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The main objects of the investigation presented in this paper are branched-continued-fraction representations of ratios of contiguous hypergeometric series and type II multiple orthogonal polynomials on the step-line with respect to linear functionals or measures whose moments are ratios of products of Pochhammer symbols. This is an interesting case study of the recently found connection between multiple orthogonal polynomials and branched continued fractions that gives a clear example of how this connection leads to considerable advances on both topics. We start by obtaining new results about generating polynomials of lattice paths and total positivity of matrices and giving new contributions to the general theory of the connection between multiple orthogonal polynomials and branched continued fractions with emphasis on its application to the analysis of multiple orthogonal polynomials. Then, we construct new branched continued fractions for ratios of contiguous hypergeometric series. We give conditions for positivity of the coefficients of these branched continued fractions and we show that the ratios of products of Pochhammer symbols are generating polynomials of lattice paths for a special case of the branched continued fractions under study. Next, we introduce a family of type II multiple orthogonal polynomials on the step-line associated with those branched continued fractions. We present a formula as terminating hypergeometric series for these polynomials, we study their differential properties, and we find an explicit recurrence relation satisfied by them. Finally, we focus the analysis of the multiple orthogonal polynomials to the cases where the corresponding branched-continued-fraction coefficients are all positive. In those cases, the orthogonality conditions can be written using measures on the positive real line involving Meijer G-functions and we obtain results about the location of the zeros and the asymptotic behaviour of the polynomials. Specialisations of the multiple orthogonal polynomials studied here include the classical Laguerre, Jacobi, and Bessel orthogonal polynomials, multiple orthogonal polynomials with respect to Nikishin systems of two measures involving modified Bessel functions, confluent hypergeometric functions, and Gauss' hypergeometric function, and multiple orthogonal polynomials with respect to Meijer G-functions used to investigate the singular values of products of Ginibre random matrices as well as a r -orthogonal polynomial sequence with constant recurrence coefficients for any positive integer r and particular instances of the Jacobi-Piñeiro polynomials. [ABSTRACT FROM AUTHOR]

Mehler-Heine asymptotics describe the behavior of orthogonal polynomials near the edges of the interval where the orthogonality measure is supported. For Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the hard edge involves Bessel functions Jα. We show that the asymptotic behavior near the endpoint of the interval of (one of) the measures for multiple orthogonal polynomials involves a generalization of the Bessel function. The multiple orthogonal polynomials considered are Jacobi-Angelesco polynomials, Jacobi-Pi˜neiro polynomials, multiple Laguerre polynomials, multiple orthogonal polynomials associated with modified Bessel functions (of the first and second kind) and multiple orthogonal polynomials associated with Meijer G-functions. [ABSTRACT FROM AUTHOR]

In the present paper we prove a Stieltjes type theorem on the convergence of a sequence of rational functions associated with a mixed type Hermite-Padé approximation problem of a Nikishin system of functions and analyze the ratio asymptotic of the corresponding Hermite-Padé polynomials. • Properties for multi-level Hermite-Padé approximants. • Ratio asymptotic theorem for multi-level Hermite-Padé approximants. • Weak asymptotic theorem for multi-level Hermite-Padé approximants. [ABSTRACT FROM AUTHOR]

We study weak asymptotic behaviour of the Christoffel–Darboux kernel on the main diagonal corresponding to multiple orthogonal polynomials. We show that under some hypotheses the weak limit of 1 n K n (x , x) d μ is the same as the limit of the normalized zero counting measure of type II multiple orthogonal polynomials. We also study an extension of Nevai's operators to our context. [ABSTRACT FROM AUTHOR]

We give the asymptotic distribution of the zeros of Jacobi-Piñeiro polynomials and multiple Laguerre polynomials of the first kind. We use the nearest neighbor recurrence relations for these polynomials and a recent result on the ratio asymptotics of multiple orthogonal polynomials. We show how these asymptotic zero distributions are related to the Fuss-Catalan distribution. [ABSTRACT FROM AUTHOR]

Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a ( r+2)-term recurrence relation connecting the type II multiple orthogonal polynomials near the diagonal (the so-called step-line recurrence relation) and there is a system of r recurrence relations connecting the nearest neighbors (the so-called nearest neighbor recurrence relations). In this paper we deal with two problems. First we show how one can obtain the nearest neighbor recurrence coefficients (and in particular the recurrence coefficients of the orthogonal polynomials for each of the defining measures) from the step-line recurrence coefficients. Secondly we show how one can compute the step-line recurrence coefficients from the recurrence coefficients of the orthogonal polynomials of each of the measures defining the multiple orthogonality. [ABSTRACT FROM AUTHOR]

The normal matrix model with a cubic potential is ill-defined and it develops a critical behavior in finite time. We follow the approach of Bleher and Kuijlaars to reformulate the model in terms of orthogonal polynomials with respect to a Hermitian form. This reformulation was shown to capture the essential features of the normal matrix model in the subcritical regime, namely that the zeros of the polynomials tend to a number of segments (the motherbody) inside a domain (the droplet) that attracts the eigenvalues in the normal matrix model. In the present paper we analyze the supercritical regime and we find that the large n behavior is described by the evolution of a spectral curve satisfying the Boutroux condition. The Boutroux condition determines a system of contours Σ 1 , consisting of the motherbody and whiskers sticking out of the domain. We find a second critical behavior at which the original motherbody shrinks to a point at the origin and only the whiskers remain. In the regime before the second criticality we also give strong asymptotics of the orthogonal polynomials by means of a steepest descent analysis of a 3 × 3 matrix valued Riemann–Hilbert problem. It follows that the zeros of the orthogonal polynomials tend to Σ 1 , with the exception of at most three spurious zeros. [ABSTRACT FROM AUTHOR]

In linear space X consider arbitrary norms || ⋅ ||j, j = 1, 2, ..., d, and linear subspaces Mj ⊂ X of dim Mj = nj, j = 1, 2, ..., d, and set n = n1 + ⋯ + nd. Then given any subspace U ⊂ X of dimension n + 1 we shall verify the existence of u ∈ U \ {0} such that for every 1 ≤ j ≤ d the element u is orthogonal to Mj in the || ⋅ ||j norm, that is, ||u||j ≤ || u - m||j, ∀m ∈ Mj, ∀1 ≤ j ≤ d. In case of polynomial approximation with respect to distinct weighted uniform norms, the above extremal element is called a multiple Chebyshev polynomial. It will be shown that for weights whose ratios are monotone there always exists a unique multiple Chebyshev polynomial of maximal degree. [ABSTRACT FROM AUTHOR]

Multiple orthogonal polynomials on the unit circle (MOPUC) were introduced by J. Mínguez and W. Van Assche for the first time in 2008. Some applications were given there and recurrence relations were obtained from a Riemann–Hilbert problem. This paper is a second contribution to this field. We first obtain a determinantal formula for MOPUC (multiple Heine’s formula) and we analyze the concept of normality, from a dynamical point of view and by presenting a first example: the combination of the Lebesgue and Rogers–Szegő measures. Secondly, we deduce recurrence relations for MOPUC without using Riemann–Hilbert analysis, only by considering orthogonality conditions. This new approach allows us to complete the recurrence relations in the situation when the origin is a zero of MOPUC, a case that was not considered before. As a consequence, we give an appropriate definition of multiple Verblunsky coefficients. A multiple version of the well known Szegő recurrence relation is also obtained. Here, the coefficients that appear in the recurrence satisfy certain partial difference equations that are used to present a recursive algorithm for the computation of MOPUC. A discussion on the Riemann–Hilbert approach that also includes the case when the origin is a zero of MOPUC is presented. Some conclusions and open questions are finally mentioned. [ABSTRACT FROM AUTHOR]

We study the convergence of type I Hermite–Padé approximation for a class of meromorphic functions obtained by adding a vector of rational functions with real coefficients to a Nikishin system of functions. [ABSTRACT FROM AUTHOR]

