CONFLUENCES OF APPELL'S <i>F</i>₂SYSTEM OF HYPERGEOMETRIC DIFFERENTIAL EQUATIONS

We consider confluences of Euler-type integrals expressing solutions to Appell's F2 system of hypergeometric differential equations, and study systems of confluent hypergeometric differential equations of rank four of two variables. Our consideration is based on a confluence transforming the abelian group (ℂ×)2 to the Jordan group of size two. For each system obtained by our study, we give its Pfaffian system with a connection matrix admitting a decomposition into four or five parts, each of which is the product of a matrix depending only on parameters and a rational 1-form in two variables. We classify these Pfaffian systems under an equivalence relation. Any system obtained by our study is equivalent to one of Humbert's Ψ1 system, Humbert's Ξ1 system, and the system satisfied by the product of two Kummer's confluent hypergeometric functions.