Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 525–531

Some finite generalizations of Euler’s pentagonal number theorem



Euler’s pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler’s pentagonal number theorem.


q-binomial coefficient q-binomial theorem pentagonal number theorem 

MSC 2010

05A17 11B65 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. E. Andrews: The Theory of Partitions. Encyclopedia of Mathematics and Its Applications, Vol. 2. Section: Number Theory. Reading, Advanced Book Program, Addison-Wesley Publishing Company, Massachusetts, 1976.Google Scholar
  2. [2]
    G. E. Andrews: Euler’s pentagonal number theorem. Math. Mag. 56 (1983), 279–284.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    A. Berkovich, F. G. Garvan: Some observations on Dyson’s new symmetries of partitions. J. Comb. Theory, Ser. A 100 (2002), 61–93.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    T. J. Bromwich: An Introduction to the Theory of Infinite Series, Chelsea Publishing Company, New York, 1991.MATHGoogle Scholar
  5. [5]
    S. B. Ekhad, D. Zeilberger: The number of solutions of X 2 = 0 in triangular matrices over GF(q). Electron. J. Comb. 3 (1996), Research paper R2, 2 pages; printed version in J. Comb. 3 (1996), 25–26.Google Scholar
  6. [6]
    M. Petkovšek, H. S. Wilf, D. Zeilberger: A = B. With foreword by Donald E. Knuth, A. K. Peters, Wellesley, 1996.MATHGoogle Scholar
  7. [7]
    D. Shanks: A short proof of an identity of Euler. Proc. Am. Math. Soc. 2 (1951), 747–749.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    S. O. Warnaar: q-hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue’s identity and Euler’s pentagonal number theorem. Ramanujan J. 8 (2004), 467–474.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceWenzhou UniversityWenzhou, ZhejiangP.R. China

Personalised recommendations