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Yoshifumi KATO* and Kazuhiko AOMOTO**§ 0. IntroductionIt is well-known that the spectral theory of a Jacobi matrix isessentially equivalent to the theory of orthogonal polynomials withrespect to its spectral density on a real interval. This is also relatedto the Fade approximation of a Stieltjes integral by rational functionswhich gives us the inverse spectral theory of Jacobi matrices (See[19] and [20]).On the other hand several people have shown that the equationsof Toda lattices are described by Lax type formalism of Jacobi ma-trices and related to its spectral densities (See [2]

Jacobi-Perron algorithms, bi-orthogonal polynomials and inverse scattering problems