Low Complexity Algorithms for Linear Recurrences

TL;DR

This paper introduces new low-complexity algorithms for solving linear recurrences and hypergeometric summation problems, significantly improving efficiency over previous methods by exploiting solution structures.

Contribution

It presents probabilistic and deterministic algorithms that reduce computational complexity for finding polynomial and rational solutions of linear recurrences and hypergeometric sums.

Findings

Probabilistic algorithm detects solutions in O(√N log^2 N) operations.

Deterministic algorithm computes solutions in O(N log^3 N) operations.

Implementation results demonstrate practical efficiency improvements.

Abstract

We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are hypergeometric over the rational numbers. The algorithms for these tasks all involve as an intermediate quantity an integer $N$ (dispersion or root of an indicial polynomial) that is potentially exponential in the bit size of their input. Previous algorithms have a bit complexity that is at least quadratic in $N$. We revisit them and propose variants that exploit the structure of solutions and avoid expanding polynomials of degree $N$. We give two algorithms: a probabilistic one that detects the existence or absence of nonzero polynomial and rational solutions in $O(\sqrt{N}\log^{2}N)$ bit operations; a deterministic one that computes a compact representation of the solution in $O(N\log^{3}N)$ bit operations. Similar speed-ups are obtained in indefinite and definite hypergeometric summation. We describe the results of an implementation.