Rational approximants defined from double power series

Rational approximants are defined from double power series in variables x and y, and it is shown that these approximants have the following properties: (i) they possess symmetry between x and y; (ii) they are in general unique; (iii) if x = 0 or y = 0, they reduce to diagonal Pad6 approximants; (iv) their definition is invariant under the group of transformations x = Au/(l - Bu), y = Av/(l - Cv) with A 5 0; (v) an approximant formed from the reciprocal series is the reciprocal of the corresponding original approx- imant. Possible variations, extensions and generalizations of these results are discussed.