15 pages.-- ArXiv pre-print available at: http://arxiv.org/abs/0911.0941 Submitted to: Inverse Problems We consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal polynomials are studied. In particular for these polynomials we consider the recursion relations, and for rank one weights its relation with multiple orthogonal polynomials of mixed type with a type II normalization and the corresponding link with a Riemann--Hilbert problem. MM thanks economical support from the Spanish Ministerio de Ciencia e Innovación, research project FIS2008-00200 and UF thanks economical support from the Spanish Ministerio de Ciencia e Innovación research projects MTM2006-13000-C03-02 and MTM2007-62945 and from Comunidad de Madrid/Universidad Carlos III de Madrid project CCG07-UC3M/ESP-3339. No publicado
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"One can also consider the bigraded extension of the extended QTH which might be included in our future work. The multicomponent 2D Toda hierarchy was considered from the point of view of the Gauss–Borel factorization problem, non-intersecting Brownian motions and matrix Riemann-Hilbert problem    . In fact the http://dx.doi.org/10.1016/j.chaos.2015.03. "
[Show abstract][Hide abstract] ABSTRACT: In this paper, we construct the Sato theory including the Hirota bilinear equations and tau function of a new q-deformed Toda hierarchy (QTH). Meanwhile the Block type additional symmetry and bi-Hamiltonian structure of this hierarchy are given. From Hamiltonian tau symmetry, we give another definition of tau function of this hierarchy. Afterwards, we extend the q-Toda hierarchy to an extended q-Toda hierarchy (EQTH) which satisfy a generalized Hirota quadratic equation in terms of generalized vertex operators. The Hirota quadratic equation might have further application in Gromov–Witten theory. The corresponding Sato theory including multi-fold Darboux transformations of this extended hierarchy is also constructed. At last, we construct the multicomponent extension of the q-Toda hierarchy and show the integrability including its bi-Hamiltonian structure, tau symmetry and conserved densities.
[Show abstract][Hide abstract] ABSTRACT: Multiple orthogonality is considered in the realm of a Gauss--Borel
factorization problem for a semi-infinite moment matrix. Perfect combinations
of weights and a finite Borel measure are constructed in terms of M-Nikishin
systems. These perfect combinations ensure that the problem of mixed multiple
orthogonality has a unique solution, that can be obtained from the solution of
a Gauss--Borel factorization problem for a semi-infinite matrix, which plays
the role of a moment matrix. This leads to sequences of multiple orthogonal
polynomials, their duals and second kind functions. It also gives the
corresponding linear forms that are bi-orthogonal to the dual linear forms.
Expressions for these objects in terms of determinants from the moment matrix
are given, recursion relations are found, which imply a multi-diagonal Jacobi
type matrix with snake shape, and results like the ABC theorem or the
Christoffel--Darboux formula are re-derived in this context (using the
factorization problem and the generalized Hankel symmetry of the moment
matrix). The connection between this description of multiple orthogonality and
the multi-component 2D Toda hierarchy, which can be also understood and studied
through a Gauss--Borel factorization problem, is discussed. Deformations of the
weights, natural for M-Nikishin systems, are considered and the correspondence
with solutions to the integrable hierarchy, represented as a collection of Lax
equations, is explored. Corresponding Lax and Zakharov--Shabat matrices as well
as wave functions and their adjoints are determined. The construction of
discrete flows is discussed in terms of Miwa transformations which involve
Darboux transformations for the multiple orthogonality conditions. The bilinear
equations are derived and the $\tau$-function representation of the multiple
orthogonality is given.
Full-text · Article · Apr 2010 · Advances in Mathematics
[Show abstract][Hide abstract] ABSTRACT: We give a Riemann-Hilbert approach to the theory of matrix orthogonal
polynomials. We will focus on the algebraic aspects of the problem, obtaining
difference and differential relations satisfied by the corresponding orthogonal
polynomials. We will show that in the matrix case there is some extra freedom
that allows us to obtain a family of ladder operators, some of them of 0-th
order, something that is not possible in the scalar case. The combination of
the ladder operators will lead to a family of second-order differential
equations satisfied by the orthogonal polynomials, some of them of 0-th and
first order, something also impossible in the scalar setting. This shows that
the differential properties in the matrix case are much more complicated than
in the scalar situation. We will study several examples given in the last years
as well as others not considered so far.
Full-text · Article · Jun 2011 · Symmetry Integrability and Geometry Methods and Applications