Appell and Lauricella Hypergeometric Functions

Appell introduced four kinds of hypergeometric series in two variables as extensions of the hypergeometric series F(a,b,c;x), and Lauricella generalized them to hypergeometric series in m variables, and they considered systems of partial differential equations satisfied by them. In this chapter, we give definitions of Appell’s and Lauricella’s hypergeometric series and state their fundamental properties such as domains of convergence, integral representations, systems of partial differential equations, fundamental systems of solutions, and transformation formulas. We define the rank and the singular locus of a system of partial differential equations, and list them for Appell’s and Lauricella’s systems. We describe Pfaffian systems, contiguity relations, monodromy representations and twisted period relations for the systems. We give their explicit forms for Lauricella’s E_D, which is the simplest among Lauricella’s systems. We also mention the uniformization of the complement of the singular locus of E_D by the projectivization of its fundamental system of solutions.