Towards a theory of multivariate generating functions

Multivariate generating functions (mvGFs) are a ubiquitous tool in many applications of combinatorics and probability. There are two main parts to the theory: determining a mvGF in terms of known functions, and (asymptotic) coefficient extraction. Neither part has been much systematized, in comparison with the univariate case. Of course the multivariate case is much harder, but not so hard as to justify the present level of neglect.

I will discuss results so far from an ongoing project with Robin Pemantle and others (see www.cs.auckland.ac.nz/~mcw/Research/mvGF/) to improve the theory of coefficient extraction for meromorphic mvGFs. There are several interesting technical problems arising. I hope that this audience will feel inspired to add their expertise to our efforts.