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

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

$$\begin{aligned} \sum _{n=0}^\infty (\pm 1)^{k(k+1)/2} p(n-r\cdot k(k+1)/2). \end{aligned}$$