Twisting q-holonomic sequences by complex roots of unity

Details Twisting q-holonomic sequences by complex roots of unity Twisting q-holonomic sequences by complex roots of unity

2012-01-16

arxiv.org/abs/1201.3353v1

Mathematics

Geometric Topology

7 pages, 4 figures

Scientific paper

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 theorem states that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $\omega q$ where $\omega$ is a complex root of unity. Our proof is constructive, works in the multivariate setting of $\partial$-finite sequences and is 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.

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