On a conjecture due to Kanade related to Nahm sums

TL;DR

This paper proves Kanade's conjecture involving the dilogarithm function related to Nahm sums and modular identities, and extends it by proposing two new conjectures with associated matrices.

Contribution

It confirms Kanade's conjecture using dilogarithm identities and introduces two new related dilogarithm identities and matrices.

Findings

Proof of Kanade's conjecture using dilogarithm identities

Extension to two new dilogarithm identities and matrices

Advancement in understanding Nahm sums and modular forms

Abstract

Kanade explored the construction of modular companions to $q$-series identities, using the asymptotics of Nahm sums, and Mizuno [Ramanujan J.\ {\bf 66} (2025), Paper No.\ 62, 31] recently obtained a generalization of Kanade's asymptotic formula for symmetrizable Nahm sums. A related conjecture from Kanade concerning the dilogarithm function and related to the work of Kur\c sung\"oz on Andrews--{G}ordon-type series [Ann.\ Comb.\ {\bf 23} (2019), 835--888] has remained open. In this paper, we prove Kanade's conjecture, through an application of dilogarithm identities due to Kirillov together with a dilogarithm ladder due to Lewin and Loxton. Inspired by Kanade's result, we extend this to conjecture two new dilogarithm identities and associated rank-2 matrices.