On Rational Recursion for Holonomic Sequences

TL;DR

This paper proves a conjecture for holonomic sequences, showing they satisfy quasi-linear algebraic difference equations, and introduces two algorithms for converting holonomic recurrences into such equations.

Contribution

It establishes the conjecture for holonomic sequences and presents two algorithms for converting recurrences into quasi-linear equations.

Findings

The conjecture holds for holonomic sequences.

Two algorithms differ in efficiency and minimality.

Holonomic sequences can be represented by rational dynamical systems.

Abstract

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.