We study the asymptotic behaviour of orthogonal polynomials inside the unit circle for a subclass of measures that satisfy Szegő's condition. We give a connection between such behavior and a Montessus de Ballore type theorem for Szegő-Padé rational approximants of the corresponding Szegő function.

Ratio asymptotics for orthogonal polynomials on the unit circle is characterized in terms of the existence of lim n |Φ n (0)| and lim, where {Φ n (0)} n≥0 denotes the sequence of reflection coefficients. The limit periodic case, that is when these limits exist for n = j mod k , j = 1, . . . , k , is also considered.

Given a solution of a high order Toda lattice we construct a one parameter family of new solutions. In our method, we use a set of Bäcklund transformations in such a way that each new generalized Toda solution is related to a generalized Volterra solution.

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