An algebraic function of the third order plays an important role in the problem of asymptotics of Hermite–Padé approximants for two analytic functions with branch points. This algebraic function appears as the Cauchy transform of the limiting measure of the asymptotic distribution of the poles of the approximants. In many cases this algebraic function can be determined by using the given position of the branch points of the functions which are approximated and by the condition that its Abelian integral has purely imaginary periods. In the present paper we obtain a hyperelliptic uniformization of this algebraic function. In the case when each approximated function has only two branch points, the genus of this function can be equal to 0, 1 (elliptic case) or 2 (ultra-elliptic case). We use this uniformization to parametrize the elliptic case. This parametrization allows us to obtain a numerical procedure for finding this elliptic curve and as a result we can describe the limiting measure of the distribution of the poles of the approximants. [ABSTRACT FROM AUTHOR]

Padé approximation has two natural extensions to vector rational approximation through the so-called type I and type II Hermite–Padé approximants. The convergence properties of type II Hermite–Padé approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite–Padé approximants for Nikishin systems of functions. [ABSTRACT FROM AUTHOR]

An alternative expression for the Christoffel–Darboux formula for multiple orthogonal polynomials of mixed type is derived from the LU factorization of the moment matrix of a given measure and two sets of weights. We use the action of the generalized Jacobi matrix J , also responsible for the recurrence relations, on the linear forms and their duals to obtain the result. [ABSTRACT FROM AUTHOR]

We investigate the average characteristic polynomial E[Πi=1N(z–xi)] where the xi's are real random variables drawn from a Biorthogonal Ensemble, i.e. a determinantal point process associated with a bounded finite-rank projection operator. For a subclass of Biorthogonal Ensembles, which contains Orthogonal Polynomial Ensembles and (mixed-type) Multiple Orthogonal Polynomial Ensembles, we provide a sufficient condition for its limiting zero distribution to match with the limiting distribution of the random variables, almost surely, as N goes to infinity. Moreover, such a condition turns out to be sufficient to strengthen the mean convergence to the almost sure one for the moments of the empirical measure associated to the determinantal point process, a fact of independent interest. As an application, we obtain from Voiculescu's theorems the limiting zero distribution for multiple Hermite and multiple Laguerre polynomials, expressed in terms of free convolutions of classical distributions with atomic measures, and then derive explicit algebraic equations for their Cauchy–Stieltjes transform. [ABSTRACT FROM AUTHOR]

We consider multiple orthogonal polynomials associated with the exponential cubic weight e − x 3 over two contours in the complex plane. We study the basic properties of these polynomials, including the Rodrigues formula and nearest-neighbor recurrence relations. It turns out that the recurrence coefficients are related to a discrete Painlevé equation. The asymptotics of the recurrence coefficients, the ratio of the diagonal multiple orthogonal polynomials and the (scaled) zeros of these polynomials are also investigated. [ABSTRACT FROM AUTHOR]

Abstract: We study the asymptotic behavior of multiple Meixner polynomials of first and second kind, respectively [6]. We use an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution. Moreover, analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics. [Copyright &y& Elsevier]

Abstract: Denote by the space of real algebraic polynomials of degree at most and consider a multi-index , of length . Then given the nonnegative weight functions , the polynomial is called a multiple orthogonal polynomial relative to and the weights , if The above orthogonality relations are equivalent to the conditions for the multiple best approximation The existence of multiple orthogonal polynomials easily follows from the solvability of the above linear system. The analogous question for the multiple best approximation, i.e., the existence of an extremal polynomial satisfying poses a more difficult nonlinear problem when . In this paper we shall address this question and verify the existence and uniqueness of multiple orthogonal polynomials under proper conditions. [Copyright &y& Elsevier]

Abstract: The zeros of orthogonal polynomials on the real line are simple, lie in the interior of the convex hull of the support of the measure and the zeros of consecutive orthogonal polynomials interlace. We study a large class of mixed type multiple orthogonal polynomials for which similar properties hold. [Copyright &y& Elsevier]

Abstract: It is well known that the zeros of orthogonal polynomials interlace. In this paper we study the case of multiple orthogonal polynomials. We recall known results and some recursion relations for multiple orthogonal polynomials. Our main result gives a sufficient condition, based on the coefficients in the recurrence relations, for the interlacing of the zeros of neighboring multiple orthogonal polynomials. We give several examples illustrating our result. [Copyright &y& Elsevier]

