Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T17:30:22.632Z Has data issue: false hasContentIssue false
  • English
  • Français

EISENSTEIN SERIES TO THE TREDECIC BASE

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  27 June 2014

SHAUN COOPER  and
DONGXI YE
SHAUN COOPER
Affiliation:
Institute of Natural and Mathematical Sciences, Massey University-Albany, Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand email [email protected]
DONGXI YE*
Affiliation:
Institute of Natural and Mathematical Sciences, Massey University-Albany, Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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.

We employ a modular method to establish the new result that two types of Eisenstein series to the tredecic base may be parametrised in terms of the eta quotients ${\it\eta}^{13}({\it\tau})/{\it\eta}(13{\it\tau})$ and ${\it\eta}^{2}(13{\it\tau})/{\it\eta}^{2}({\it\tau})$. The method can also be used to give short and simple proofs for the analogous cubic, quintic and septic theories.

Keywords

Eisenstein seriesDedekind’s eta function

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight
Secondary: 11F20: Dedekind eta function, Dedekind sums 11F03: Modular and automorphic functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 1 , February 2015 , pp. 19 - 28
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Andrews, G. E. and Berndt, B. C., Ramanujan’s Lost Notebook, Part III (Springer, New York, 2012).CrossRefGoogle Scholar
Apostol, T. M., Modular Functions and Dirichlet Series in Number Theory, 2nd edn (Springer, New York, 1990).CrossRefGoogle Scholar
Berndt, B. C., Ramanujan’s Notebooks, Part III (Springer, New York, 1991).CrossRefGoogle Scholar
Berndt, B. C., Bhargava, S. and Garvan, F. G., ‘Ramanujan’s theories of elliptic functions to alternative bases’, Trans. Amer. Math. Soc. 347 (1995), 4163–4244.Google Scholar
Borwein, J. M. and Borwein, P. B., ‘A cubic counterpart of Jacobi’s identity and the AGM’, Trans. Amer. Math. Soc. 323 (1991), 691–701.Google Scholar
Chan, H. H. and Liu, Z.-G., ‘Analogues of Jacobi’s inversion formula for the incomplete elliptic integral of the first kind’, Adv. Math. 174 (2003), 69–88.CrossRefGoogle Scholar
Chan, H. H. and Liu, Z.-G., ‘Elliptic functions to the quintic base’, Pacific J. Math. 226 (2006), 53–64.CrossRefGoogle Scholar
Chan, H. H. and Cooper, S., ‘Eisenstein series and theta functions to the septic base’, J. Number Theory 128 (2008), 680–699.CrossRefGoogle Scholar
Cooper, S., ‘Cubic elliptic functions’, Ramanujan J. 11 (2006), 355–397.CrossRefGoogle Scholar
Cooper, S., ‘Construction of Eisenstein series for Γ0(p)’, Int. J. Number Theory 5 (2009), 765–778.CrossRefGoogle Scholar
Cooper, S. and Ye, D., ‘The Rogers–Ramanujan continued fraction and its level 13 analogue’, J. Approx. Theory to appear. doi:10.1016/j.jat.2014.01.008.Google Scholar
Dobbie, J. M., ‘A simple proof of some partition formulae of Ramanujan’, Q. J. Math. 6(2) (1955), 193–196.CrossRefGoogle Scholar
Hirschhorn, M. D., ‘A simple proof of an identity of Ramanujan’, J. Aust. Math. Soc., Ser. A 34 (1983), 31–35.CrossRefGoogle Scholar
Kolberg, O., ‘Note on the Eisenstein series of Γ0(p)’, Arbok Univ. Bergen Mat.-Natur. Ser. 1968 (1968), 20 pp.Google Scholar
Liu, Z.-G., ‘Some Eisenstein series identities related to modular equation of the seventh order’, Pacific J. Math. 209 (2003), 103–130.Google Scholar
Ramanujan, S., Notebooks (2 Volumes) (Tata Institute of Fundamental Research, Bombay, 1957).Google Scholar
Ramanujan, S., The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988).Google Scholar
Venkatachaliengar, K. and Cooper, S., Development of Elliptic Functions According to Ramanujan (World Scientific, Singapore, 2012).Google Scholar