Twisting q-holonomic sequences by complex roots of unity

TL;DR

This paper proves that $q$-holonomicity of sequences is preserved under twisting by roots of unity and rational powers, with constructive proofs and applications to quantum topology invariants.

Contribution

It establishes the preservation of $q$-holonomicity under twisting and substitution, with constructive proofs and implementation in Mathematica for multivariate sequences.

Findings

Preservation of $q$-holonomicity under twisting by roots of unity.

Implementation of results in Mathematica package HolonomicFunctions.

Application to quantum topology invariants like colored Jones polynomials.

Abstract

A sequence $f_n(q)$ is $q$-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in $q$ and $q^n$. Our main theorems state that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $\omega q$ where $\omega$ is a complex root of unity, and under the substitution $q \to q^{\alpha}$ where $\alpha$ is a rational number. Our proofs are constructive, work in the multivariate setting of $\partial$-finite sequences and are implemented in the Mathematica package HolonomicFunctions. Our results are illustrated by twisting natural $q$-holonomic sequences which appear in quantum topology, namely the colored Jones polynomial of pretzel knots and twist knots. The recurrence of the twisted colored Jones polynomial can be used to compute the asymptotics of the Kashaev invariant of a knot at an arbitrary complex root of unity.