On Rational Recursion for Holonomic Sequences

It was recently conjectured that every component of a discrete-time rational dynamical system is a solution to an algebraic difference equation, that is, linear in its highest-shift term (a quasi-linear equation). We prove that the conjecture holds in the special case of holonomic sequences, which can straightforwardly be represented by rational dynamical systems. We propose two algorithms for converting holonomic recurrence equations into such quasi-linear equations. The two algorithms differ in their efficiency and the minimality of orders in their outputs.