The spectral transformation technique for symmetric R_{II} polynomials is developed. Use of this technique reveals that the nonautonomous discrete modified KdV (nd-mKdV) lattice is directly connected with the R_{II} chain. Hankel determinant solution
s to the semi-infinite nd-mKdV lattice are also presented.
A generalized eigenvalue algorithm for tridiagonal matrix pencils is presented. The algorithm appears as the time evolution equation of a nonautonomous discrete integrable system associated with a polynomial sequence which has some orthogonality on t
he support set of the zeros of the characteristic polynomial for a tridiagonal matrix pencil. The convergence of the algorithm is discussed by using the solution to the initial value problem for the corresponding discrete integrable system.
A connection between the finite ultradiscrete Toda lattice and the box-ball system is extended to the case where each box has own capacity and a carrier has a capacity parameter depending on time. In order to consider this connection, new carrier rul
es size limit for solitons and recovery of balls, and a concept expansion map are introduced. A particular solution to the extended system of a special case is also presented.
