Multi-dimensional q-summations and multi-colored partitions

  • Shane Chern
  • Shishuo Fu
  • Dazhao TangEmail author


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.


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

Mathematics Subject Classification

05A17 11P84 



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.


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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

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