The ABC of Creative Telescoping - Algorithms, Bounds, Complexity

Creative telescoping is an algorithmic principle that has been developed since the 1990s in combinatorics and computer algebra, notably since Doron Zeilberger's work, in order to compute with parametrised sums and integrals, whether it be to find explicit forms or to justify identities in sums or integrals. The process is particularly suited to a large family of functions and sequences given by linear differential and recurrence equations, whether it be special functions of analysis, sequences of combinatorics, or families of orthogonal polynomials. In the present memoir, I shall recount the evolution of algorithms and of my contributions in adapting the approach to larger and larger classes of functions, from the initial framework of hypergeometric sequences, given by first-order recurrences, to the case of functions given by higher-order equations, then to functions given by positive-dimensional ideals. The difficulty to obtain fast implementations in all these cases stems from the computation of a certificate that justifies the applicability of creative telescoping, this certificate being naturally of large size. This motivated me in the complexity study of the process. Several tracks of improvement have been explored, first by trying and maintaining the certificate in compact form, then in obtaining algorithms that are validated without computing any certificate. As often, estimating the arithmetical sizes of objects involved in creative telescoping has at the same time guided the design of new, more efficient algorithms and made it possible to estimate their theoretical complexity. Finally, I shall briefly indicate the new direction taken in my works, towards formal proofs, which reveal new tracks for a better justification of the use of creative telescoping.