Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials

High-order non symmetric difference operators with complex coefficients are considered. The correspondence between dynamics of the coefficients of the operator defined by a Lax pair and its resolvent function is established. The method of investigation is based on the analysis of the moments for the operator. The solution of a discrete dynamical system is studied. We give explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for the integrable systems

Coherent pairs of linear functionals on the unit circle

In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle: it is established that if ([mu]0,[mu]1) is a coherent pair of measures on the unit circle, then [mu]0 is a semi-classical measure. Moreover, we obtain that the linear functional associated with [mu]1 is a specific rational transformation of the linear functional corresponding to [mu]0. Some examples are given.http://www.sciencedirect.com/science/article/B6WH7-4S2MJ0D-1/1/61050bb3811832f373ff40a48b7461d

Lax-type pairs in the theory of bivariate orthogonal polynomials

Sequences of bivariate orthogonal polynomials written as vector polynomials of increasing size satisfy a couple of three term relations with matrix coefficients. In this work, introducing a time-dependent parameter, we analyse a Lax-type pair system for the coefficients of the three term relations. We also deduce several characterizations relating the Lax-type pair, the shape of the weight, Stieltjes function, moments, a differential equation for the weight, and the bidimensional Toda-type systems

The Darboux transformation and the complex Toda lattice

Abstract It is well known that each solution of the Toda lattice can be represented by a tridiagonal matrix J(t). Under certain restrictions, it is possible to obtain some new solution by using the Darboux transformation of J(t) − CI. Our goal is the extension of this fact, which is known for the real lattice, to high order complex Toda lattices as well as to the bi-infinite Toda lattice. In this latter case, we use the factorization LU for block-tridiagonal matrices. The Toda lattice We study the construction of some solutions { from another given solution {α n (t) , λ n (t)} , n ∈ Z. We consider: 1. the semi-infinite problem: S = N, λ 1 = 0, 2. the infinite problem: In [6] the semi-infinite complex problem was analyzed. In the real, infinite case, sufficient conditions for the existence of a new solution were given in The problem: obtain a similar result to the complex infinite Toda lattice. The generalized Toda lattice In a more general way, when S = N we consider the generalized Toda where we denote by J i,j (t) (respectively J p i,j (t)) the entry in the (i+1)-row and (j + 1)-column of matrix J(t) (respectively (J(t)) The generalized Toda lattice admits a Lax pair representation, i.e. a formulation in terms of the commutator of two operators, J(t) = [J(t), K(t)] = J(t)K(t) − K(t)J(t) , where we prove the existence of is another solution of (2), and Γ(t) is a solution of the Volterra lattice: 3 Relation between the generalized Toda lattice and some polynomials The matrix J(t) t defines the sequence of polynomials given by The main tools in the proof of [2, Th. 1.3]: a. We have established the dynamic behavior of P n (t, z), where C ∈ C verifies (3). The sequence Q (C) n (t, C) satisfies a threeterm recurrence relation whose coefficients define the new generalized solution J(t) = J(t, C) The new solutions and the Darboux transformation If we define and C ∈ C verifies (3), then there exist The new solution is defined by the Darboux transformation of J (1) (t) − CI, this is, The infinite Toda lattice Let us consider (1) with S = Z and take the infinite matrix The infinite Toda lattice admits also a Lax pair representation. However, in this case it is not possible to use directly the sequences of polynomials associated to J. , n ∈ N , it is possible to change the infinite recurrence relation to a semi-infinite recurrence relation, where E m , V m , m ∈ N , are 2 × 2-finite matrices. In this way, we can study the infinite case as a semi-infinite vectorial case. The vectors R n are not polynomials, but we can prove where the sequence {C n } of 2 × 2 matrices verifies and for each i = 1, 2, 3, 4, c ni is a polynomial in z , deg c ni ≤ n − 1. Taking I −1 := 1 0 0 −1 , W n := I −1 V n , n ∈ N , we can shoẇ , n = 2, 3, . . . This is, {W n , E n } is a solution of a semi-infinite matricial Toda lattice, like (1). The infinite Toda lattice and the Darboux transformation We defin

Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

In this work we characterize a high-order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to study the solution of this dynamical system, we give explicit expressions for the Weyl function, generalized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high-order Toda lattice. We also present a Lax-type theorem for the point spectrum of the Jacobi operator associated with a Toda-type lattic

Second-order differential equations in the Laguerre-Hahn class

Laguerre–Hahn families on the real line are characterized in terms of second-order differential equations with matrix coefficients for vectors involving the orthogonal polynomials and their associated polynomials, as well as in terms of second-order differential equation for the functions of the second kind. Some characterizations of the classical families are derived

Coherent pairs of linear functionals on the unit circle

In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle. Some examples are given

Riemann-Hilbert problem associated with Angelesco systems

Angelesco systems of measures with Jacobi-type weights are considered. For such systems, strong asymptotics for the related multiple orthogonal polynomials are found as well as the Szego-type functions. In the procedure, an approach from the Riemann Hilbert problem plays a fundamental role

Asymptotics of orthogonal polynomials for a weight with a jump on [−1,1]

We consider the orthogonal polynomials on [-1, 1] with respect to the weight w(c)(x) = h(x)(1 - x)(alpha) (1+ x)beta Xi(c)(x), alpha, beta > -1, where h is real analytic and strictly positive on [-1, 1] and Xi(c) is a step-like function: Xi(c)(x) = 1 for x is an element of [-1, 0) and Xi(c) (x) = c(2), c > 0, for x is an element of [0, 1]. We obtain strong uniform asymptotics of the monic orthogonal polynomials in C, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as n -> infinity. In particular, we prove for w(c) a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel-Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit clock behavior. For the asymptotic analysis we use the steepest descent method of Deift and Zhou applied to the noncommutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The local analysis at x = 0 is carried out in terms of confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest.Junta de Andalucía-Spain- FQM-229 and P06- FQM-01735.Ministry of Science and Innovation of Spain - MTM2008-06689-C02-01FCT -SFRH/BD/29731/200