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Research Article | Open Access

Volume 2021 |Article ID 6614572 | https://doi.org/10.1155/2021/6614572

B. R. Srivatsa Kumar, Shruthi, D. Anu Radha, "Relation between Borweins’ Cubic Theta Functions and Ramanujan’s Eisenstein Series", Journal of Applied Mathematics, vol. 2021, Article ID 6614572, 6 pages, 2021. https://doi.org/10.1155/2021/6614572

# Relation between Borweins’ Cubic Theta Functions and Ramanujan’s Eisenstein Series

Academic Editor: Bruno Carpentieri
Revised24 Feb 2021
Accepted15 Apr 2021
Published10 May 2021

#### Abstract

Two-dimensional theta functions were found by the Borwein brothers to work on Gauss and Legendre’s arithmetic-geometric mean iteration. In this paper, some new Eisenstein series identities are obtained by using (, )-parametrization in terms of Borweins’ theta functions.

#### 1. Introduction

The series , , and by Eisenstein are defined below, primarily mentioned by Ramanujan in his second notebook [1].

Borwein brothers [2, 3] established the consecutive three two-dimensional theta functions on cubic; as parallel to Jacobi’s theta function identity and the arithmetic-geometric mean iteration of Guass and Legendre, they are stated as follows for :

One can easily verify that and . By using Euler binomial theorem [2, 3], Borwein brothers found an expression for and in terms of infinite products, namely,

where is defined as

Proofs of (5)–(7) can be found in [2, 3]. Berndt et al. [4] demonstrated the following identities by using Ramanujan’s elliptic functions in the theory of signature 3, namely,

In the current work, using -parameters introduced by Alaca et al. [57], the new Eisenstein series identities are obtained which connects , , and ; these are the examples of sum-to-product identities. Using the classical theory of elliptic functions and modular equations of degree 3, Chan [8] proved (9) and (12). Moreover, Liu [9, 10] provided alternate proofs of (9)–(12) by using the simple elliptic functions. The identities of Ramanujan’s Eisenstein series of weight 2 found in [10, 11] have been proved by Bhuvan [12]. Xia and Yao [11] also obtained several Eisenstein series identities that includes Borweins’ theta functions containing , , , , , and by using (, )-parametrization. Further, in 2020, Shruthi and Srivatsa Kumar [13] obtained new Eisenstein series involving Borweins’ cubic theta functions. This paper is classified as follows. Section 2 is devoted for preliminary results which are required to prove our main results in Section 3.

#### 2. Preliminaries

Alaca and Williams [14] ascertained the parametric representations for , , , and for , , and in terms of parameters and , namely, where

Since , one can verify and . The sequential parametric representations in terms of and given by Alaca et al. [6, 7] and Alaca and Williams [14] for for and for are listed below.

The following parametric representations in terms of the parameters and are given by Alaca et al. [5].

Now, we explain the concept of -parametrization as explained in [11]. Using the identities listed above, we derive the representations for in terms of and . The said representation in terms of is a polynomial in and , and we want to show that where every is a rational number and is the product involving or . Substituting the representations for in terms of and in (17), then both sides of (17) are functions of and . Further on equating the coefficients of on both sides of (17), a set of linear equations in are deduced. If these equations have a solution, then by using Maple, we solve the equations and evaluate the values of , which gives the Eisenstein series identities involving Borweins’ cubic theta functions.

#### 3. Main Results

Theorem 1. One has

Proof. If suppose to be the case

On equating the various powers of on both sides of the above, it is observed that

On solving the above five equations, one can see that

Substituting (21) in (19) and using (1), the desired result is obtained. In the same way, applying this technique, the following identities are also derived:

Theorem 2. One has

Proof. If suppose to be the case

On equating the various powers of on both sides of the above, it is observed that

On solving the above, it is easy to see that

Substituting (26) in (24) and using (2), the desired result is obtained. In the same way, applying this technique, the following identities are deduced:

#### 4. Conclusion and Application

Employing the technique developed by Xia and Yao [11], one can derive several relations between Eisenstein Series and Borwiens’ cubic theta functions using (, )-parametrization. As an application, one can evaluate convolution sums involving divisor sum.

#### Data Availability

The data used to support the findings of the study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### References

1. S. Ramanujan, Notebooks (2 Volumes), Tata Institute of Fundamental Research, Bombay, 1957.
2. J. M. Borwein and P. B. Browein, “A cubic counterpart of Jacobi’s identity and the AGM,” Transactions of the American Mathematical Society, vol. 323, no. 2, pp. 691–701, 1991. View at: Google Scholar
3. J. M. Borwein, P. B. Borwein, and F. G. Garvan, “Some cubic modular identities of Ramanujan,” Transactions of the American Mathematical Society, vol. 343, no. 1, pp. 35–47, 1994. View at: Publisher Site | Google Scholar
4. B. C. Berndt, S. Bhargava, and F. G. Garvan, “Ramanujan’s theories of elliptic functions to alternative bases,” Transactions of the American Mathematical Society, vol. 347, no. 11, pp. 4163–4244, 1995. View at: Google Scholar
5. A. Alaca, S. Alaca, and K. S. Williams, “On the two-dimensional theta functions of the Borweins,” Acta Arithmetica, vol. 124, no. 2, pp. 177–195, 2006. View at: Publisher Site | Google Scholar
6. A. Alaca, S. Alaca, and K. S. Williams, “Evaluation of the convolution sums and ,” Advances in Theoretical and Applied Mathematics, vol. 1, no. 1, pp. 27–48, 2006. View at: Google Scholar
7. A. Alaca, S. Alaca, and K. S. Williams, “Evaluation of the convolution sums and ,” International Mathematical Forum. Journal for Theory and Applications, vol. 2, no. 14, pp. 45–68, 2007. View at: Google Scholar
8. H. H. Chan, “On Ramanujan’s cubic transformation formula for ,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 124, no. 2, pp. 193–204, 1998. View at: Google Scholar
9. Z. -G. Liu, “The Borweins’ cubic theta function identity and some cubic modular identities of Ramanujan,” The Ramanujan Journal, vol. 4, no. 1, pp. 3–50, 2000. View at: Publisher Site | Google Scholar
10. Z. -G. Liu, “Some Eisenstein series identities,” Journal of Number Theory, vol. 85, no. 2, pp. 231–252, 2000. View at: Publisher Site | Google Scholar
11. E. X. W. Xia and O. X. M. Yao, “Eisenstein series identities involving the Borweins’ cubic theta functions,” Journal of Applied Mathematics, vol. 2012, Article ID 181264, 14 pages, 2012. View at: Publisher Site | Google Scholar
12. E. Bhuvan, “On some Eisenstein series identities associated with Borweins’ cubic theta functions,” Indian Journal of Pure and Applied Mathematics, vol. 49, no. 4, pp. 698–703, 2018. View at: Google Scholar
13. Shruthi and B. R. S. Kumar, “Some new Eisenstein series containing the Borweins’ cubic theta functions and convolution sum ,” Afrika Mathematica, vol. 31, no. 5-6, pp. 971–982, 2020. View at: Google Scholar
14. S. Alaca and K. S. Williams, “Evaluation of the convolution sums and ,” Journal of Number Theory, vol. 124, no. 2, pp. 491–510, 2007. View at: Google Scholar

Copyright © 2021 B. R. Srivatsa Kumar et al. 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.

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