Exploring multivariate Pade approximants for multiple hypergeometric series

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions F-i, i = 1,..., 4, by multivariate Pade approximants. Section 1 reviews the results that exist for the projection of the F-i onto x = 0 or y = 0, namely, the Gauss function F-2(1)( a, b; c; z), since a great deal is known about Pade approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Pade approximants. In section 3 we prove that the table of homogeneous multivariate Pade approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F-1( a, b, b'; c; x, y) with a = b = b' = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Pade approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Pade approximants in this context and discussing directions for future work.