On Some Eisenstein Series Identities Associated with Borwein’s Cubic Theta Functions

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

“…Moreover, Liu [9,10] provided alternate proofs of ( 9)-( 12) by using the simple elliptic functions. The identities of Ramanujan's Eisenstein series P n 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 bðqÞ, bðq 2 Þ, bðq 4 Þ, cðqÞ, cðq 2 Þ, and cðq 4 Þ by using (p, k)-parametrization.…”

“…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. 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.…”