The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and Riemann–Hilbert problems (original) (raw)
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Advances in Mathematics, 2010
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 τ-function representation of the multiple orthogonality is given.
Properties of matrix orthogonal polynomials via their Riemann–Hilbert characterization
2011
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.
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Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice
Advances in Applied Mathematics, 2019
A general family of matrix valued Hermite type orthogonal polynomials is introduced and studied in detail by deriving Pearson equations for the weight and matrix valued differential equations for these matrix polynomials. This is used to derive Rodrigues formulas, explicit formulas for the squared norm and to give an explicit expression of the matrix entries as well to derive a connection formula for the matrix polynomials of Hermite type. We derive matrix valued analogues of Burchnall formulas in operational form as well explicit expansions for the matrix valued Hermite type orthogonal polynomials as well as for previously introduced matrix valued Gegenbauer type orthogonal polynomials. The Burchnall approach gives two descriptions of the matrix valued orthogonal polynomials for the Toda modification of the matrix weight for the Hermite setting. In particular, we obtain a non-trivial solution to the non-abelian Toda lattice equations. − B (α,ν) n−1 consists of a product of diagonal matrices and A *. Since we have T as a diagonal matrix, the commutation shows that [T, A * ] = 0. This means that all diagonal elements of T are equal, and the result follows.
Painlevé V and a Pollaczek–Jacobi type orthogonal polynomials
We study a sequence of polynomials orthogonal with respect to a one-parameter family of weights w(x) := w(x, t) = e −t/x x α (1 − x) β , t ≥ 0, defined for x ∈ [0, 1]. If t = 0, this reduces to a shifted Jacobi weight. Our ladder operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients. For t > 0, the factor e −t/x induces an infinitely strong zero at x = 0. With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a system of difference equations. These, when suitably combined with a pair of Toda-like equations derived from the orthogonality principle, show that the auxiliary quantities are particular Painlevé V and/or allied functions. It is also shown that the logarithmic derivative of the Hankel determinant,