A high-order linear q-difference equation with polynomial coefficients having q-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation coincides with the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when are studied. Indeed, the difference equation for Hahn multiple orthogonal polynomials given in Lee [J. Approx. Theory (2007), doi: 10.1016/j.jat.2007.06.002] is obtained as a limiting case. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 τ-function representation of the multiple orthogonality is given. [Copyright &y& Elsevier]

Abstract: K. Mahler introduced the concept of perfect systems in the theory of simultaneous Hermite–Padé approximation of analytic functions. Recently, we proved that Nikishin systems, generated by measures with bounded support and non-intersecting consecutive supports contained on the real line, are perfect. Here, we prove that they are also perfect when the supports of the generating measures are unbounded or touch at one point. As an application, we give a version of the Stieltjes theorem in the context of simultaneous Hermite–Padé approximation. [Copyright &y& Elsevier]

Abstract: We study the asymptotic zero distribution of type II multiple orthogonal polynomials associated with two Macdonald functions (modified Bessel functions of the second kind). On the basis of the four-term recurrence relation, it is shown that, after proper scaling, the sequence of normalized zero counting measures converges weakly to the first component of a vector of two measures which satisfies a vector equilibrium problem with two external fields. We also give the explicit formula for the equilibrium vector in terms of solutions of an algebraic equation. [ABSTRACT FROM AUTHOR]

Abstract: For any pair of compact intervals of the real line , , with , we obtain two probability measures , , supported on and respectively, such that the Nikishin system has a sequence of monic multiple orthogonal polynomials which satisfy a four term recurrence relation with constant coefficients of period 2. The measures are obtained from the functions which give the ratio asymptotic of multiple orthogonal polynomials with respect to an arbitrary Nikishin system on , , such that a.e. on , . The role of , is symmetric in the sense that the same construction is possible on , , with . [ABSTRACT FROM AUTHOR]

We consider n nonintersecting Brownian motion paths with p prescribed starting positions at time t=0 and q prescribed ending positions at time t=1. The positions of the paths at any intermediate time are a determinantal point process, which in the case p=1 is equivalent to the eigenvalue distribution of a random matrix from the Gaussian unitary ensemble with external source. For general p and q, we show that if a temperature parameter is sufficiently small, then the distribution of the Brownian paths is characterized in the large n limit by a vector equilibrium problem with an interaction matrix that is based on a bipartite planar graph. Our proof is based on a steepest descent analysis of an associated ( p+ q)×( p+ q) matrix-valued Riemann-Hilbert problem whose solution is built out of multiple orthogonal polynomials. A new feature of the steepest descent analysis is a systematic opening of a large number of global lenses. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the model of non-intersecting squared Bessel processes with parameter , in the confluent case where all particles start, at time , at the same positive value , remain positive, and end, at time , at the position . The positions of the paths have a limiting mean density as which is characterized by a vector equilibrium problem. We show how to obtain this equilibrium problem from different considerations involving the recurrence relations for multiple orthogonal polynomials associated with the modified Bessel functions. We also extend the situation by rescaling the parameter , letting it increase proportionally to as increases. In this case we also analyze the recurrence relation and obtain a vector equilibrium problem for it. [Copyright &y& Elsevier]

We study the logarithmic and ratio asymptotics of linear forms constructed from a Nikishin system which satisfy orthogonality conditions with respect to a system of measures generated by a second Nikishin system. This construction combines type I and type II multiple orthogonal polynomials. The logarithmic asymptotics of the linear forms is expressed in terms of the extremal solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotics is described by means of a conformal representation of an appropriate Riemann surface of genus zero onto the extended complex plane. [ABSTRACT FROM AUTHOR]

Abstract: This work treats the Mellin transform of multiple Jacobi–Piñeiro polynomials. This allows us to put a number of irrationality and -linear independence proofs into the framework of Hermite–Padé approximation. A similar approach is presented for the -analogue: the -Mellin transform of multiple little -Jacobi polynomials and its applications in irrationality proofs. [Copyright &y& Elsevier]

