Rational approximants defined from power series in N variables

Abstract

Rational approximants in N variables z r ( r = 1, 2, ..., N ) are defined from power series in these variables. They are generalizations of the two-variable approximants defined recently, and have the properties: ( i ) they possess symmetry between the N variables; ( ii ) they exist and are in general unique: ( iii ) if any k (< N ) variables are equated to zero, the approximants reduce to approximants in ( N — k ) variables formed from the corresponding reduced power series; in particular, if k = N — 1, they reduce to diagonal Padé approximants; ( iv ) their definition is invariant under the group of transformations z r = Aω r /(1 — B r ω r ) provided A ≠ 0, for all r = 1, 2, ..., N ; this group of homographic transformations preserves the origin z r = 0 ( r = 1, 2, ..., N ) but does not allow changes in the relative scales of the variables z r ; ( v ) an approximant formed from the reciprocal series is the reciprocal of the corresponding original approximant; ( vi ) if the series is the product of two power series in mutually exclusive sets of variables, the approximant is the product of the corresponding approximants formed from the two series; ( vii ) if the series is the sum of two power series in mutually exclusive sets of variables, the approximant is the sum of the corresponding approximants formed from the two series. Rigorous proofs of the properties ( i ) and ( iii ) to ( vii ) are given, based on complex variable methods. We discuss the possible use of the approximants in practical problems, especially in theoretical physics, and their possible importance in the theory of functions of several variables.