The Factorial-Basis Method for Finding Definite-Sum Solutions of Linear Recurrences With Polynomial Coefficients

TL;DR

This paper introduces a new algorithm, implemented in SageMath, for finding definite-sum solutions to linear recurrences with polynomial coefficients using factorial bases, advancing the solution of a three-decade-old problem.

Contribution

The authors present a novel factorial-basis method and an algorithm that computes recurrences for solutions of linear recurrences with polynomial coefficients, including implementation in SageMath.

Findings

Algorithm successfully computes recurrences for solution coefficients.

Implementation in SageMath enables practical application of the method.

Method generalizes to m-sieved bases for more complex solutions.

Abstract

The problem of finding a nonzero solution of a linear recurrence $Ly = 0$ with polynomial coefficients where $y$ has the form of a definite hypergeometric sum, related to the Inverse Creative Telescoping Problem of [14][Sec. 8], has now been open for three decades. Here we present an algorithm (implemented in a SageMath package) which, given such a recurrence and a quasi-triangular, shift-compatible factorial basis $\mathcal{B} = \langle P_k(n)\rangle_{k=0}^\infty$ of the polynomial space $\mathbb{K}[n]$ over a field $\mathbb{K}$ of characteristic zero, computes a recurrence satisfied by the coefficient sequence $c = \langle c_k\rangle_{k=0}^\infty$ of the solution $y_n = \sum_{k=0}^\infty c_kP_k(n)$ (where, thanks to the quasi-triangularity of $\mathcal{B}$, the sum on the right terminates for each $n \in \mathbb{N}$). More generally, if $\mathcal{B}$ is $m$-sieved for some $m \in \mathbb{N}$, our algorithm computes a system of $m$ recurrences satisfied by the $m$-sections of the coefficient sequence $c$. If an explicit nonzero solution of this system can be found, we obtain an explicit nonzero solution of $Ly = 0$.