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 ( p , k )-parametrization in terms of Borweins’ theta functions.
Highlights
The series LðqÞ, MðqÞ, and NðqÞ by Eisenstein are defined below, primarily mentioned by Ramanujan in his second notebook [1]
In the current work, using ðp, kÞ-parameters introduced by Alaca et al [5,6,7], the new Eisenstein series identities are obtained which connects aðqÞ, bðqÞ, and cðqÞ; these are the examples of sum-to-product identities
Employing the technique developed by Xia and Yao [11], one can derive several relations between Eisenstein Series and Borwiens’ cubic theta functions using (p, k)-parametrization
Summary
The series LðqÞ, MðqÞ, and NðqÞ by Eisenstein are defined below, primarily mentioned by Ramanujan in his second notebook [1]. Berndt et al [4] demonstrated the following identities by using Ramanujan’s elliptic functions in the theory of signature 3, namely, MðqÞ ≔ aðqÞÀa3ðqÞ + 8c3ðqÞÁ, ð9Þ. In the current work, using ðp, kÞ-parameters introduced by Alaca et al [5,6,7], the new Eisenstein series identities are obtained which connects aðqÞ, bðqÞ, and cðqÞ; these are the examples of sum-to-product identities. The identities of Ramanujan’s Eisenstein series Pn of weight 2 found in [10, 11] have been proved by Bhuvan [12]. Xia and Yao [11] obtained several Eisenstein series identities that includes Borweins’ theta functions containing bðqÞ, bðq2Þ, bðq4Þ, cðqÞ, cðq2Þ, and cðq4Þ by using (p, k)-parametrization. Kumar [13] obtained new Eisenstein series involving Borweins’ cubic theta functions.
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