The ɛ-algorithm and multivariate Padé-approximants

In the univariate case the ɛ-algorithm of Wynn is closely related to the Padé-table in the following sense: if we apply the ɛ-algorithm to the partial sums of the power series $$f(x) = \sum\limits_{i = 0}^\infty {c_i x^i } $$ then ε 2 − is the (l, m) Padé-approximant tof(x) wherel is the degree of the numerator andm is the degree of the denominator [1 pp. 66–68]. Several generalizations of the ɛ-algorithm exist but without any connection with a theory of Padé-approximants. Also several definitions of the Padé-approximant to a multivariate function exist, but up till now without any connection with the ɛ-algorithm. In this paper, we see that the multivariate Padé-approximants introduced in [3], satisfy the same property as the univariate Padé-approximants: if we apply the ɛ-algorithm to the partial sums of the power series $$f\left( {x_1 ,...,x_n } \right) = \sum\limits_{i_1 + ... + i_n = 0}^\infty {c_{i_1 ...i_n } x_1^{i_1 } ...x_n^{i_n } } $$ then ε 2 (−) is the (l, m) multivariate Padé-approximant tof(x 1, ...,x n ).