The Ramanujan Journal

, Volume 29, Issue 1–3, pp 185–198

Two-dimensional series evaluations via the elliptic functions of Ramanujan and Jacobi

Article

DOI: 10.1007/s11139-011-9351-9

Cite this article as:
Berndt, B.C., Lamb, G. & Rogers, M. Ramanujan J (2012) 29: 185. doi:10.1007/s11139-011-9351-9

Abstract

We evaluate in closed form, for the first time, certain classes of double series, which are remindful of lattice sums. Elliptic functions, singular moduli, class invariants, and the Rogers–Ramanujan continued fraction play central roles in our evaluations.

Keywords

Two-dimensional lattice sums Ramanujan’s theta functions Class invariants Singular moduli Rogers-Ramanujan continued fraction Cubic continued fraction Jacobian elliptic functions Hypergeometric functions 

Mathematics Subject Classification (2000)

33E05 11F03 11F27 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.TucsonUSA
  3. 3.Département de Mathématiques et StatistiqueUniversité de MontréalMontréalCanada

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