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We study multiple orthogonal polynomials exploiting their explicit determinantal representation in terms of moments. Our reasoning follows that applied to solve the Hermite–Padé approximation and interpolation problems. We also study families of multiple orthogonal polynomials obtained by variation of the measures known from the theory of discrete‐time Toda lattice equations. We present determinantal proofs of certain fundamental results of the theory, obtained earlier by other authors in a different setting. We also derive quadratic identities satisfied by the polynomials, which are new elements of the theory. Resulting equations allow to present multiple orthogonal polynomials within the theory of integrable systems. [ABSTRACT FROM AUTHOR]

For a general number p\geq 2 of measures, we provide explicit expressions for the Jacobi–Piñeiro and Laguerre of the first kind multiple orthogonal polynomials of type I, presented in terms of generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]

This paper studies a new family of Angelesco multiple orthogonal polynomials with shared orthogonality conditions with respect to a system of weight functions, which are complex analogs of Pascal distributions on a legged star-like set. The emphasis is placed on the algebraic properties, such as the raising operators, the Rodrigues-type formula, the explicit expression of the polynomials, and the nearest neighbor recurrence relations. [ABSTRACT FROM AUTHOR]

We study two families of type II discrete multiple orthogonal polynomials on an -legged star-like set with respect to weight functions of Charlier (Poisson distributions) and Meixner (negative binomial distributions), respectively. We focus our attention on the structural properties such as the raising operators, the Rodrigues-type formulas, and the explicit expressions of the polynomial families as well as the coefficients of the nearest neighbor recurrence relation. A limit relation between these families is given. [ABSTRACT FROM AUTHOR]

For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect to w allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples. [ABSTRACT FROM AUTHOR]

We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights (w1,w2)$(w_1,w_2)$ on the positive real line, with w1(x)=xαe−x$w_1(x)=x^\alpha e^{-x}$ the gamma density and w2(x)=xαEν+1(x)$w_2(x) = x^\alpha E_{\nu +1}(x)$ a density related to the exponential integral Eν+1$E_{\nu +1}$. We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed‐type multiple orthogonal polynomials. [ABSTRACT FROM AUTHOR]

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. These type of matrices are oscillatory. In this paper the Lima–Loureiro hypergeometric multiple orthogonal polynomials and the Jacobi–Piñeiro multiple orthogonal polynomials are discussed at the light of this bidiagonal factorization for tetradiagonal matrices. The Darboux transformations of tetradiagonal Hessenberg matrices is studied and Christoffel formulas for the elements of the bidiagonal factorization are given, i.e., the bidiagonal factorization is given in terms of the recursion polynomials evaluated at the origin. [ABSTRACT FROM AUTHOR]

In this paper, we consider N non-intersecting Bessel paths starting at x = a ≥ 0 , and conditioned to end at the origin x = 0 . We derive the explicit formula of the distribution function for the maximum height. Depending on the starting point a > 0 or a = 0 , the distribution functions are also given in terms of the Hankel determinants associated with the multiple discrete orthogonal polynomials or discrete orthogonal polynomials, respectively. [ABSTRACT FROM AUTHOR]

This paper studies a new family of Angelesco multiple orthogonal polynomials with shared orthogonality conditions with respect to a system of weight functions, which are complex analogs of Pascal distributions on a legged star-like set. The emphasis is placed on the algebraic properties, such as the raising operators, the Rodrigues-type formula, the explicit expression of the polynomials, and the nearest neighbor recurrence relations.

Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and one Hessenberg, respectively, and the Gauss–Borel factorization of the moment matrix is given. It is shown a hidden freedom exhibited by the spectral system related to the multiple orthogonal polynomials. Pearson equations are discussed, a Laguerre–Freud matrix is considered, and differential equations for type I and II multiple orthogonal polynomials, as well as for the corresponding linear forms are given. The Jacobi–Piñeiro multiple orthogonal polynomials of type I and type II are used as an illustrating case and the corresponding differential relations are presented. A permuting Christoffel transformation is discussed, finding the connection between the different families of multiple orthogonal polynomials. The Jacobi–Piñeiro case provides a convenient illustration of these symmetries, giving linear relations between different polynomials with shifted and permuted parameters. We also present the general theory for the perturbation of each weight by a different polynomial or rational function aka called Christoffel and Geronimus transformations. The connections formulas between the type II multiple orthogonal polynomials, the type I linear forms, as well as the vector Stieltjes–Markov vector functions is also presented. We illustrate these findings by analyzing the special case of modification by an even polynomial. [ABSTRACT FROM AUTHOR]

