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Generalized m-th Order Jacobi Theta Functions Pee Choon TOHNational University of Singapore,Department of Mathematics,2 Science Drive 2, Singapore 117543mattpc@nus.edu.sg We describe a m-th order generalization of Jacobi’s theta functions and use these functions to construct classes of theta function identitiesin multiple variables. These identities are equivalent to the Macdonaldidentities for the seven infinite families of irreducible affine root systems.
They are also equivalent to some elliptic determinant evaluations provenrecently by H. Rosengren and M. Schlosser.
Keywords: determinants, theta functions, elliptic functions, Macdonald identi-ties2000 Mathematics Subject Classification: 11F20, 14K25, 15A15, 33D52, 33E05 (1 − qn)(1 + aqn)(1 + a−1qn−1)(1 − a2q2n−1)(1 − a−2q2n−1), is an important identity with a very rich history. Beginning from G.N. Watsonin 1929 [22] till the present, many different proofs of this identity have beenpublished. Surveys of the various proofs of the quintuple product identity canbe found in [1, p.83] and [6].
The Hirschhorn-Farkas-Kra septagonal numbers identity, (−1)nq(5n2−n)/2(a5n−2 + a−5n−1) (1 − qn)2(1 − aqn)(1 − a−1qn−1)(1 − a2qn)(1 − a−2qn−1).
is an analogue of the quintuple product. It was first discovered by M.D. Hirschhorn[11] and later rediscovered independently by H.M. Farkas and I. Kra [8]. In theirimportant work, Farkas and Kra approached the problem from a function the-oretic perspective and developed an entire theory of function spaces of N -thorder θ-function (with rational characteristics) [8], [9, p.129-145, p.268-273].
More recently, H.H. Chan, Z.-G. Liu and S.T. Ng [3] used the theory of elliptic functions to give elegant proofs of (1.1) and (1.2). In this present article,we generalize the idea used in [3] to derive classes of identities involving a m-thorder generalization of the classical Jacobi theta functions. These are essentiallyidentical to the functions used by Farkas and Kra, but our approach has adifferent flavour. Moreover, these identities involve multiple variables and arestronger than those derived by Farkas and Kra. As an illustration, (1.1) and(1.2) are special cases of the following, e(4n +2 +2j−1)izk + e−(4n +2 +2j−1)izk The infinite product on the right hand side of the above identity turns out to be equivalent to the infinite product that appears in the Macdonald identityfor the affine root system BCn [17].
This development lead to a literature search which revealed that our iden- tities have been anticipated by H. Rosengren and M. Schlosser [19]. Althoughthe results and proofs are very similar, the approaches to constructing theseidentities are different. Rosengren and Schlosser approached the problem froma determinant evaluation point of view. Their aim was to study the ellipticanalogues of the Weyl denominator formulas and they used the affine root sys-tems as their starting point, defining a special theta function for each affine rootsystem.
Our approach, on the other hand, was to study identities constructed from m-th order theta functions. This construction does not make use of the affine root systems in any way. For each m, there are essentially four different thetafunctions, each satisfying a different transformation formula. (See (2.1) for thefour transformations satisfied by the classical Jacobi theta functions. This is thecase where m = 1.) By grouping the m-th order theta functions into odd andeven functions, and considering the parity of m, we end up with sixteen classesof identities. Each of these identities has an infinite product corresponding toone of the seven infinite families of irreducible affine root systems described byMacdonald [17]. In a sense, our work can be viewed as an elementary approachto the Macdonald identities that is independent of root systems. For anotherelementary approach to the Macdonald identities for the infinite families, see[21].
In Section 2, we will define the classical Jacobi theta functions and list some of their properties which are crucial in this work. Then we will describe a m-th order generalization of the theta functions. In Section 3, we will use thism-th order theta function to construct classes of identities equivalent to theMacdonald identities. Examples of well known identities which occur as specialcases of the main theorems, as well as some new formulas for powers of (q)∞will be given in Section 4.
Let q = eπiτ where Im(τ ) > 0. The classical Jacobi theta functions are definedby Jacobi’s theta functions are functions of one complex variable z and a parameterτ . Throughout this article, we do not explore the “modular” properties of thetheta functions and all identities can be considered as formal q-series identities.
We list some important properties, the proofs of which can all be found in [23,Ch. 21].
The Jacobi theta functions satisfies the following transformation: θ1(z + πτ |τ ) = −q−1e−2izθ1(z|τ ), θ2(z + πτ |τ ) = q−1e−2izθ2(z|τ ), θ3(z + πτ |τ ) = q−1e−2izθ3(z|τ ), θ4(z + πτ |τ ) = −q−1e−2izθ4(z|τ ).
Hence they are quasi-elliptic with (quasi) periods π and πτ and it suffices tostudy their values in the fundamental parallelogram, Π = {aπ + bπτ | 0 ≤ a < 1, 0 ≤ b < 1}.
