Milner 317

Wednesdays, 12:30-1:30 PM

Texas A&M University

**Wednesday, April 11
Milner 317, 12:30 PM**

**Title:** Hypergeometric
differential equations in two variables

**Abstract:** Given a system
of two hypergeometric equations in two variables, the most natural
estimate for their "holonomic rank" (i.e. the dimension of the space of
complex holomorphic solution in a neighbourhood of a regular point)
when the parameters are generic, is the product of the orders of the
differential equations. Unfortunately, this does not work, as a
system of two hypergeometric differential equations in two variables of
order 2 can have holonomic rank 3 or 4 depending on what the equations
actually are. However, one can get a correct formula if one subtracts
some correction terms from the product of the orders. These correction
terms can be easily read off from the equations.

The goal of the talk is to write down this rank formula, then write a
better (easier to generalize) rank formula, and then explain why these
two formulas work, by considering three illustrative examples: The
systems for the Appell function F_1, the Appell function F_4 and the
Horn function G_3.

The results in this talk are contained in two papers, one joint with
Alicia Dickenstein and Timur Sadykov, the other joint with Alicia
Dickenstein and Ezra Miller.