The list of physically motivated urn models that can be solved in terms of classical orthogonal polynomials is very small. It includes a model proposed by D. Bernoulli and further analyzed by S. Laplace and a model proposed by P. and T. Ehrenfest and eventually connected with the Krawtchouk and Hahn polynomials. This connection was reversed recently in the case of the Jacobi polynomials where a rather contrived, and later a simpler urn model was proposed. Here we consider an urn model going with the Jacobi-Piñeiro multiple orthogonal polynomials. These polynomials have recently been put forth in connection with a stochastic matrix. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe a Matlab package for computing the simultaneous Gaussian quadrature rules associated with a variety of multiple orthogonal polynomials. Multiple orthogonal polynomials can be considered as a generalization of classical orthogonal polynomials, satisfying orthogonality constraints with respect to r different measures, with r ≥ 1. Moreover, they satisfy (r + 2) -term recurrence relations. In this manuscript, without loss of generality, r is considered equal to 2. The so-called simultaneous Gaussian quadrature rules associated with multiple orthogonal polynomials can be computed by solving a banded lower Hessenberg eigenvalue problem. Unfortunately, computing the eigendecomposition of such a matrix turns out to be strongly ill-conditioned and the Matlab function balance.m does not improve the condition of the eigenvalue problem. Therefore, most procedures for computing simultaneous Gaussian quadrature rules are implemented with variable precision arithmetic. Here, we propose a Matlab package that allows to reliably compute the simultaneous Gaussian quadrature rules in floating point arithmetic. It makes use of a variant of a new balancing procedure, recently developed by the authors of the present manuscript, that drastically reduces the condition of the Hessenberg eigenvalue problem. [ABSTRACT FROM AUTHOR]

Abstract The main aim of this paper is to define and investigate more general multiple Charlier polynomials on the linear lattice ω N = { 0 , ω , 2 ω , … } $\omega \mathbb{N} = \{ 0,\omega ,2\omega ,\ldots \} $ , ω ∈ R $\omega \in \mathbb{R}$ . We call these polynomials ω-multiple Charlier polynomials. Some of their properties, such as the raising operator, the Rodrigues formula, an explicit representation and a generating function are obtained. Also an ( r + 1 ) $( r+1 )$ th order difference equation is given. As an example we consider the case ω = 3 2 $\omega =\frac{3}{2}$ and define 3 2 $\frac{3}{2}$ -multiple Charlier polynomials. It is also mentioned that, in the case ω = 1 $\omega =1$ , the obtained results coincide with the existing results of multiple Charlier polynomials.

We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval, the positive real line, or the unit circle and the multiple orthogonal polynomials are generalizations of the Jacobi, Laguerre or Bessel orthogonal polynomials. We give explicit formulas for the type I and type II multiple orthogonal polynomials and study some of their properties. In particular, we describe the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials via the free convolution. Essential to our overall approach is the use of the Mellin transform. Finally, we discuss two applications. First, we show that the multiple orthogonal polynomials appear naturally in the study of the squared singular values of (mixed) products of truncated unitary random matrices and Ginibre matrices. Secondly, we use the multiple orthogonal polynomials to simultaneously approximate certain hypergeometric series and to provide an explicit proof of their Q -linear independence. [ABSTRACT FROM AUTHOR]

Laurie in [Math. Comp., 65(1996), pp. 739–747] introduced anti-Gaussian quadrature rule, that gives an error equal in magnitude but of opposite sign to that of the corresponding Gaussian quadrature rule. Guided by that idea, in this paper we consider a set of anti-Gaussian quadrature rules for the optimal set of quadrature rules in Borges' sense (see [Numer. Math., 67(1994), pp. 271–288]), as well as the corresponding class of multiple orthogonal polynomials. The main properties of such quadrature rules and multiple orthogonal polynomials are proved and numerical methods for their constructions are presented. Some numerical examples are also included. [ABSTRACT FROM AUTHOR]

A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval (0,1) is studied. This type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total‐positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson‐type differential equation. The focus is on the polynomials whose indices lie on the step‐line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. We obtain Rodrigues‐type formulas for type I polynomials and functions, while a more detailed characterization is given for the type II polynomials (aka 2‐orthogonal polynomials) that include an explicit expression as a terminating hypergeometric series, a third‐order differential equation, and a third‐order recurrence relation. The asymptotic behavior of their recurrence coefficients mimics those of Jacobi–Piñeiro polynomials, based on which their asymptotic zero distribution and a Mehler–Heine asymptotic formula near the origin are given. Particular choices of the parameters and confluence relations give some known systems such as special cases of the Jacobi–Piñeiro polynomials, Jacobi‐type 2‐orthogonal polynomials, components of the cubic decomposition of threefold symmetric Hahn‐classical polynomials, and multiple orthogonal polynomials with respect to confluent hypergeometric functions of the second kind. [ABSTRACT FROM AUTHOR]

