Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices J. Math. Phys. 48, 113506 (2007) Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices x:Z N →R M , NрM , whose elementary quadrilaterals are planar. Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogs of the Laplace, Combescure, Lévy, radial, and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between ''transformations'' and ''discretizations'' is also investigated for quadrilateral lattices. We finally interpret these results within the ‫ץ‬ formalism.

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...

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Given a matrix polynomial W(x), matrix bi-orthogonal polynomials with respect to the sesquilinear formwhere µ(x) is a matrix of Borel measures supported in some infinite subset of the real line, are considered. Connection formulas between the sequences of matrix bi-orthogonal polynomials with respect to •, • W and matrix polynomials orthogonal with respect to µ(x) are presented. In particular, for the case of nonsingular leading coefficients of the perturbation matrix polynomial W(x) we present a generalization of the Christoffel formula constructed in terms of the Jordan chains of W(x). For perturbations with a singular leading coefficient several examples by Durán et al are revisited. Finally, we extend these results to the non-Abelian 2D Toda lattice hierarchy. CONTENTS 1991 Mathematics Subject Classification. 42C05,15A23. Key words and phrases. Matrix orthogonal polynomials, Block Jacobi matrices, Darboux-Christoffel transformation, Block Cholesky decomposition, Block LU decomposition, quasi-determinants, non-Abelian Toda hierarchy. GA thanks financial support from the Universidad Complutense de Madrid Program "Ayudas para Becas y Contratos Complutenses Predoctorales en España 2011".MM & FM thanks financial support from the Spanish "Ministerio de Economía y Competitividad" research project MTM2012-36732-C03-01, Ortogonalidad y aproximación; teoría y aplicaciones.