  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.
