# Journal of the Ramanujan Mathematical Society

Volume 37, Issue 3, September 2022 pp. 257–272.

On Ramanujan's Eisenstein series of level 5 and 7

**Authors**:
K. Pushpa and K. R. Vasuki

**Author institution:**Department of Studies in Mathematics, University of Mysore,
Manasagangotri, Mysuru 570 006, India

**Summary: **
Ramanujan recorded eight Eisenstein series identities of level 5 and 7 (Four
in each level). These identities have been proved for the first time by S.
Raghavan and S. S. Rangachari using the theory of modular forms. B. C. Berndt
et al have constructed the proofs of these identities in the spirit of
Ramanujan's work. In fact, they acknowledged the usage of Mathematica often
times for algebraic manipulations in their proofs. S. Cooper proved quintic
level identities by using the parametrization k = r(q)r{2}(q{2}), where r(q) is
the famous Rogers-Ramanujan continued fraction. Z.-G. Liu found proofs of
septic level identities using the complex variable theory of elliptic
functions. Our objective of this article is to give a simple proofs of these
identities by using only P - Q theta function identities recorded by Ramanujan.

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