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Bisected theta series, least r-gaps in partitions, and polygonal numbers

  • Cristina BallantineEmail author
  • Mircea Merca
Article
  • 15 Downloads

Abstract

The least r-gap, \(g_r(\lambda )\), of a partition \(\lambda \) is the smallest positive integer that does not appear at least r times as a part of \(\lambda \). In this article, we introduce two new partition functions involving least r-gaps. We consider a bisection of a classical theta identity and prove new identities relating Euler’s partition function p(n), polygonal numbers, and the new partition functions. To prove the results, we use an interplay of combinatorial and q-series methods. We also give a combinatorial interpretation for
$$\begin{aligned} \sum _{n=0}^\infty (\pm 1)^{k(k+1)/2} p(n-r\cdot k(k+1)/2). \end{aligned}$$

Keywords

Partitions Least gap Polygonal numbers Theta series 

Mathematics Subject Classification

05A17 11P83 

Notes

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Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceCollege of The Holy CrossWorcesterUSA
  2. 2.Academy of Romanian ScientistsBucharestRomania

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