In this paper, we give an elementary proof of Eisenstein series of level 3 and 6, recorded by Ramanujan. Further, as an application of these Eisenstein series identities, we establish certain incomplete elliptic integrals of the first kind, involving theta-functions, which are analogous to integrals recorded by Ramanujan in his Lost Notebook.
Keywords and Phrases
Dedekind eta-function, Theta-functions, Eisenstein series, Bilateral basic hypergeometric series, Incomplete elliptic integrals.
A.M.S. subject classiﬁcation
11F11, 11F20, 11M36, 33D15, 33E05.