Asymptotics and sign patterns for coefficients in expansions of Habiro elements

TL;DR

This paper investigates the asymptotic behavior and sign patterns of coefficients in Habiro ring expansions, with applications to generalized Fishburn numbers and their positivity, extending previous results on Fishburn numbers.

Contribution

It introduces new asymptotic formulas and sign pattern analyses for coefficients in Habiro ring expansions, linking them to generalized Fishburn numbers and knot invariants.

Findings

Established asymptotics for coefficients in Habiro ring expansions.

Analyzed sign patterns and positivity of generalized Fishburn numbers.

Extended Zagier's asymptotic results to new classes of numbers.

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.