An application of hypergeometric functions to a construction in several complex variables

The paper is devoted to the investigations of holomorphic functions on complete n-circular domains G of ℂn which are solutions of some partial differential equations in G. Our considerations concern a collection MGk, k ≥ 2, of holomorphic solutions of equations corresponding to planar Sakaguchi’s conditions for starlikeness with respect to k-symmetric points. In an earlier paper of the first author some embedding theorems for MGk were given. In this paper we solve the problem of finding some sharp estimates of m-homogeneous polynomials in a power series expansion of f from MGk. We obtain a formula of the extremal function which includes some special functions. Moreover, its construction is based on properties of hypergeometric functions and (j, k)-symmetric functions. The (j, k)-symmetric functions were considered in several papers of the second author and his co-author, J. Połubiński.