Journal of Applied Mathematics

Volume 2012 (2012), Article ID 181264, 14 pages

doi:10.1155/2012/181264

Research Article

Eisenstein Series Identities Involving the Borweins' Cubic Theta Functions

Ernest X. W. Xia and Olivia X. M. Yao

Department of Mathematics, Jiangsu University, Jiangsu, Zhenjiang 212013, China

Received 28 March 2012; Revised 15 May 2012; Accepted 23 May 2012

Academic Editor: Ferenc Hartung

Copyright © 2012 Ernest X. W. Xia and Ferenc Hartung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Based on the theories of Ramanujan's elliptic functions and the (p, k)-parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins' cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the formulas given by Alaca et al.

1. Introduction

Let and denote the sets of positive integers and complex numbers, respectively. Throughout the paper, we always assume that and .

In their paper, J. M. Borwein and P. B. Borwein [1] introduced the following three functions:

where . These functions are now called the Borweins' cubic theta functions. The Borwein brothers [1] derived representations for and in terms of infinite products, namely,

where

The function has the following representation derived by the Borwein brothers [1] and Berndt [2]:

Elementary proofs of (1.2), (1.3), and (1.5) can be found in [3]. The Borwein brothers [1] also proved the following well-known relation for , and , namely:

In his second notebook [4], Ramanujan gave the definitions of the Eisenstein series , , and , namely,

Utilizing Ramanujan's elliptic functions in the theory of signature 3, Berndt et al. [5] proved the following representations for , , , and , namely:

Chan [6] proved (1.10) and (1.11) by employing the classical theory of elliptic functions and modular equations of degree 3. Based on the fundamental theory of elliptic functions, Liu [7, 8] provided different proofs of (1.10), (1.11), (1.12), and (1.13). He also discovered some striking Eisenstein series identities.

In this paper, using the parameters and introduced by Alaca et al. [9–11] (see (2.1)), we deduce some Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. These identities are examples of sum-to-product identities. Some of these identities were proved by Liu [7, 8] based on the fundamental theory of elliptic function and some of them may be new.

This paper is organized as follows. In Section 2, we gave the -parametrization of , and for , , which are due to Alaca et al. [9–11]. Section 3 is devoted to deriving some Eisenstein series identities involving the Borweins' cubic theta functions. As corollaries of our results, in Section 4, we derive new representations for the convolution sums and and contrast them with the known evaluations due to Alaca and Williams [11] and Alaca et al. [10].

2. Eisenstein Series , and Parameters ,

In this section, we gave the parametric representations for , , and for , and in terms of the parameters and first defined by Alaca and Williams [11], namely,

Alaca and Williams [11] derived the representations of , , and in terms of and . Equations (3.69), (3.70), (3.71), (3.72), (3.84), (3.87), and (3.89) in [11] are

respectively. Alaca et al. [12, 13] also deduced the representations of , and in terms of and . Equations (3.12), and (3.19) in [12] and (3.13) in [13] are

respectively. Alaca et al. [9] also derived the following parametric representations for and in terms of and . From Theorems 1, 2, and 4 in [9], we have

We now describe our approach. Let be a function, where or . Utilizing the representations for , , , , and in terms of and , we derive the representations for in terms of and . We select suitable such that is a polynomial in and . We want to show that

where each is a rational number and is a product involving and . Substituting the representation for in terms of and into (2.5), then both sides of (2.5) are functions in and . Equating the coefficients of on both sides of (2.5), we obtain some linear equations in . If these equations have a solution, then we can use computer to solve the equations and determine the values of the . We then obtain some Eisenstein series identities involving the Borweins' cubic theta functions.

3. Some Eisenstein Series Identities

In this section, we derive some Eisenstein series identities. In fact, utilizing our method, we can obtain many identities, here we just list some of them. Our main theorem can be stated as follows.

Theorem 3.1. One has

Remark 3.2. The identities (1.16) and (1.17) in [8] contain typos, they should be (3.1) and (3.5), respectively.

Proof. We first prove the formula (3.1) by our method. We assume that

Equating the coefficients of , , , , and on both sides of (3.24), we obtain the following five equations:

Solving the above five equations, we obtain

Substituting the above values into (3.24), from (1.2) and (1.7), we obtain (3.1).

Similarly, utilizing the same method, we also derive the following formulas:

From the above identities and (1.2), (1.3), (1.7) and (1.8), we may derive other formulas.

4. Convolution Sums and

Let . The divisor function is defined by

where runs through the positive divisors of . If is not a positive integer, set . As usual, we write for . For all , the convolution has been evaluated explicitly for , and , see [10–19]. In this section, we also derive the representations for the convolution sums and from the identities in Theorem 3.1. Our representations are different from those derived in [11, 12]. In fact, we can derive many formulas for the convolution sums and , and here we just list two of them.

Theorem 4.1. Let be a positive integer. One has

where

Remark. Alaca and Williams [11] derived the representation for

where

Alaca et al. [12] also derived the representation for

where

Proof. From (3.9), we have

For , equating the coefficients of on both sides of (4.10), we obtain

where is defined by (4.4). From (4.11), utilizing the convolution sum

we can derive the formula (4.2).

From (3.23), we have

For , equating the coefficients of on both sides of (4.13), we obtain

where is defined by (4.5). From (4.2), (4.12), and (4.14), we derive the formula (4.3).

The advantage of the formulas of Alaca et al. is that the values of and are often very small. Numerical evidence suggests that and . The advantage of our formulas is that the signs of and appear to have periodicity. Numerical evidence suggests that for , when and when, and . Therefore, we conjecture that, for , we have

Acknowledgments

The authors would like to thank the anonymous referee very much for valuable suggestions, corrections, and comments which resulted in a great improvement of the original paper. This work was supported by the National Natural Science Foundation of China and the Jiangsu University Foundation Grants 11JDG035 and 11JDG036.

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