This thesis shows how computer algebra makes it possible to manipulate a large class of sequences and functions that are solutions of linear operators, namely that of holonomic functions. This class contains numerous special functions, in one or several variables, as well as numerous combinatorial sequences. First, a theoretical framework is introduced in order to give algorithms for the closure properties of the holonomic class, to permit a zero test in this class, and to unify differential calculations with functions and calculations of recurrences with sequences. These methods are based on calculations by an extension of the theory of Gröbner bases in a framework of non-commutative polynomials, namely Ore polynomials. Two kinds of algorithms for symbolic definite and indefinite summation and integration are then developed, whose theoretical justification appeals to the theory of holonomic D-modules. The former resort to non-commutative polynomial elimination by Gröbner bases; the latter to algorithms to solve linear functional systems for their rational function solutions. Much more than the search for closed forms, the aim is to be able to continue to compute with the implicit representation of holonomic objects even when no explicit form is available. In particular, this type of calculation makes the automatic proof of summatory and integral identities possible. An implementation of these algorithms for the computer algebra system Maple has made it possible to give the first automatic proof of identities so far unreachable by computer algebra.
