfies the functional equations z + π τ = z τ and z + πτ τ = q 1 e iz z τ. It follows that w z π τ w + z + π τ = w z τ w + z τ

3 DETERMINANT IDENTITIES FOR THETA FUNCTIONS 3 and 3 w z πτ τ w + z + πτ τ = q e 4iz w z τ w + z τ. Fix w 1, w, w 3, z and z 3 and consider the erminant as a function of z 1, i.e., let Furthermore, let F z 1 = w j z k τ w j + z k τ 1 j,k 3. Gz 1 = z + z 1 τ z z 1 τ. Then, 3 and elementary properties of the erminant imply 4 F z 1 + π = F z 1, F z 1 + πτ = q e 4iz 1 F z 1. Clearly, by and 3 we also have 5 Gz 1 + π = Gz 1, Gz 1 + πτ = q e 4iz 1 Gz 1. If z 1 = ±z or z 1 = ±z 3 then the matrix in the definition of F z 1 will have two identical columns, and so F z 1 From this, and using 4, it follows that the function F z 1 has zeros at z 1 = ±z + mπ + nπτ and at z 1 = ±z 3 + mπ + nπτ, where m and n are any integers and possibly at at other points, too. It is known [10, pp. 470] that the zeros of z τ are all simple and occur precisely at the points z = mπ + nπτ, where m and n are integers 1. Thus, Gz 1 has simple zeros at z 1 = ±z + mπ + nπτ, and these are the only zeros of G. It follows that the quotient F z 1 /Gz 1 is an entire doubly periodic function with periods π and πτ. By Liouville s theorem it is constant. If we set z 1 = z 3, we find that the value of the constant is zero. Thus F z 1 0 and this completes the first proof of 1. Remark.1. Recently, the second author [9] used similar techniques to construct several infinite families of erminant identities involving theta functions, which are equivalent to the Macdonald identities. 3. Second proof of the Gosper-Schroeppel identity For the second proof, we rely on two simple lemmas. As noted at the beginning of the previous section, it suffices to prove Theorem 1.1 for the case r = s = 1. Lemma 3.1. w z τ w + z τ = ϑ 3 w τϑ z τ ϑ w τϑ 3 z τ. 47]. 1 This is an immediate consequence of Jacobi s triple product identity [10, pp. 469,

4 4 SHAUN COOPER AND PEE CHOON TOH Proof. By the definition of we have w z τ w + z τ = 1 m+n q m +n +m+n+ 1 e m+n+1iw+n miz. m= n= Now apply the series rearrangement 6 c m,n = j c j+k,k j + j k k c j+k+1,k j to complete the proof. Lemma 3.. Let r i, s i, t i and u i be complex variables, where 1 i 3. Then r j s k + t j u k 1 j,k 3 Proof. Expand the erminant and observe that all terms cancel. We are now ready for the second proof of Theorem 1.1. Second proof of Theorem 1.1. By Lemmas 3.1 and 3., we have w j z k τ w j + z k τ 1 j,k 3 = ϑ 3 w j τϑ z k τ ϑ w j τϑ 3 z k τ 1 j,k 3 Remark 3.3. Lemma 3.1 may also be used to give an alternative proof of Corollary 1.. A sketch of the argument is as follows. When s = 1, apply Lemma 3.1 to each matrix entry, expand the erminant, and factor the resulting expression. When s =, 3 or 4, use the translational properties, as explained at the beginning of Section. 4. Cubic theta functions Let q be a complex number which satisfies q < 1, and let x and y be nonzero complex numbers. The cubic theta function ax, y; q is defined by ax, y; q = m q m +mn+n x m n y m+n n

