Advertisement

Multi-dimensional q-summations and multi-colored partitions

  • Shane Chern
  • Shishuo Fu
  • Dazhao TangEmail author
Article
  • 34 Downloads

Abstract

Motivated by Alladi’s recent multi-dimensional generalization of Sylvester’s classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different colors. This new identity encompasses a handful of classical results as special cases, such as Cauchy’s identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan.

Keywords

Sylvester’s identity Cauchy’s identity Multiple summations Multi-colored partitions Combinatorial proof 

Mathematics Subject Classification

05A17 11P84 

Notes

Acknowledgements

We would like to acknowledge our gratitude to Ae Ja Yee for her helpful suggestions, which strengthen our original version of Theorem 1.2. We also want to thank the referee for the careful reading and useful comments.

References

  1. 1.
    Alladi, K.: A multi-dimensional extension of Sylvester’s identity. Int. J. Number Theory 13(10), 2487–2504 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, vol. 2. Addison-Wesley Publishing Co., Reading (1976) (Reprinted: Cambridge University Press, London and New York, 1984)Google Scholar
  3. 3.
    Andrews, G.E.: J. J. Sylvester, Johns Hopkins and Partitions. In: A Century of Mathematics in America, Part I, pp. 21–40. American Mathematical Society, Providence (1988)Google Scholar
  4. 4.
    Dousse, J., Kim, B.: An overpartition analogue of the \(q\)-binomial coefficients. Ramanujan J. 42(2), 267–283 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fu, S., Tang, D.: Multiranks and classical theta functions. Int. J. Number Theory 14(2), 549–566 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Sylvester, J.J.: A constructive theory of partitions, arranged in three acts, an interact and an exodion. Am. J. Math. 5(1–4), 251–330 (1882)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.College of Mathematics and StatisticsChongqing UniversityChongqingPeople’s Republic of China
  3. 3.College of Mathematics and StatisticsChongqing UniversityChongqingPeople’s Republic of China

Personalised recommendations