The Generation of Chisholm Rational Polynomial Approximants to Power Series in Two Variables

Abstract

The generation of successive Chisholm rational polynomial approximants f_m/_m of f(x, y), a power series in two variables, is discussed. A necessary and sufficient condition for the non-degeneracy of f_m/_m is given. It is shown that the non-degeneracy of the diagonal Padé approximants of order m in each variable separately is a necessary condition for the non-degeneracy of f_m/_m. In the case of a symmetric function, it is proved that the Chisholm approximant f_m/_m is symmetric and non-degenerate if and only if all the diagonal Padé approximants of order up to m in one variable are non-degenerate. The generation of successive Chisholm approximants to symmetric functions is also considered. The computational scheme, called the prong method, extends to cover the computation of Chisholm approximants in N-variables (Chisholm & McEwan, 1974).