Holonomic Functions in Computer Algebra (abstract)
Frédéric Chyzak
Abstract:
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 
-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.
Keywords:
computer algebra, holonomic functions, symbolic
integration, symbolic summation, polynomial elimination, Gröbner
bases, -modules, Zeilberger's algorithm, Ore operators.