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

TL;DR

This paper introduces new partition functions based on least r-gaps, explores their connections with theta identities, polygonal numbers, and Euler's partition function, using combinatorial and q-series methods.

Contribution

It presents novel partition functions involving least r-gaps and establishes new identities linking these functions with classical theta identities, polygonal numbers, and Euler's partition function.

Findings

Derived new identities relating partition functions and polygonal numbers

Provided combinatorial interpretations for complex q-series sums

Connected theta series, partition functions, and polygonal numbers in novel ways

Abstract

The least $r$-gap, $g_r(\lambda)$, of a partition $\lambda$ is the smallest part of $\lambda$ appearing less than $r$ times. 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 $$\sum_{n=0}^\infty (\pm 1)^{k(k+1)/2} p(n-r\cdot k(k+1)/2).$$