Some Eisenstein Series Identities Associated with the Borwein Functions
In this note, we show how certain Eisenstein series identities associated with the Borweins’ theta-functions can be derived from a well-known identity involving theta-functions and one identity of Ramanujan. We employ the theory of elliptic functions to derive some related theta-function identities. These theta-function identities give a different approach to the Eisenstein series identities. By using some Eisenstein series identities of this note, we provide completely new proofs of the Borweins’ cubic theta function identity and the well-known Jacobi identity in the theory of modular forms.
KeywordsThe Borwein functions elliptic functions Eisenstein series The Borweins’ cubic theta function identity Jacobi identity
Unable to display preview. Download preview PDF.
- S. Ramanujan, On certain arithmetical functions, Trans. Camb. Phil. Soc. 22 (1916), 159–184.Google Scholar
- S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bom-bay, 1957.Google Scholar
- S Ramanujan, Collected papers, Chelsea, New York, 1966.Google Scholar
- S. Ramanujan, The Lost Notebook of Other Unpublished papers, Narosa, New Delhi, 1988.Google Scholar
- E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed, Cambridge University Press, Cambridge, 1966.Google Scholar