5 DETERMINANT IDENTITIES FOR THETA FUNCTIONS 5 where the sums are over all integer values of m and n. The Hirschhorn- Garvan-Borwein cubic theta functions may be defined by ax; q = ax, 1; q, bx; q = aω, x; q, where ω = expπi/3, cx; q = q 1/3 ax, q; q, dx; q = a1, x; q. Explicitly, we have ax; q = q m +mn+n x m n, bx; q = q m +mn+n ω m n x n, cx; q = q m m+ 1 3 n n+ 1 3 x m n, dx; q = q m +mn+n x n. The results for ax; q and cx; q follow immediately from the definitions. For the function bx; q, put m = j and n = j + k to get bx; q = q m +mn+n ω m n x m+n = q j +jk+k ω j k x k j k = q j +jk+k ω j k x k, j and the result for dx; q is obtained similarly. When x = 1, write k aq = a1; q = d1; q, bq = b1; q and cq = c1; q. The cubic analogue of Theorem 1.1 that we shall prove is Theorem 4.1. Let x 1, x, x 3, y 1, y and y 3 be complex variables. Then ax 1, y 1 ; q ax 1, y ; q ax 1, y 3 ; q ax, y 1 ; q ax, y ; q ax, y 3 ; q ax 3, y 1 ; q ax 3, y ; q ax 3, y 3 ; q The proof of Theorem 4.1 relies on the following lemma. Lemma 4.. Let w and z be complex numbers and put x = e iw and y = e iz. Recall that q = e iπτ. Then ax, y; q = ϑ w τϑ z + ϑ 3 w τϑ 3 z. The functions ax; q and bx; q defined here correspond to the functions aq, x and bq, x, respectively, in [8]. The function cx; q defined here differs from the function cq, x in [8] by a factor of q 1/3, and our function dx; q is the same as the function a q, x in [8].

6 6 SHAUN COOPER AND PEE CHOON TOH Proof. This follows immediately from the series rearrangement 6 with c m,n = q m +mn+n x m n y m+n. We are now ready to prove Theorem 4.1. Proof of Theorem 4.1. For 1 j 3, let w j and z j be any complex numbers for which x j = e iw j and y j = e iz j. Applying Lemma 4. and then Lemma 3., we get ax j, y k ; q 1 j,k 3 = ϑ w j τϑ z k + ϑ 3 w j τϑ 3 z k 1 j,k 3 Remark 4.3. The proof of Theorem 4.1 that we have given is analogous to the proof in Section 3. It is also possible to give a proof which is similar to the one in Section. The relevant functional equations are ax, y; q = qx aqx, y; q = qy ax, q 3 y; q. These can be proved by replacing m, n with m+1, n 1 or m+1, n+1, respectively, in the definition of ax, y; q. The next goal is to give a analogue of Theorem 4.1. Theorem 4.4. Let w j and z j be complex variables, where j = 1 or. Let x j = e iw j and y j = e iz j, and recall that q = e iπτ. Then ax1, y 1 ; q ax 1, y ; q ax, y 1 ; q ax, y ; q = w1 w τ w1 + w τ z1 z z1 + z Proof. If we apply Lemma 4., expand the resulting erminant and simplify, we obtain. ax j, y k ; q 1 j,k = ϑ w j τϑ z k + ϑ 3 w j τϑ 3 z k 1 j,k = ϑ 3 w 1 τϑ 3 z 1 ϑ w τϑ z +ϑ 3 w τϑ 3 z ϑ w 1 τϑ z 1 ϑ 3 w 1 τϑ 3 z ϑ w τϑ z 1 ϑ 3 w τϑ 3 z 1 ϑ w 1 τϑ z.

