Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-23T21:18:55.454Z Has data issue: false hasContentIssue false
  • English
  • Français

A Proof of some identities of Ramanujan using modular forms

Published online by Cambridge University Press:  18 May 2009

Anthony J. F. Biagioli
Anthony J. F. Biagioli
Affiliation:
The University of Missouri at Rolla, Rolla, Missouri 65401, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1974 B. J. Birch [1] published a description of some manuscripts of Ramanujan which contained, among other things, a list of forty identities involving the Rogers-Ramanujan functions

At that time nine of these had been proven, and since then twenty-two more of them have been proven, fifteen of them by David Bressoud in his thesis [2]. Bressoud gives a synopsis of the extant proofs, where he attributes proofs to H. B. C. Darling [3], L. J. Rogers [4], L. J. Mordell [5], and G. N. Watson [6].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 3 , September 1989 , pp. 271 - 295
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

1.Birch, B. J., A look back at Ramanujan's manuscripts, Math. Proc. Camb. Phil. Soc, 78 (1975), 73–79.CrossRefGoogle Scholar
2.Bressoud, David, Proof and generalization of certain identities conjectured by Ramanujan (thesis, Temple university 1980).Google Scholar
3.Darling, H. B. C., Proofs of certain identities and congruences enunciated by S. Ramanujan, Proc. London Math. Soc. (2), 19 (1921), 350–372.CrossRefGoogle Scholar
4.Rogers, L. J., On a type of modular relation, Proc. London Math. Soc. (2), 19 (1921), 387–397.CrossRefGoogle Scholar
5.Mordell, L. J., Note on certain modular relations considered by Messrs. Ramanujan, Darling and Rogers, Proc. London Math. Soc. (2), 20 (1922), 408–416.CrossRefGoogle Scholar
6.Watson, G. N., Proof of certain identities in combinatory analysis, J. Indian Math. Soc, 20 (1933), 57–69.Google Scholar
7.Rankin, Robert, Modular forms and functions, (Cambridge University Press 1977).CrossRefGoogle Scholar
8.Schoeneberg, Bruno, Elliptic modular functions, (Springer-Verlag 1974).CrossRefGoogle Scholar
9.Knopp, Marvin I., Modular functions in analytic number theory, (Markham 1970).Google Scholar
10.Rademacher, Hans, Topics in analytic number theory, (Springer-Verlag 1972).Google Scholar
11.Berndt, Bruce C., Biagioli, Anthony J. F. and Purtilo, James M., Ramanujan's modular equations of “large” prime degree, Journal of the Indian Math. Soc. 51 (1987).Google Scholar
12.Berndt, Bruce C., Biagioli, Anthony J. F., and Purtilo, James M., Ramanujan's modular equations of degrees 7 and 11, Indian Journal of Mathematics 29 (1987).Google Scholar
13.Berndt, Bruce C., Biagioli, Anthony J. F., and Purtilo, James M., Ramanujan's ”mixed” modular equations, J. Ramanujan Math. Soc. 1 (1 & 2) 1986, 1–25.Google Scholar