On certain identities involving Nahm-type sums with double poles

TL;DR

This paper proves new Nahm-type sum identities related to Andrews-Gordon, Andrews-Bressoud, and Rogers' false theta functions, motivated by recent work on double pole series and defect Schur's indices.

Contribution

It introduces novel Nahm-type sum representations for several classical identities, expanding the understanding of series with double poles.

Findings

Proved Nahm-type sum identities for Andrews-Gordon and Andrews-Bressoud identities.

Established connections between double pole series and defect Schur's indices.

Extended the theory of false theta functions with new sum representations.

Abstract

We prove certain Nahm-type sum representations for the (odd modulus) Andrews-Gordon identities, the (even modulus) Andrews-Bressoud identities, and Rogers' false theta functions. These identities are motivated on one hand by a recent work of C. Jennings-Shaffer and one of us on double pole series, and, on the other hand, by C\'ordova, Gaiotto and Shao's work on defect Schur's indices.