Application of the discrete integrable systems on the semi-infinite lattice to the system of the bi-orthogonal polynomials

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The purpose of this research is to construct the theory of the spectral transformations of the bi-orthogonal polynomials and the continued fractions from the point of the view of the discrete integrable systems on the semi-infinite lattice. We mainly study the relationship between the bi-orthogonal polynomials which have the three-term relations and the Toda-type discrete integrable systems. We give main results:・The R1 type and the R2 type discrete integrable systems associated with the generalized eigen value problemsWe study the bilinear equations of the R1 type and the R2 type discrete integrable systems. Then we clarify the relationship with the Toda type discrete systems including the nonautonomous discrete Toda equation and the relativistic discrete Toda equation, and derive their Backlund transformations.・The discrete integrable system associated with the rational interpolation functionsWe derive a discrete integrable system associated with Frobenius-Stickelberger's pioneering research on the rational interpolation functions and Thiele's theory on the Pade approximations. Moreover we give the bilinear equations of this integrable system and show that these bilinear equations are derived from the discrete KP equation and the two-dimensional discrete Toda equation. A generalised Lotka-Volterra equation and a generalized epsilon algorithm are presented.・The nonautonomous discrete Toda equatioFrom the research on the bilinear equation of the R1 chain and the R2 chain, we have new results on the nonautonomous discrete Toda equation. The Darboux transformations of the nonautonomous discrete Toda equation are derived from the Darboux transformation of the discrete KP equation and the two-dimensional discrete Toda equation.