A Limiting form of the q-Dixon 4φ3 Summation and Related Partition Identities

  • Krishnaswami Alladi
  • Alexander Berkovich
Part of the Developments in Mathematics book series (DEVM, volume 8)


By considering a limiting form of the q-Dixon 4φ3 summation, we prove a weighted partition theorem involving odd parts differing by ≥ 4. A two parameter refinement of this theorem is then deduced from a quartic reformulation of Göllnitz’s (Big) theorem due to Alladi, and this leads to a two parameter extension of Jacobi’s triple product identity for theta functions. Finally, refinements of certain modular identities of Alladi connected to the Göllnitz-Gordon series are shown to follow from a limiting form of the q-Dixon 4φ3 summation.


q-Dixon sum q-Dougall Sum weighted partition identities Göllnitz’s (Big) theorem quartic reformulation Jacobi’s triple product GöllnitzGordon identities modular relations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    K. Alladi, A combinatorial correspondence related to Göllnitz’s (Big) partition theorem and applications, Trans. Amer. Math Soc. 349 (1997), 2721–2735.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    K. Alladi, On a partition theorem of Göllnitz and quartic transformations (with an appendix by B. Gordon), J. Num. Th. 69 (1998), 153–180.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    K. Alladi and G. E. Andrews, A new key identity for Göllnitz’s (Big) partition theorem, Contemp. Math. 210 (1998), 229–241.MathSciNetCrossRefGoogle Scholar
  4. [4]
    G. Gasper and M. Rahman, Basic hyper-geometric series, Encyclopedia of Mathematics and its Applications, Vol.10, Cambridge (1990).Google Scholar
  5. [5]
    H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew Math. 225 (1967), 154–190.MathSciNetzbMATHGoogle Scholar
  6. [6]
    B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J. 32 (1965), 741–748.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    L. J. Slater, Further identities of Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147–167.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Krishnaswami Alladi
    • 1
  • Alexander Berkovich
    • 1
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

Personalised recommendations