The mixed mock modularity of a new $U$-type function related to the Andrews-Gordon identities

TL;DR

This paper introduces a new $U$-type function linked to Andrews-Gordon identities, expressing related sums as mixed mock modular forms involving Appell and theta functions, thus advancing understanding of partition generating functions.

Contribution

It resolves a question about a vector-valued $U$-type function and expresses related sums as mixed mock modular forms, connecting partition identities with modular objects.

Findings

Derived an explicit expression for the $U$-type function

Connected the function to mixed mock modular forms involving Appell and theta functions

Enhanced understanding of partition generating functions in the context of modular forms

Abstract

In this paper we resolve a question by Bringmann, Lovejoy, and Rolen on a new vector-valued $U$-type function. We obtain an expression for a corresponding family of Hecke-Appell-type sums in terms of mixed mock modular forms; that is, we express the sum in terms of Appell functions and theta functions. This $U$-type function appears from considering the special polynomials related to generating functions for the partitions occurring in Gordon generalization of the Rogers-Ramanujan identities.