Studies on the orthogonal polynomials, the continued fractions and the discrete integrable systems

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In the last decades, discrete integrable system has been getting a lot of attention from the viewpoints of difference scheme and algorithm. Already there have been many discussions about relations between continuous-time integrable system and orthogonal polynomials. There are only a few studies on the relations between full-discretised (discrete time and space) systems and orthogonal polynomials fully clarified, such as the discrete integrable system (Toda, Lotka-Volterra) and classical orthogonal polynomials by Spiridonov and Zhedanov. Hence the purpose of this studies is to clarify the relations between the discrete integrable system and the orthogonal polynomials and to develop the Numerical algorithms related to the orthogonality. At first, we pick up the discrete hungry Lotka-Volterra equation and the coupled KP equation. We probe them by means of Hirota's tau-function and develop the relations.