Padé-type approximants for multivariate series of functions

Abstract

Padé-type approximants (PTA) for multivariate series of functions $f(t) = \mathop \Sigma \limits_{p = o}^{ + \infty } \mathop \Sigma \limits_{\left| i \right| = p} c_i g_i (t) (t \in \mathbb{R}^k ,i \in \mathbb{N}^k )$ are formally defined as c(P(.,t)) where P(x,t) are interpolating polynomials in IR^k, in the sense of Hakopian, of the generating functions $\mathop \Sigma \limits_{p = o}^\infty \mathop \Sigma \limits_{\left| p \right| = i} (\begin{array}{*{20}c}p \\i \\\end{array} ) x^i g_i (t)$ and where c is the linear form, associated with f, defined by $c(x^i ) = c_i /(\begin{array}{*{20}c}p \\i \\\end{array} )$ for |i| = p. When g(x,t)=(1−x.t)^−k (x.t = scalar product of × and t in IR^k), we get rational functions whose singularities are hyperplanes in IR^k. Some properties of these PTA are given together with some computational remarks and an example.