# zbMATH — the first resource for mathematics

On rational de Rham cohomology associated with the generalised confluent hypergeometric functions. I: $$\mathbb{P}^ 1$$ case. (English) Zbl 0883.33009
The author introduces some characters on a maximal abelian subgroup of $$GL(n+1)$$ and defines the confluent hypergeometric function on a Zariski open subset $$M^\ast$$ of the space of $$2\times(n+1)$$-matrices as an integration over a cycle in $$\mathbb{P}^1$$ of the character. The confluent hypergeometric function so defined transforms according to the character under the action of the abelian subgroup on $$M^\ast$$ by right multiplication. The author then defines the rational de Rham cohomology groups associated with the confluent hypergeometric functions using the character. It is proved that only the top cohomology group does not vanish and it is a vector space of dimension $$n-1$$. An explicit basis of the vector space is constructed.
 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 33C15 Confluent hypergeometric functions, Whittaker functions, $${}_1F_1$$