Exact formula for 1-lower run overpartitions

TL;DR

This paper derives an exact formula for 1-lower run overpartitions, revealing its generating function as a mixed mock-modular form and employing an extended Circle Method with advanced bounds.

Contribution

It introduces a novel exact formula for 1-lower run overpartitions and applies an extended Circle Method to analyze its mixed mock-modular generating function.

Findings

Exact formula for 1-lower run overpartitions

Generating function identified as mixed mock-modular form

Application of extended Circle Method with bounds on Kloosterman sums

Abstract

We are going to show an exact formula for lower $1$-run overpartitions. The generating function is of mixed mock-modular type with an overall weight $0.$ We will apply an extended version of the classical Circle Method. The approach requires bounding modified Kloosterman sums and Mordell integrals.