Each θi vanishes at exactly one point in Π and we have [23, p.465] Furthermore, each θi can be expressed as infinite products. We adopt the θ1(z|τ ) = 2q 4 sin z(q2; q2)∞(q2e2iz; q2)∞(q2e−2iz; q2)∞, θ2(z|τ ) = 2q 4 cos z(q2; q2)∞(−q2e2iz; q2)∞(−q2e−2iz; q2)∞, θ3(z|τ ) = (q2; q2)∞(−qe2iz; q2)∞(−qe−2iz; q2)∞, θ4(z|τ ) = (q2; q2)∞(qe2iz; q2)∞(qe−2iz; q2)∞.
We now construct a m-th order generalization of Jacobi’s theta functions.
Definition 2.1 Let m, j ∈ Z and l = 0 or 1, define It is easy to check that the following identities hold, Definition 2.2 Let m be fixed. Define V l (z), satisfying the following transformation It is clear that for each m, there are only four distinct spaces, determined by the parity of l and j. We can check that the functions T l Proof. Follow the treatment in [9, p.129-135].
has exactly m zeroes in Π, the fundamental Proof. Follow the method in [23, p.465].
The following corollary of the transformation formula (2.4) is useful for lo- We have seen that for each m, we have four vector spaces V l into subspaces of odd and even functions with {Ol (z)} as the respective basis elements. The dimensions of the odd and even subspaces now also depend on the parity of m, and can be tabulated with thehelp of (2.3).
Dimensions of odd and even subspaces of V l For example, consider the space V 0 . Since both m and j are even, this space can be decomposed into an odd subspace of dimension 2, with basis{O0 (z), O0 (z)} and an even subspace of dimension 4, with basis {E0 (z), For each of these sixteen subspaces, we now construct a multi-variable theta function identity with the basis elements. The most natural way to combinethese elements symmetrically into an equation is via the determinant function.
Once a choice of determinant function is written down, the right hand sideof the identity is determined by the location of the zeroes that appear in thefundamental parallelogram, Π. To simplify notation, we let n denote the numberof basis elements.
Theorem 3.1 Let zj be arbitrary complex variables, then the following identi-ties hold: θ1(z |τ )θ2(z |τ )θ3(z |τ )θ4(z |τ ).
We provide a few remarks before giving the proof of one case in detail. The other cases will follow from similar arguments. First of all, since all the fourθi(z|τ ) are equivalent up to a half period transform [23, p. 464], some of theabove identities are equivalent. For example (3.5) can be obtained from (3.1)by replacing zj with zj + πτ /2 for all j.
Secondly, the infinite product on the right hand side of each identity corre- sponds to an affine root system. In fact, these identities are equivalent to theMacdonald identities [17]. Specifically, we have: Identities appearing in the same row are equivalent and the identity in the firstcolumn is the one that is most easily recognized from the affine root system.
The identities for An will be discussed in Theorem 3.2.
The third remark is that all of the above theorems have been independently discovered by Rosengren and Schlosser [19, Prop 3.4]. They have also shown intheir paper, the equivalence of these determinant identities to the Macdonaldidentities.
Let F (z1, . . . , zn) denote the determinant expression in (3.15). We first assumethat {zj|j = 1} are fixed, distinct complex numbers in the fundamental par-allelogram Π, that are different from 0, π , πτ and π+πτ . Then, F (z can be considered as a function of z1, i.e F (z1, . . . , zn) = F (z1). Since it is alinear combination of O0 1) for j = 1 to n, it satisfies the transformation Moreover as a function of z1, F (z1) is odd. Corollary 2.5 allows us to con- clude that F (z1) = 0 at each of the four values 0, π , πτ and π+πτ . It is also evident that F (±zj) = 0, 2 ≤ j ≤ n. This accounts for all the 2n + 2 zeroes ofF (z1) in Π. (The points −zj are not in Π but their equivalent points −zj +π+πτare.) θ1(z |τ )θ2(z |τ )θ3(z |τ )θ4(z |τ ).
From the formulas in (2.1), we can see that as a function of z1, i.e. P (z1, . . . , zn) =P (z1) satisfies the same transformation formula as F (z1), and has the samezeroes. Thus the quotient F (z1)/P (z1) is elliptic and entire. Appealing to Li-ouville’s theorem, F (z1)/P (z1) is a “constant” expression c(z2, . . . , zn, τ ) thatis independent of z1.
We can repeat the same argument for each of the zj and conclude that the quotient F (z1, . . . , zn)/P (z1, . . . , zn) equals a constant c(τ ) that is dependentonly on τ . The principle of analytic continuation then allows us to concludethat the identity holds for all zj.
We now calculate c(τ ). Let zk = πk/(2n + 2) and let w denote the primitive (q4n+4, −q2n+2+2j , −q2n+2−2j ; q4n+4) where we have used the Jacobi triple product (2.2) to convert the series intoinfinite products.
To evaluate the determinant expression explicitly, we use [12, identity (2.3)] Next, we evaluate the right hand side of (3.15) for the same values of zk.
