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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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