View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ScholarBank@NUS Elliptic Functions, Theta Functions and Identities Ng Say Tiong Department of Mathematics National University of Singapore 2004 Elliptic Functions, Theta Functions and Identities Ng Say Tiong A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE. Supervisor: A/P. Chan Heng Huat Department of Mathematics National University of Singapore 2004 Acknowledgements I would like to thank my family members and friends for giving me support in the comple- tion of this thesis. Also I would like to thank all the lecturers that have given me guidance in my courses in NUS. Last but not least, I would like to thank my supervisor, Associate Professor Chan Heng Huat for his patience and guidance throughout the duration of this thesis. Without his support, the completion of this thesis in such a short amount of time would not be possible. ii Statement of Author’s Contributions 1. The proof to Lemma 3.3.2, which is given as an exercise in [16], is given in detail by the author. 2. Together with Associate Professor Chan Heng Huat, and Professor Liu Zhi Guo, the author provides original proofs for the quintuple product identity in Theorem 4.1.1, the septuple product identity in Theorem 4.2.1 and the Winquist identity in Theorem 4.3.1, based on the theories of elliptic functions and the theta functions. 3. The author provides original proof for the theta function identity in Theorem 5.1.1 based on the theories of elliptic functions and the theta functions. iii Contents 1 Introduction 1 2 Elliptic Functions 3 2.1 Definition of elliptic functions . 3 2.2 Properties of elliptic functions . 4 3 The Theta Functions 8 3.1 Definition of the theta functions . 8 3.2 The zeros of the theta functions . 12 3.3 Infinite products expressions for the theta functions . 14 4 The Quintuple, Septuple Product Identities and the Winquist Identity 23 4.1 The quintuple product identity . 24 4.2 The septuple product identity . 30 4.3 The Winquist identity . 35 5 A Theta Function Identity and Its Application 43 5.1 The theta-function identity . 43 5.2 A conjecture on the relation between Dixon’s doubly-periodic functions and the theta functions . 47 iv Chapter 1 Introduction The first function related to theta functions is perhaps the function ∞ Y (1 − xnz)−1, n=1 defined by L. Euler in his studies of the properties of partitions [6]. However, it was C.G.J. Jacobi who defined the more general theta functions in [11], and he developed the theory of theta functions from the theory of elliptic functions. His results were later reprinted in [12]. The theta functions are great tools for solving q-series identities. In latter chapters we will use the theta functions to systematically proved the quintuple product identity [3, p.g 80], the septuple product identity [7] and the Winquist identity [17]. The quintuple product identity has a long history, and is best summarized in [3, p.g 83]. The septuple product identity was first discovered by M.D. Hirschhorn [9] in 1983, and later rediscovered by H.M.Farkas and I.Kra in 1999 [7]. In 1999, D.Foata and G.-N. Han showed a connection between the quintuple product identity and the septuple product identity [8], using the Jacobi triple product identity [2, p.g 91] and the quintuple product identity to prove the septuple product identity. The Winquist identity was discovered by L.Winquist in [17], and was used to derive a double series identity for the product series ∞ Y (1 − xn)10, n=1 1 and a proof for the partition function congruence p (11n + 6) ≡ 0 (mod 11), where p (n) denotes the number of unrestricted partitions of n [1, p.g 1]. Later in 1997, S.-Y. Kang gave a new proof of the Winquist identity based on elementary methods [13]. In [5], A.C.Dixon defined two doubly-periodic functions smu and cmu using the curve x3 + y3 − 3αxy = 1, where α is a parameter of the cubic curve. Dixon showed that these two functions shared many similar properties to the Jacobian elliptic functions snu and cnu [16, p.g 491]. Using a theta function identity proved by Z.-G. Liu in [14], we conjecture possible algebraic expressions for smu and cmu based on the theta functions. 2 Chapter 2 Elliptic Functions We will begin our discussion with elliptic functions. The properties of elliptic functions that are discussed in this chapter will play an important part in our proofs of theorems in latter chapters. 