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(Nlog2N) bit
operations; a deterministic one that computes a compact representation of the
solution in O(Nlog3N) bit operations. Similar speed-ups are obtained in
indefinite and definite hypergeometric summation. We describe the results of an
implementation.Comment: This is the author's version of the work. It is posted here by
permission of ACM for your personal use. Not for redistributio