7 DETERMINANT IDENTITIES FOR THETA FUNCTIONS 7 If we factorize the last expression and then apply Lemma 3.1, we get ax1, y 1 ; q ax 1, y ; q ax, y 1 ; q ax, y ; q = ϑ 3 w 1 τϑ w τ ϑ 3 w τϑ w 1 τ ϑ 3 z 1 ϑ z ϑ 3 z ϑ z 1 w1 w = τ w1 + w τ z1 z z1 + z The following identities are ready consequences of Theorems 4.1 and 4.4. Corollary 4.5. Let w 1, w, w 3 and z be complex variables and recall that q = e iπτ. Then ae iw 1, e iz w w 3 ; q τ w + w 3 τ + ae iw, e iz w3 w 1 ; q τ w3 + w 1 τ + ae iw 3, e iz w1 w ; q τ w1 + w τ = 0 and ae iz, e iw 1 ; q w w 3 + ae iz, e iw ; q w3 w 1 + ae iz, e iw 3 ; q w1 w w + w 3 w3 + w 1 w1 + w Proof. Expand the 3 3 erminant in Theorem 4.1 along the first column to get ae iw 1, e iz 1 ae iw, e ; q iz ; q ae iw, e iz 3 ; q ae iw 3, e iz ; q ae iw 3, e iz 3 ; q ae iw, e iz 1 ae iw 1, e ; q iz ; q ae iw 1, e iz 3 ; q ae iw 3, e iz ; q ae iw 3, e iz 3 ; q + ae iw 3, e iz 1 ae iw 1, e ; q iz ; q ae iw 1, e iz 3 ; q ae iw, e iz ; q ae iw, e iz 3 ; q Now apply Theorem 4.4 to each of the erminants, and cancel the common factor of z z 3 z +z 3 that arises. This proves the first result. The second result can be obtained in a similar way, by expanding along the first row..

8 8 SHAUN COOPER AND PEE CHOON TOH 5. Applications In this section we consider some special cases of results in the previous section. In Theorem 5. we obtain an identity that is equivalent to a result of Hirschhorn, Garvan and Borwein [8, 1.1]. By specializing further, and making use of infinite product formulas for bq and cq, we obtain a fundamental result from the theory of elliptic functions in signature 3 in Theorem 5.5. Theorem 5.1. Let F x, y; q = x y1 xy xy 1 q j xy1 q j x 1 y 1 1 q j xy 1 1 q j x 1 y. The Hirschhorn-Garvan-Borwein functions satisfy the identities ax; q cx; q = q ay; q cy; q 1/3 F x, y; q 1 q j 4, bx; q dx; q by; q dy; q = 3qF x, y; q 3 1 q 3j 4. Proof. The infinite product for [10, p. 470] may be given as z τ = iq 1/8 x 1/ x 1/ 1 q j x1 q j x 1 1 q j, where x = e iz. Using this and the definition of F we find that Theorem 4.4 may be written in the form ax1, y 1 ; q ax 1, y ; q ax, y 1 ; q ax, y ; q = qf x 1, x ; qf y 1, y ; q 3 1 q j 1 q 3j. The results in Theorem 5.1 follow from this by taking x 1, x, y 1, y = x, y, 1, q and x 1, x, y 1, y = ω, 1, x, y, respectively, and using the results qf 1, q; q 3 1 q j 1 q 3j = 1 q 3j and F ω, 1; q = 3 1 q j. The first result in the next theorem is equivalent to formula 1.1 in [8]. Theorem 5.. ax; q cx; q aq cq = q 1/3 x x 1 1 q j x 1 q j x 1 1 q j 4

9 DETERMINANT IDENTITIES FOR THETA FUNCTIONS 9 and bx; q dx; q = 3q x x bq aq 1 1 q 3j x 1 q 3j x 1 1 q 3j 4. Proof. Take y = 1 in Theorem 5.1. The next goal is to let x 1 in the results in Theorem 5.. We will need Lemma 5.3. ax; q x = q d x=1 dq aq, bx; q x = x=1 3 q d dq bq, cx; q x = q d x=1 dq cq and dx; q x = x=1 3 q d dq aq. Proof. Clearly 7 m q m +mn+n = n q m +mn+n. If we replace m, n with m, m n we get 8 n q m +mn+n = m + n q m +mn+n. From 7 and 8 it follows that 9 m q m +mn+n = mnq m +mn+n. Using 7 and 9 we have ax; q x = m n q m +mn+n x=1 = 3 m q m +mn+n = m + mn + n q m +mn+n = q d dq aq. This proves the first result. The other results may be proved using the same procedure. The only significant difference is that for the result involving cq, we replace m, n with m, m n 1 to obtain the analogue of 8. We omit the ails. Theorem 5.4. Let D be the differential operator defined by Df = q df dq. Then aq cq = q Daq Dcq 1/3 1 q j 8

