Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 525-531, 2017

Some finite generalizations of Euler's pentagonal number theorem

Ji-Cai Liu

Received February 10, 2016.   First published March 1, 2017.

Ji-Cai Liu, College of Mathematics and Information Science, Wenzhou University, 276 Xueyuan Middle Road, Wenzhou, 325027, Zhejiang, P. R. China, e-mail:

Abstract: 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.

Keywords: $q$-binomial coefficient; $q$-binomial theorem; pentagonal number theorem

Classification (MSC 2010): 05A17, 11B65

DOI: 10.21136/CMJ.2017.0063-16

Full text available as PDF.

  [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). DOI 10.1002/zamm.19790590632 | MR 0557013 | Zbl 0655.10001
  [2] G. E. Andrews: Euler's pentagonal number theorem. Math. Mag. 56 (1983), 279-284. DOI 10.2307/2690367 | MR 0720648 | Zbl 0523.01011
  [3] A. Berkovich, F. G. Garvan: Some observations on Dyson's new symmetries of partitions. J. Comb. Theory, Ser. A 100 (2002), 61-93. DOI 10.1006/jcta.2002.3281 | MR 1932070 | Zbl 1016.05003
  [4] T. J. Bromwich: An Introduction to the Theory of Infinite Series. Chelsea Publishing Company, New York (1991). Zbl 0901.40001
  [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. MR 1364064 | Zbl 0851.15010
  [6] M. PetkovÅ¡ek, H. S. Wilf, D. Zeilberger: $A=B$. With foreword by Donald E. Knuth, A. K. Peters, Wellesley (1996). MR 1379802 | Zbl 0848.05002
  [7] D. Shanks: A short proof of an identity of Euler. Proc. Am. Math. Soc. 2 (1951), 747-749. DOI 10.1090/S0002-9939-1951-0043808-6 | MR 0043808 | Zbl 0044.28403
  [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. DOI 10.1007/s11139-005-0275-0 | MR 2130521 | Zbl 1066.05023

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at
Subscribers of Springer need to access the articles on their site, which is

[Previous Article] [Next Article] [Contents of This Number] [Contents of Czechoslovak Mathematical Journal]