Rational approximants defined from double power series

Abstract

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 x = 0 or y = 0 y = 0 , they reduce to diagonal Padé approximants; (iv) their definition is invariant under the group of transformations x = A u / ( 1 − B u ) , y = A v / ( 1 − C v ) x = Au/(1 - Bu),y = Av/(1 - Cv) with A ≠ 0 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.