Holonomic Functions in Several Variables

This is a continuation of the 6 January seminar of Philippe Flajolet. We will say that a multivariate power series is D-finite if its partial derivatives span a finite dimensional vector space over the field of complex rational functions. The definition of holonomic function is closely related, but more technical. Most of the special functions encountered as generating functions in analyses of algorithms are holonomic. We will survey work of L. Lipshitz and D. Zeilberger on the closure properties of D-finite and holonomic functions and discuss an algorithm of Zeilberger to decide series identities. This algorithm is based on a normal form for holonomic functions.