2.1 Definition of elliptic functions Definition 2.1.1 Let ω1, ω2 be any two numbers (real or complex) whose ratio is complex. A function which satisfies the equations f(z + 2ω1) = f(z), f(z + 2ω2) = f(z), for all values of z for which f(z) exists, is called a doubly-periodic function of z, with periods 2ω1, 2ω2. A doubly-periodic function which has no singularities other than poles in the finite part of the complex plane is called an elliptic function. To study elliptic functions, we need the geometrical representation of the complex plane afforded by the Argand diagram [16, p.g 9]. Suppose that in the complex plane, we mark out the points 0, 2ω1, 2ω2, 2ω1 + 2ω2, and generally, all the points whose complex coordi- nates are of the form 2mω1 + 2nω2, where m and n are integers. By joining in succession consecutive points of the set 0, 2ω1, 2ω1 + 2ω2, 2ω2, 0, we will obtain a parallelogram. If 3 there is no point ω inside or on the boundary of this parallelogram (the vertices excepted) such that f(z + ω) = f(z) for all values of z, this parallelogram is called a fundamental period-parallelogram [16, p.g 430] for an elliptic function with periods 2ω1, 2ω2. It is clear that the complex plane can be covered with a network of parallelograms equal to the fundamental period-parallelogram and similarly situated, each of the points 2ω1 + 2ω2 being a vertex of four parallelograms. These parallelograms are called period- parallelograms or meshes [16, p.g 430], and for all values of z, the points z, z + 2ω1, ... z + 2mω1 + 2nω2, ... manifestly occupy corresponding positions in the meshes. Any of such pair of points are said to be congruent to one another [16, p.g 430]. Due to the fundamental periodic properties of elliptic functions, it follows that an elliptic function takes the same value at every one of a set of congruent points, and so its values in any mesh are just a mere repetition of its values in any other mesh. However, for integration purposes, it is not convenient to deal with the actual meshes if they have singularities of the integrand on their boundaries. Because of the periodic prop- erties of elliptic functions, nothing is lost by taking a contour, which is not an actual mesh, but a parallelogram obtained by translating a mesh without rotation in such a way that none of the poles of the integrands considered are on the boundaries of the parallelogram, and all the poles considered are inside the parallelogram. Such a parallelogram is called a cell [16, p.g 430]. Again the values assumed by an elliptic function in a cell are just a mere repetition of its value in any mesh. We will now prove some important facts about elliptic functions. 2.2 Properties of elliptic functions Theorem 2.2.1 i) The number of poles of an elliptic function in any cell is finite. ii) The number of zeros of an elliptic function in any cell is finite. 4 iii) The sum of the residues of an elliptic function, f(z), at its poles in any cell is zero. iv) An elliptic function, f(z), with no pole in a cell is merely a constant. Proof i) Suppose there exists an elliptic function such that its number of poles in a cell is infinite, then the poles would have a limit point in the cell, which is an essential singularity of the function. However this contradicts the definition of an elliptic function. ii) Suppose there exists an elliptic function f(z) such that its number of zeros in a cell 1 1 is infinite, we consider the reciprocal function f(z) . Since f(z) is elliptic, so is f(z) , and it has an infinite number of poles in a cell by the property of f(z). This is a contradiction, by part(i). iii) Let C be a contour formed by edges of a cell, whose corners are t, t+2ω1, t+2ω1 +2ω2 and t + 2ω2, t being an arbitrary complex number. The sum of the residues of f(z) at its poles inside C is 1 Z f(z) dz 2πi C 1 Z t+2ω1 Z t+2ω1+2ω2 Z t+2ω2 Z t = f(z) dz + f(z) dz + f(z) dz + f(z) dz 2πi t t+2ω1 t+2ω1+2ω2 t+2ω2 1 Z t+2ω1 Z t+2ω2 Z t Z t = f(z) dz + f(z + 2ω1) dz + f(z + 2ω2) dz + f(z) dz 2πi t t t+2ω1 t+2ω2 1 Z t+2ω1 1 Z t+2ω2 = {f(z) − f(z + 2ω2)} dz − {f(z) − f(z + 2ω1)} dz 2πi t 2πi t = 0, and the theorem is established. iv) Let f(z) be an elliptic function such that it has no pole in a cell.