Monodromy Properties of A-hypergeometric Functions

Abstract

Hypergeometric functions started out as generalizations of classical elementary functions like the square root, logarithm, or arcsine. And over the years many types of hypergeometric functions appeared, for example, Appell hypergeometric functions, Horn hypergeometric functions, and Lauricella hypergeometric functions. All of these generalizations can be expressed in terms of A-hypergeometric functions.

One problem with A-hypergeometric functions is their multivaluedness. One way to study this multivaluedness is through monodromy groups. In 2016, Beukers gives a method to compute these monodromy groups under very strict conditions using Mellin-Barnes integrals.

Part one of this thesis is interested in the invariant Hermitian form over monodromy groups for A-hypergeometric functions. We give an explicit construction of such an invariant Hermitian form under strict conditions. In particular, the monodromy groups we use are with respect to a Mellin-Barnes basis of solutions, constructed through Beukers' method.

The construction of this Hermitian form does not explicitly use Mellin-Barnes integrals. This makes it possible to extend Beukers' construction without assuming the existence of Mellin-Barnes bases. Thus we want to construct a matrix group that has an invariant Hermitian form following the one we constructed. We call the resulting invariant Hermitian form the virtual Hermitian form. It is often the case that the constructed matrix group, which we call the virtual monodromy group, is a subgroup of the full monodromy group. We give conditions and algorithms to determine whether such a virtual extension exists.

In 2008, Beukers gives a combinatoric criterion to determine whether all solutions of an A-hypergeometric system are algebraic. This criterion is based on the properties of apexpoints. We give new combinatoric criteria for when the number of apexpoints is maximal. Then we link Beukers' criterion to the definiteness of the invariant Hermitian form.

Part two of this thesis is concerned with relations between classical hypergeometric functions. In 1933, Bailey published an identity where Appell's F4 factors into two Gauss hypergeometric functions. Later Vidunas (2008) and Beukers (2013) independently found a similar factorization for Appell's F2. We show how these factorizations follow through properties of monodromy groups and we construct Bailey-type identities for Horn functions H1, H4, and H5.