Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation

Complex high order Toda and Volterra lattices1

“…be a formal power series at z = ∞ and let f n (z) = P (1) n (z)/P n (z) be the n-diagonal Padé approximant of W (z) , n ∈ N. It is well known that the sequences of polynomials {P n (z)} and {P (1) n (z)} verify the same three-term recurrence relation, whose coefficients define a tridiagonal matrix J. Moreover, we have…”

“…Assume that {J(t)} , t ∈ R, is a generalized Toda solution. The system (5) was described in [1] as representation in Lax paiṙ…”

“…COMPLEX HIGH ORDER TODA AND VOLTERRA LATTICES 3 Note that, for p = 1 and n ∈ N, (5) and (7) coincide, respectively, with (1) and (3).…”

“…Moreover, for each solution of the Toda lattice, a family of new solutions was obtained for this lattice, each one associated with a different solution of the Volterra lattice (3) by a Bäcklund transformation like (2a)-(2b), where it is possible that C = 0. In this paper, we generalize the analysis given in [7] and [3] to the kind of Toda and Volterra lattices studied in [1] and [2]. For this purpose we consider the family {J(t)} , t ∈ R , of tridiagonal infinite matrices given by…”

Jacobi Operators and Completely Integrable Nonlinear Lattices

“…In case of the Toda equation on the half line a different approach using the spectral measure of H+ is introduced and extended in [28] - [32] (see also [15]). It turns out that the time evolution of the spectral measure can be computed explicitly as the Freud transform of the initial measure.…”