Power-series unification and reversion

摘要

In analysis, it is sometimes necessary to unite a pair of power series into a single power series. If x = x(z) = Σjajzj and y = y(z) = Σjbjzj are given power series, then by eliminating the common parameter z, the power-series unification is obtained: y = y(x) = Σkckxk, where the coefficients ck are to be determined in terms of the given power-series coefficients aj and bj. In a special case that y = z, the power-series reversion is obtained: z = z(x) = Σkdkxk, where the coefficients dk are to be expressed in terms of the original power-series coefficients aj. 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.