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. further developments on the Gauss-Borel factorization and multi-component 2D Toda hierarchy see [7] and [29]. This motivated our initial research in relation with this paper; i.e., the construction of an appropriate Gauss-Borel factorization in the group of semi-infinite matrices leading to multiple orthogonality and integrability in a simultaneous manner. The main advantage of this approach lies in the application of different techniques based on the factorization problem used frequently in the theory of integrable systems. The key finding of this paper is, therefore, the characterization of a semi-infinite moment matrix whose Gauss-Borel factorization leads directly to multiple orthogonality. This makes sense when factorization can be performed, which is the case for perfect combinations (µ, w 1 , w 2 ), which allows us to consider some sets of multiple orthogonal polynomials (called ladders) very much in the same manner as in the (non multiple) orthogonal polynomial setting. The Gauss-Borel factorization of this moment matrix leads, when one takes into account the Hankel type symmetry of the moment matrix, to results like: 1. Recursion relations, 2. ABC theorems and 3. Christoffel-Darboux formulas. Th...

We consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hänkel 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.

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Abstract. We discuss the theoretical convergence and numerical evaluation of simultaneous interpolation quadrature formulas which are exact for rational functions. Basically, the problem consists in integrating a single function with respect to different measures by using a common set of quadrature nodes. Given a multi-index n, the nodes of the integration rule are the zeros of the multiorthogonal Hermite-Padé polynomial with respect to (S, α, n), where S is a collection of measures, and α is a polynomial which modifies S. The theory is based on the connection between Gausslike simultaneous quadrature formulas of rational type and multipoint Hermite-Padé approximation. As for the numerical treatment we present a procedure based on the technique of modifying the integrand by means of a change of variable when the integrand has real poles close to the integration interval. The output of some tests show the power of this approach when compared to other ones.Key words. simultaneous integration rule, Gauss-like rational quadrature formula, meromorphic integrand AMS subject classifications. Primary 41A55. Secondary 41A28, 65D321. Introduction. Simultaneous integration is a problem which arises in computer graphics to determinate the color of light which emanates from a given point on a surface toward the viewer (see Borges [4] and the bibliography therein). In an abstract setting, this problem was earlier studied by Nikishin [19] in connection with Hermite-Padé approximation. Simultaneous integration means that we are integrating a single function f : I → R, with respect to m distinct measures ds 1 , ..., ds m on I, respectively.A reference index Φ = (Ov + 1)/N u (performance ratio) is considered in [4] to state the efficiency of the procedure. Here Ov is the overall degree of exactness and N u is the number of integrand evaluations. Borges [4] remarked that when m ≥ 3 the use of the m corresponding Gaussian rules of polynomial type yields Φ < 1, which indicates a low performance. Instead he suggests quadrature rules whose nodes are the zeros of multi-orthogonal polynomials for which Φ > 1 holds.Geometric rate of convergence has been proved by the authors in [8] for polynomials methods when the integrand is analytic in a neighborhood of I. Nevertheless, instability shows up when the corresponding numerical method is applied to meromorphic integrands with poles close to I. Several methods to integrate functions with singularities on or near the integration interval have been developed in the past few years.An interesting approach, whose starting point seems to be [13], is that based on the use of rational Gaussian integration rules connected with multipoint Padé

Abstract. K. Mahler introduced the concept of perfect systems in the general theory he developed for the simultaneous Hermite-Padé approximation of analytic functions. We prove that Nikishin systems are perfect providing, by far, the largest class of systems of functions for which this important property holds. As consequences, in the context of Nikishin systems, we obtain: an extension of Markov's theorem to simultaneous Hermite-Padé approximation, a general result on the convergence of simultaneous quadrature rules of Gauss-Jacobi type, the logarithmic asymptotics of general sequences of multiple orthogonal polynomials, and an extension of the Denisov-Rakhmanov theorem for the ratio asymptotics of mixed type multiple orthogonal polynomials.