On a conjecture due to Kanade related to Nahm sums

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.