Partial Differential Approximants for Multivariable Power Series. III. Enumeration of Invariance Transformations

A partial differential approximant (or an ordinary rational approximant) which approximates a function, f(x,y), of two variables may be invariant under certain changes of variable (x,y) -> (bar{x},bar{y}). We prove that, under fairly mild conditions, such invariance is possible for only five classes of variable transformations, namely: the Euler transformations bar{x} = Ax/(1 + Bx) and bar{y} = Cy/(1 + Dy); homogeneous linear transformations of x and y; 'Euler-rotations' bar{x} = (A_1x + A_2y)/(1 + Cx + Dy) and bar{y}=(B_1x + B_2y)/(1 + Cx + Dy); the skew Euler transformations bar{x} = Ay/(1 + By) and bar{y} = Cx/(1 + Dx); and transformations such as bar{x} = Ax and {y} = C(x) + By, where C(x) is a polynomial. This result also holds for ordinary two-variable rotational approximants.