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 Padé approximants; (iv) their definition is invariant under the group of transformations $x = Au/(1 - Bu),y = Av/(1 - Cv)$ with $A \ne 0$; (v) an approximant formed from the reciprocal series is the reciprocal of the corresponding original approximant. Possible variations, extensions and generalisations of these results are discussed.