10 10 SHAUN COOPER AND PEE CHOON TOH and aq bq Daq Dbq = 9q 1 q 3j 8. Proof. Divide the first result in Theorem 5. by 1 x and let x 1 to obtain x ax; q cx; q lim x 1 1 x = q aq cq 1/3 1 q j 8. Interchange the rows and apply L Hôpital s rule twice to get 1 aq cq ax; q = q 1/3 1 q j 8. cx; q x x Now apply Lemma 5.3 to obtain the first result in Theorem 5.4. The proof of the second result in Theorem 5.4 is similar. We omit the ails, except to say that the negative sign arises from interchanging the two columns in the matrix as well as interchanging the two rows. Theorem 5.4 leads to a simple proof of the following fundamental formula from Ramanujan s theory of elliptic functions in signature 3. x=1 Theorem 5.5. Let z = aq and x = c3 q a 3 q. Then q dx dq = z x1 x. Proof. The proof uses the Borweins cubic identity 10 a 3 q = b 3 q + c 3 q as well as the infinite products 1 q j 3 11 bq = 1 q 3j and cq = 3q 1/3 1 q 3j 3 1 q j. Many proofs of 10 and 11 have been published. For example, the proofs in [6] are simple and self-contained. Let us write a, b and c for aq, bq and cq, respectively. By 11, the first identity in Theorem 5.4 is equivalent to aq dc da cq dq dq = b3 c 3. Multiply by 3c /a 4 and use 10 to get q d c 3 c dq a 3 = a 3 b 3 c a 3 a 3 = a 3 1 a 3 c3 a 3. This is equivalent to the identity in the statement of the theorem.

11 DETERMINANT IDENTITIES FOR THETA FUNCTIONS 11 Other proofs of Theorem 5.5, by a variety of methods, have been given in [1, 4.4], [, 4.4], [3, 4.7], [4, 11.13] and [5, Thm. 4.1]. The proof we have given above is to observe that the identity is essentially a special case of a matrix erminant. References [1] B. C. Berndt, Ramanujan s Notebooks, Part V, Springer-Verlag, New York, [] B. C. Berndt, S. Bhargava and F. G. Garvan, Ramanujan s theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc [3] H. H. Chan, On Ramanujan s cubic transformation for F 1 1, ; 1; z, Math. Proc. 3 3 Cambridge Phil. Soc., , [4] S. Cooper, Cubic elliptic functions, Ramanujan Journal, , [5] S. Cooper, Inversion formulas for elliptic functions, Preprint. [6] F. Garvan, Cubic modular identities of Ramanujan, hypergeometric functions and analogues of the arithmetic-geometric mean iteration, The Rademacher legacy to mathematics University Park, PA, 199, 45 64, Contemp. Math., 166, Amer. Math. Soc., Providence, RI, [7] R. W. Gosper and R. Schroeppel, Somos sequence near-addition formulas and modular theta functions, arxiv:math.nt/ v1 15 Mar 007. [8] M. Hirschhorn, F. Garvan and J. Borwein, Cubic analogues of the Jacobian theta function ϑz, q, Canadian Journal of Mathematics , [9] P. C. Toh, Generalized m-th order Jacobi theta functions and the Macdonald identities, Int. J. Number Theory, to appear. [10] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 4th edition, 197. Institute of Information and Mathematical Sciences, Massey University- Albany, Private Bag 10904, North Shore Mail Centre, Auckland, New Zealand s.cooper@massey.ac.nz Department of Mathematics, National University of Singapore, Science Drive, Singapore mattpc@nus.edu.sg