On Rational Recursion for Holonomic Sequences
Bertrand Teguia Tabuguia, James Worrell
TL;DR
This paper investigates whether holonomic (P-recursive) sequences arising from discrete-time rational dynamical systems satisfy a quasi-linear recurrence where the highest-shift term appears linearly, i.e., $a(n+l)=r(a(n),\dots,a(n+l-1))$ with $r\in\mathbb{K}(x_1,\dots,x_l)$. It develops a difference-algebra framework and proves that, for a ratrec sequence, there exists a holonomic $p\in\mathbb{D}_s$ of order at most $k$ with $p(a(n))=0$, reframing the goal as finding $q\in\langle \sigma^{\infty}(p)\rangle$ that is linear in the highest shift $s(n+\mathrm{ord}(q))$. Two constructive algorithms are proposed to obtain simple ratrec from holonomic recurrences: a Gröbner-bases elimination approach (GB) that yields minimal order but can be slow, and a linear-algebra approach (LA) that is faster in practice, with an upper bound of order $\mathrm{ord}(p)+\mathrm{deg}(p)$ for holonomic equations of order $\mathrm{ord}(p)$ and degree $\mathrm{deg}(p)$. The methods are demonstrated on examples and implemented in Maple, with GitHub materials illustrating the trade-off between minimality and efficiency and enabling generation of Somos-like sequences from holonomic inputs. Overall, the work provides principled procedures to automatically convert holonomic recurrences into quasi-linear rational recurrences, advancing the automatic synthesis of simple rational recurrences for holonomic sequences.
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.