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A discrete-time integrable system, called the DSF chain, is derived from the compatibility condition of spectral equations for biorthogonal polynomials associated with the discrete Schur flow. The DSF algorithm for computing eigenvalues of a type of matrix eigenvalue problem is designed. A discrete analogue of the Miura transformation between the discrete Schur flow and the qd-algorithm is constructed. Two special solutions to the DSF chain related to q -special functions are presented.

The integrable chain of RII type by Spiridonov–Zhedanov is studied by using the bilinear method. Bilinear equations of the system are derived by applying appropriate-dependent variable transformations. A particular solution on a semi-infinite lattice is explicitly given in terms of the Casorati-type determinants. It is shown that the RII chain and the Toda-type integrable systems are connected by Backlund transformations.

A determinant structure of the RI type discrete integrable system by Vinet–Zhedanov on a semi-infinite lattice is studied using the bilinear method. Bilinear equations of the RI type discrete integrable system are derived by applying appropriate dependent variable transformations. It is shown that a particular solution for the bilinear equations on a semi-infinite lattice is given in terms of Casorati-type determinants. It is also discussed how the RI type discrete integrable system relates to the discrete relativistic Toda lattice.

An integrable discretization of the modified Korteweg-de Vries (mKdV) equation is considered. The mKdV algorithm is designed in terms of the resulting discrete mKdV equation. It is shown that a class of algebraic equations can be solved numerically in terms of the algorithm.

The finite Toda molecule over finite fields is introduced whose dynamics are completely classified by conserved quantities. The trajectories blow up in a finite time or are periodic. A BCH-Goppa decoding algorithm is designed. The number of Toda particles is congruent with the maximal number of errors to be decoded.

A relationship between orthogonal rational functions and discrete integrable systems is studied by an approach based on Schur-type symmetric functions. A system of orthogonal rational functions is constructed using a multiparameter deformation of the Schur functions. Spectral equations for the orthogonal rational functions are derived by using properties of the Schur-type symmetric functions. The compatibility condition of the spectral equations induces a discrete dressing chain which is a Toda-type discrete integrable system describing dressing transformations for orthogonal rational functions.

It is shown that a nonautonomous version of the discrete KP equation has particular solutions described by Newton-type interpolation.

It is shown that the RII chain has a solution described by Schur type functions indexed by rectangular partitions.

A max‐plus version of the Stieltjes function is studied using the ultra‐discrete limit. Several features including interlacing properties on its zeros and poles are presented.

We construct $N$-soliton solution for the non-autonomous discrete-time Toda lattice equation, which is a generalization of the discrete-time Toda equation such that the lattice interval with respect to time is an arbitrary function in time., 8 pages. Title has been changed. Some comments and discussions have been added

A class of factorial-type Schur functions is introduced from a viewpoint of a Newton-type interpolation. Their basic properties are presented. It is shown that the factorial-type Schur functions can be used to describe a particular solution to a nonautonomous version of the discrete KP equation.

Generalized factorial Schur functions are introduced through a Newton-type interpolation. They are applied to construct a class of biorthogonal rational functions and a special solution to its spectral transformation chain, the RII chain. A two-parameter deformation of the Schur-type functions is also discussed.

A τ-function on the inhomogeneous lattice is introduced by a deformation of that of the Toda lattice hierarchy. An associated discrete-time integrable system is constructed in Sato and Lax formalisms. It is shown that a one-dimensional reduction of the system gives a Backlund transformation of the classical isotropic Heisenberg spin chain, which is equivalent to the RII-chain proposed by Spiridonov and Zhedanov.