Hypergeometric Solutions of Linear Difference Systems

TL;DR

This work extends Petkovšek's hypergeometric-solution framework from scalar difference operators to systems $\tau(Y)=MY$ with $M\in GL_n(\mathbb{C}(x))$, introducing the Bronstein-Petkovšek strategy to bound rational candidates via $\denom(M)$ and $\denom(M^{-1})$ and to control duplicates using local-type information. It develops three algorithm versions that progressively prune the search space by exploiting unramified generalized exponents, local types, and simple-form reductions, culminating in a robust method that computes hypergeometric solutions efficiently even for high-dimensional systems. The paper also integrates a Beke-Bronstein-based factoring approach using exterior algebra to factor operators, with implementations and experiments showing dramatic improvements over cyclic-vector methods for large problems. The practical impact lies in enabling scalable factoring and Liouvillian-style analyses of high-dimensional difference systems, with publicly available code and demonstrated performance on challenging operators. All mathematical constructs are rigorously connected through generalized exponents, local types, and gauge-transform techniques to ensure correct and provable candidate reduction and solution retrieval.

Abstract

We extend Petkovšek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems $τ(Y) = M Y$, with $M \in {\rm GL}_n(C(x))$, where $τ$ is the shift operator. Hypergeometric solutions are solutions of the form $γP$ where $P \in C(x)^n$ and $γ$ is a hypergeometric term over $C(x)$, i.e. ${τ(γ)}/γ \in C(x)$. Our contributions concern efficient computation of a set of candidates for ${τ(γ)}/γ$ which we write as $λ= c\frac{A}{B}$ with monic $A, B \in C[x]$, $c \in C^*$. Factors of the denominators of $M^{-1}$ and $M$ give candidates for $A$ and $B$, while another algorithm is needed for $c$. We use the super-reduction algorithm to compute candidates for $c$, as well as other ingredients to reduce the list of candidates for $A/B$. To further reduce the number of candidates $A/B$, we bound the so-called type of $A/B$ by bounding local types. Our algorithm has been implemented in Maple and experiments show that our implementation can handle systems of high dimension, which is useful for factoring operators.

Theorems & Definitions (33)

• Definition 1: Hypergeometric
• Theorem 2
• proof
• Example 3
• Remark 4
• Lemma 5
• proof
• Proposition 6
• Definition 7
• Remark 8: Algorithms