Power-series unification and reversion

In analysis, it is sometimes necessary to unite a pair of power series into a single power series. If x = x(z) = Σ_ja_jz^j and y = y(z) = Σ_jb_jz^j are given power series, then by eliminating the common parameter z, the power-series unification is obtained: y = y(x) = Σ_kc_kx^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_kd_kx^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.