We use the following identity [23, p. 488], θ1(z|τ )θ2(z|τ )θ3(z|τ )θ4(z|τ ) = q 4 (q2)3∞θ1(2z|τ ), and the infinite product formulas (2.2) to obtain θ1(z |τ )θ2(z |τ )θ3(z |τ )θ4(z |τ ) (q2wj−k, q2wk−j , q2wj+k, q2w−j−k; q2)∞.
We first observe that the products involving only powers of w is equal to the determinant evaluation in (3.18) up to a factor of (−1) 2 four infinite products, we set k as j and j as k for the second product, k as k + 1 for the third product, j as n − j and k as n − k + 1 for the last product to get (q2wj−k, q2wk−j , q2wj+k, q2w−j−k; q2)∞.
Substituting (3.21) into (3.20) and combining with (3.19), we have a simpli- fied expression for the right hand side of identity (3.15). Comparing with theexpression (3.17), we can conclude that the constant Theorem 3.1 yielded six classes of identities corresponding to all the infi- nite families of affine root systems except for An. Since we have exhausted allthe possibilities for El (z), it is natural to reconsider the basic Theorem 3.2 Let zj be arbitrary complex variables, then the following identityholds: The infinite product on the right hand side of (3.22) has an extra theta factor when compared to the Macdonald identity for An−1, nevertheless thetwo identities are equivalent. Other proofs for the An−1 Macdonald identitiescan be found in [5], [18] and [21].
Theorems 3.1 and 3.2 fall into the class of elliptic determinant evaluations.
A survey of recent progress in this area can be found in [13].
We will conclude by giving several examples of well known identities that canbe constructed.
This is an alternate form of the quintuple product identity and has appeared in[3], [16] and [20]. To obtain (1.1), replace e2iz by a, q2 by q, and use the infiniteproduct expansions of θi(z|τ ) listed in (2.2).
Example 4.2 (Septagonal numbers identity) θ3(x|τ )θ4(x|τ )θ2(y|τ )θ3(y|τ )θ4(y|τ ).
Setting y = 0 gives the Hirschhorn-Farkas-Kra septagonal numbers identity(1.2). Farkas and Kra discussed a similar m-th ordered generalization in somedetail for even functions, odd m and odd j [9, p.268-271]. In their method,they needed some properties of a specific automorphic form on Γ(m). F.G.
Garvan [10] has also studied the same problem and using a different method,managed to construct a different class of explicit identities using only the Jacobitriple product identity. However, the identities constructed by both Garvan andFarkas and Kra are identities involving only a single complex variable z.
θ3(x|τ )θ4(x|τ )θ3(y|τ )θ4(y|τ ).
Setting y = 0 gives an identity studied by Ewell [7]. Two different proofs of this identity were given recently in [4].
1(x + y|τ )θ1(x − y|τ )θ1(x|τ )θ1(y|τ ).
This is equivalent to Winquist’s identity [24]. Recently, H.H. Chan, S. Cooperand the author has used a variant of this identity to generate interesting rep-resentations for η10(τ )Gk, the product of the tenth power of the Dedekind’seta-function and some Eisenstein series. (See [2] and the bibliography there forreferences to Winquist’s identity.) Example 4.5 (Representations of (q)2n2−n One main application of the Macdonald identities was to construct representa-tion of powers of η(τ ) or equivalently, powers of (q)∞. We illustrate this usingidentity (3.11), the Dn case. For each j, we apply ( ∂ )2(j−1) to the identity and set zj = 0. This will turn each θ1(zj ± zk|τ ) into θ (0|τ ) which is equal to 2q 4 (q2)3∞. Writing q2 as q, we get the following explicit formula for (q)2n2−n (−1)k−1 ((2n − 2) + j − 1)2k−2 q(n−1) 2+(j−1) A different formula can be obtained from identity (3.2), the BCn case.
For each j, apply ( ∂ )2j−1 to identity (3.5) (the B (−1) +k−1 ((4n − 2) + 2j − 1)2k−1 q((2n−1) 2+(2j−1) )/2 A different formula can be obtained from identity (3.15), the Cn case.
Due to the extra theta factor in identity (3.22) (our An−1 case), we obtain a (−1)n ((2n + 2j − n)i)k−1 q(n 2+(2j−n) )/2, (−1)n ((2n + 2j − n)i)n q(n 2+(2j−n) )/2.
Besides Macdonald’s original paper [17], other representations for powers of (q)∞ were also given by V.E. Leininger and S.C. Milne in [14] and [15]. Inparticular, [15] also contains a representation of (q)n2+2 The author would like to thank Professor H.H. Chan for his advice and invalu-able insights and Professor S. Cooper for helpful suggestions and for bringingthe work of Rosengren and Schlosser to the author’s attention. The authorwould also like to thank Professor S.-S Huang for providing a copy of his jointwork with S.-D. Chen and W.-Y. Chen.
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