Orthogonal polynomials and the finite Toda lattice

The choice of a finitely supported distribution is viewed as a degenerate bilinear form on the polynomials in the spectral parameter z and the matrix representing multiplication by z in terms of an orthogonal basis is constructed. It is then shown that the same induced time dependence for finitely supported distributions which gives the ith KP flow under the dual isomorphism induces the ith flow of the Toda hierarchy on the matrix. The corresponding solution is an N particle, finite, nonperiodic Toda solution where N is the cardinality of the support of c plus the sum of the orders of the highest derivative taken at each point. (C) 1997 American Institute of Physics.