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|Title:||Eisenstein series in Ramanujan's lost notebook|
|Keywords:||Dedekind eta function|
Differential equations for Eisenstein series
Ramanujan's lost notebook
|Source:||Berndt, B.C.,Chan, H.H.,Sohn, J.,Son, S.H. (2000-03). Eisenstein series in Ramanujan's lost notebook. Ramanujan Journal 4 (1) : 81-114. ScholarBank@NUS Repository.|
|Abstract:||In his lost notebook, Ramanujan stated without proofs several beautiful identities for the three classical Eisenstein series (in Ramanujan's notation) P(q), Q(q), and R(q). The identities are given in terms of certain quotients of Dedekind eta-functions called Hauptmoduls. These identities were first proved by S. Raghavan and S.S. Rangachari, but their proofs used the theory of modular forms, with which Ramanujan was likely unfamiliar. In this paper we prove all these identities by using classical methods which would have been well known to Ramanujan. In fact, all our proofs use only results from Ramanujan's notebooks.|
|Source Title:||Ramanujan Journal|
|Appears in Collections:||Staff Publications|
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