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. [5–7], 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.