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Discrete-time analysis for the integrable discrete Toda equations and the discrete Lotka-Volterra system

The discrete autonomous/non-autonomous Toda equations and the discrete Lotka-Volterra system are important integrable discrete systems in fields such as mathematical physics, mathematical biology and statistical physics. They also have applications to numerical linear algebra. In this paper, we first simultaneously obtain their general solutions. Then, we show the asymptotic behavior of the solutions for any initial values as the discrete-time variables go to infinity. Our two main techniques for understanding the distinct integrable systems are to introduce two types of discrete-time variables and to examine properties of a restricted infinite sequence, its associated determinants and polynomials.