Abstract: This contribution deals with multiple orthogonal polynomials of type II with respect to q-discrete measures (q-Hahn measures). In addition, we show that this family of multiple orthogonal polynomials has a lowering operator, and raising operators, as well as a Rodrigues type formula. The combination of lowering and raising operators leads to a third order q-difference equation when two orthogonality conditions are considered. An explicit expression of this q-difference equation will be given. Indeed, this q-difference equation relates polynomials with a given degree evaluated at four consecutive non-uniformed distributed points, which makes these polynomials interesting from the point of view of bispectral problems. [Copyright &y& Elsevier]

Abstract: We consider equilibrium problems for the logarithmic vector potential related to the asymptotics of the Hermite–Padé approximants. Solutions of such problems can be expressed by means of algebraic functions. The goal of this paper is to describe a procedure for determining the algebraic equation for this function in the case when the genus of this algebraic function is equal zero. Using the coefficients of the equation we compute the extremal cuts of the Riemann surfaces. These cuts are attractive sets for the poles of the Hermite–Padé approximants. We demonstrate the method by an example of the equilibrium problem related to a special system that is called the Angelesco system. [Copyright &y& Elsevier]

We obtain the system of recurrence relations for rational approximations of the Euler constant generalizing the recurrence relations obtained earlier by Aptekarev with coauthors. The leading coefficient of the recurrence relations of this system is 1, which can be used to verify that the generated numbers are integers. [ABSTRACT FROM AUTHOR]

Abstract: We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system is such that for each , has a constant sign on its compact support consisting of an interval , on which almost everywhere, and a discrete set without accumulation points in . If denotes the smallest interval containing , we assume that , . The second Nikishin system is a perturbation of the first by means of rational functions , , whose zeros and poles lie in . [Copyright &y& Elsevier]

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We introduce multiple orthogonal polynomials on the unit circle. We show how this is related to simultaneous rational approximation to Caratheodory functions (two-point Hermite-Pade approximation near zero and near infinity). We give a Riemann-Hilbert problem for which the solution is in terms of type I and type II multiple orthogonal polynomials on the unit circle, and recurrence relations are obtained from this Riemann-Hilbert problem. Some examples are given to give an idea of the behavior of the zeros of type II multiple orthogonal polynomials. [ABSTRACT FROM AUTHOR]

Abstract: We consider n one-dimensional Brownian motions, such that Brownian motions start at time in the starting point a and end at time in the endpoint b and the other Brownian motions start at time at the point and end at time in the point , conditioned that the n Brownian paths do not intersect in the whole time interval . The correlation functions of the positions of the non-intersecting Brownian motions have a determinantal form with a kernel that is expressed in terms of multiple Hermite polynomials of mixed type. We analyze this kernel in the large n limit for the case . We find that the limiting mean density of the positions of the Brownian motions is supported on one or two intervals and that the correlation kernel has the usual scaling limits from random matrix theory, namely the sine kernel in the bulk and the Airy kernel near the edges. [Copyright &y& Elsevier]

Abstract: We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures such that for each k, has constant sign on its support consisting on an interval , on which almost everywhere, and a set without accumulation points in . [Copyright &y& Elsevier]

Abstract: For discrete multiple orthogonal polynomials such as the multiple Charlier polynomials, the multiple Meixner polynomials, and the multiple Hahn polynomials, we first find a lowering operator and then give a th order difference equation by combining the lowering operator with the raising operator. As a corollary, explicit third order difference equations for discrete multiple orthogonal polynomials are given, which was already proved by Van Assche for the multiple Charlier polynomials and the multiple Meixner polynomials. [Copyright &y& Elsevier]

A generating function is obtained for multiple Hermite and multiple Laguerre polynomials from the Rodrigues formula, and then interesting recurrence relations are given. We also find a relation between multiple orthogonal polynomials and ordinary orthogonal polynomials such as multiple Hermite polynomials with classical Hermite polynomials and multiple Laguerre polynomials with classical Laguerre polynomials. Lastly, we find a differential equation for the multiple orthogonal polynomials by the generating function, which is a reconstruction of the works by Aptekarev et al. [Aptekarev, A.I., Branquinho, A., Van Assche, W., 2003, Multiple orthogonal polynomials for classical weights. Transaction of American Mathematical Society, 355, 3887-3914.]. [ABSTRACT FROM AUTHOR]