Twisting q-holonomic sequences by complex roots of unity

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 w*q where w is a complex root of unity, and under the substitution q -> 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.