The method of creative telescoping
Journal of Symbolic Computation
Hypergeometric solutions of linear recurrences with polynomial coefficients
Journal of Symbolic Computation - Special issue on symbolic computation in combinatorics
On polynomial solutions of linear operator equations
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Modern computer algebra
Special formal series solutions of linear operator equations
Discrete Mathematics
Applicability of Zeilberger's algorithm to hypergeometric terms
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Shiftless decomposition and polynomial-time rational summation
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Modular Algorithms In Symbolic Summation And Symbolic Integration (Lecture Notes in Computer Science)
Fast algorithms for polynomial solutions of linear differential equations
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Algebraic Complexity Theory
Partial denominator bounds for partial linear difference equations
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
A refined denominator bounding algorithm for multivariate linear difference equations
Proceedings of the 36th international symposium on Symbolic and algebraic computation
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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(√N log2 N) bit operations; a deterministic one that computes a compact representation of the solution in O(N log3 N) bit operations. Similar speedups are obtained in indefinite and definite hypergeometric summation. We describe the results of an implementation.