The present document collects the slides from a series of talks given in Winter 1999 at the Combinatorics Seminar held by Peter Paule in RISC-Linz (Austria). The series is mainly devoted to a presentation of the doctoral thesis which I prepared in the Algorithms Project, INRIA-Rocquencourt (France), and with which the series shares its title. Its main goal is to present algorithms for symbolic summation and integration, and also for other algorithmic manipulations of special functions and combinatorial sequences. But it also provides the detailed background on holonomy required in algorithmic applications. The material consists in an assembled, more mature, and polished version of the talks I have given during the years 1996 to 1998. It is structured in the following way:

- Motivation;
- Implicit Representations of Special Functions and Sequences;
- Gröbner Bases in Algebras of Linear Operators;
- Holonomy and Holonomic Functions;
- Holonomic Summation and Integration.

I thank Peter Paule for accepting to devote roughly a third of a semester in his weekly seminar to the presentation of my work, and to give me the opportunity to think about these topics in great details again. I also thank the audience of the Combinatorics Seminar for their many questions and stimulating remarks during the talks.

Part of this work was supported by SFB grant F1305 of the Austrian Science Foundation (FWF).

March 1999, Frédéric Chyzak