On a Ramanujan’s Eisenstein series identity of level fifteen

On pages 255–256 of his second notebook, Ramanujan recorded an Eisenstein series identity of level 15 without offering a proof. Previously, Berndt (Ramanujan’s Notebooks: Part III ( 1991 ) (New York: Springer)) proved this identity using the theory of modular forms. In this paper, we give an element...

Full description

Saved in:
Bibliographic Details
Published inProceedings of the Indian Academy of Sciences. Mathematical sciences Vol. 129; no. 4; pp. 1 - 18
Main Authors Bhuvan, E N, Vasuki, K R
Format Journal Article
LanguageEnglish
Published New Delhi Springer India 01.09.2019
Springer Nature B.V
Subjects
Analytic functions
Mathematics
Mathematics and Statistics
modular equation
Eisenstein series
11M36
Dedekind eta-function
11F20
33C05
Online AccessGet full text

Cover

More Information
Summary:On pages 255–256 of his second notebook, Ramanujan recorded an Eisenstein series identity of level 15 without offering a proof. Previously, Berndt (Ramanujan’s Notebooks: Part III ( 1991 ) (New York: Springer)) proved this identity using the theory of modular forms. In this paper, we give an elementary proof of this identity. In the process, we also give an elementary proof of three Ramanujan’s P - Q identities of level 15. Further, using the P - Q identities we prove four Ramanujan type Eisenstein series of level 15 due to Cooper and Ye ( Trans. Am. Math. Soc. 368 ( 2016 ) 7883–7910), where they have proved using the theory of modular forms.
ISSN:0253-4142
0973-7685
DOI:10.1007/s12044-019-0498-4