39 research outputs found
The Darboux transformation and the complex Toda lattice
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
Quadratic decomposition of bivariate orthogonal polynomials
We describe the relation between the systems of bivariate
orthogonal polynomial associated to a symmetric weight function and
associated to some particular Christoffel modifications of the quadratic
decomposition of the original weight. We analyze the construction of a
symmetric bivariate orthogonal polynomial sequence from a given one,
orthogonal to a weight function defined on the first quadrant of the
plane. In this description, a sort of Backlund type matrix transformations
for the involved three term matrix coefficients plays an important
role. Finally, we take as a case study relations between the classical
orthogonal polynomials defined on the ball and those on the simplex.publishe
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
Bidiagonal factorization of the recurrence matrix for the Hahn multiple orthogonal polynomials
This paper explores a factorization using bidiagonal matrices of the
recurrence matrix of Hahn multiple orthogonal polynomials. The factorization is
expressed in terms of ratios involving the generalized hypergeometric function
and is proven using recently discovered contiguous relations.
Moreover, employing the multiple Askey scheme, a bidiagonal factorization is
derived for the Hahn descendants, including Jacobi-Pi\~neiro, multiple Meixner
(kinds I and II), multiple Laguerre (kinds I and II), multiple Kravchuk, and
multiple Charlier, all represented in terms of hypergeometric functions. For
the cases of multiple Hahn, Jacobi-Pi\~neiro, Meixner of kind II, and Laguerre
of kind I, where there exists a region where the recurrence matrix is
nonnegative, subregions are identified where the bidiagonal factorization
becomes a positive bidiagonal factorization.Comment: 14 pages, 2 figure
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
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
Construction and implementation of asymptotic expansions for Jacobi-type orthogonal polynomials
We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to ∞. These are defined on the interval [−1, 1] with weight function: w(x)=(1−x)α(1+x)βh(x),α,β>−1 and h(x) a real, analytic and strictly positive function on [−1, 1]. This information is available in the work of Kuijlaars et al. (Adv. Math. 188, 337–398 2004), where the authors use the Riemann–Hilbert formulation and the Deift–Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in O(n) operations, rather than O(n2) based on the recurrence relation
Characterizations of Δ-Volterra lattice: a symmetric orthogonal polynomials interpretation
In this paper we introduce the Δ-Volterra lattice which is interpreted in terms of symmetric orthogonal polynomials. It is shown that the measure of orthogonality associated with these systems of orthogonal polynomials evolves in t like (1+x2)1−tμ(x)(1+x2)1−tμ(x) where μ is a given positive Borel measure. Moreover, the Δ-Volterra lattice is related to the Δ-Toda lattice from Miura or Bäcklund transformations. The main ingredients are orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Δ and the characterization of the point spectrum of a Jacobian operator that satisfies a Δ-Volterra equation (Lax type theorem). We also provide an explicit example of solutions of Δ-Volterra and Δ-Toda lattices, and connect this example with the results presented in the paper
Microbial diversity of the traditional Iranian cheeses Lighvan and Koozeh, as revealed by polyphasic culturing and culture-independent approaches
Characterizations of Δ-Volterra lattice: A symmetric orthogonal polynomials interpretation
In this paper we introduce the Δ-Volterra lattice which is interpreted in terms of symmetric orthogonal polynomials. It is shown that the measure of orthogonality associated with these systems of orthogonal polynomials evolves in t like (1+x^2)^1−t μ(x) where μ is a given positive Borel measure. Moreover, the Δ-Volterra lattice is related to the Δ-Toda lattice from Miura or Bäcklund transformations. The main ingredients are orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Δ and the characterization of the point spectrum of a Jacobian operator that satisfies a Δ-Volterra equation (Lax type theorem). We also provide an explicit example of solutions of Δ-Volterra and Δ-Toda lattices, and connect this example with the results presented in the paper