Solution of systems with Toeplitz matrices generated by rational functions

We consider Toeplitz matrices T_n = (t_i−j)ⁿ_i,j=0, where Σ^∞_−∞t_jz^j is a formal Laurent series of a rational function R(z). A criterion is given for T_n to be invertible, in terms of the nonvanishing of a determinant D_n involving the zeros of R(z), and of order and form independent of n; i.e., n enters into D_n as a parameter, and not so as to complicate D_n as n increases. Explicit formulas involving similar determinants are given for the solution of the system T_nX = Y in the case where T_n is invertible. Formulas are also given for T⁻¹_n in the case where T_n−1 and T_n are both convertible Suggestions concerning possible computational procedures based on the results are included.