Asymptotics and sign patterns for coefficients in expansions of Habiro elements

Abstract

We prove asymptotics and study sign patterns for coefficients in expansions of elements in the Habiro ring which satisfy a strange identity. As an application, we prove asymptotics and discuss positivity for the generalized Fishburn numbers which arise from the Kontsevich-Zagier series associated to the colored Jones polynomial for a family of torus knots. This extends Zagier's result on asymptotics for the Fishburn numbers.

17 pages, to appear in Mathematische Zeitschrift

References in corpus (4)

• q-series and L-functions related to half-derivatives of the Andrews--Gordon identity
• Quantum modularity of partial theta series with periodic coefficients
• Generalized Fishburn numbers and torus knots
• Congruences for Fishburn numbers modulo prime powers