Power-series unification and reversion

Abstract

In analysis, it is sometimes necessary to unite a pair of power series into a single power series. If x = x( z) = Σ j a j z j and y = y( z) = Σ j b j z j are given power series, then by eliminating the common parameter z, the power-series unification is obtained: y = y( x) = Σ k c k x k , where the coefficients c k are to be determined in terms of the given power-series coefficients a j and b j . In a special case that y = z, the power-series reversion is obtained: z = z( x) = Σ k d k x k , where the coefficients d k are to be expressed in terms of the original power-series coefficients a j . In this paper, explicit and recurrent formulas for the desired coefficients are derived. A simple technique of matrix formulation is developed for simplicity of computation. Finally, a complete computer program with a typical example is presented.