Partial differential approximants for multivariable power series. II. Invariance properties

Abstract

The invariance properties of partial differential approximants for power series in two (or more) variables are investigated. Certain classes of approximants are shown to exhibit desirable properties such as : ( a ) Eulerian invariance under the change of variables x => ̅x = Ax /(1+ Bx ) and/or y => ̅y = Cy / (1+ Dy ); ( b ) ‘rotational’ invariance under homo­geneous linear transformations of x and y ; ( c ) covariance under exponentiation of the original series. Similar results are demonstrated for constrained, multipoint, and higher order differential approximants. Specializing to functions of one variable provides extensions of known invariance properties of ordinary Padé approximants and inhomogeneous differential approximants.