Abstract In this study, we introduce a new family of discrete multiple orthogonal polynomials, namely ω-multiple Meixner polynomials of the first kind, where ω is a positive real number. Some structural properties of this family, such as the raising operator, Rodrigue’s type formula and an explicit representation are derived. The generating function for ω-multiple Meixner polynomials of the first kind is obtained and by use of this generating function we find several consequences for these polynomials. One of them is a lowering operator which will be helpful for obtaining a difference equation. We give the proof of the lowering operator by use of new technique which is a more elementary proof than the proof of Lee in (J. Approx. Theory 150:132–152, 2008). By combining the lowering operator with the raising operator we obtain the difference equation which has the ω-multiple Meixner polynomials of the first kind as a solution. As a corollary we give a third order difference equation for the ω-multiple Meixner polynomials of the first kind. Also it is shown that, for the special case ω = 1 $\omega = 1$ , the obtained results coincide with the existing results for multiple Meixner polynomials of the first kind. In the last section as an illustrative example we consider the special case when ω = 1 / 2 $\omega =1/2$ and, for the 1 / 2 $1/2$ -multiple Meixner polynomials of the first kind, we state the corresponding result for the main theorems.

We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was recently discovered. In this paper, we consider Angelesco systems formed by two analytic weights and obtain asymptotics of the recurrence coefficients and strong asymptotics of MOPs along all directions (including the marginal ones). These results are then applied to show that the essential spectrum of the related Jacobi matrix is the union of intervals of orthogonality. [ABSTRACT FROM AUTHOR]

In this study, a new family of discrete multiple orthogonal polynomials, namely, ω-multiple Meixner polynomials of the second kind, where ω is a positive real number which is introduced. Some structural properties for these polynomials such as raising operator, Rodrigue's type formula and explicit representation are obtained. Generating function for ω-multiple Meixner polynomials of the second kind and several consequences using this generating function for these polynomials are derived. A lowering operator for ω-multiple Meixner polynomials of the second kind which will be helpful for obtaining difference equation is derived. By combining the lowering operator and the raising operator the difference equation having the ω-multiple Meixner polynomials of the second kind as a solution is obtained. A third-order explicit difference equation for ω-multiple Meixner polynomials of the second kind is given as a corollary. It is proven that when ω = 1 , the obtained results coincide with the existing results for multiple Meixner polynomials of the second kind. In the last section, the case when ω = 5 / 3 is studied and for the 5/3-multiple Meixner polynomials of the second kind the explicit form, generating function and the third-order difference equation is given. [ABSTRACT FROM AUTHOR]

We study a family of type II multiple orthogonal polynomials. We consider orthogonality conditions with respect to a vector measure, in which each component is a q-analogue of the binomial distribution. The lowering and raising operators as well as the Rodrigues formula for these polynomials are obtained. The difference equation of order r + 1 is studied. The connection via limit relation between four types of Kravchuk polynomials is discussed. [ABSTRACT FROM AUTHOR]

We study the logarithmic asymptotic of multiple orthogonal polynomials arising in a mixed-type Hermite–Padé approximation problem associated with the rational perturbation of a Nikishin system of functions. The formulas obtained allow to give exact estimates of the rate of convergence of the corresponding Hermite–Padé approximants. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to define and investigate more general multiple Charlier polynomials on the linear lattice ω N = { 0 , ω , 2 ω , ... } , ω ∈ R . We call these polynomials ω-multiple Charlier polynomials. Some of their properties, such as the raising operator, the Rodrigues formula, an explicit representation and a generating function are obtained. Also an (r + 1) th order difference equation is given. As an example we consider the case ω = 3 2 and define 3 2 -multiple Charlier polynomials. It is also mentioned that, in the case ω = 1 , the obtained results coincide with the existing results of multiple Charlier polynomials. [ABSTRACT FROM AUTHOR]

New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations in terms of the Mellin–Barnes and Laplace integrals. A notion of the composition polynomial orthogonality is introduced. The corresponding advantages of this orthogonality to discover new sequences of polynomials and their relations to the corresponding multiple orthogonal polynomial ensembles are shown. [ABSTRACT FROM AUTHOR]

The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they are 2-orthogonal polynomials whose the sequences of their derivatives are also 2-orthogonal polynomials. Based only on this property, a new class of classical 2-orthogonal polynomials is obtained as particular solution of the non-linear system governing the coefficients involved in the recurrence relation fulfilled by these polynomials. A differential-recurrence relation as well as a third-order differential equation satisfied by the resulting polynomials are given. Many interesting subcases are highlighted and explicitly presented with special reference to some connected results that exist in the literature. The integral representations of their associated linear functionals will be exhaustively discussed in a forthcoming publication. [ABSTRACT FROM AUTHOR]

Here, we investigate type II multiple big q-Jacobi orthogonal polynomials. We provide their explicit formulae in terms of basic hypergeometric series, raising and lowering operators, Rodrigues formulae, third-order q-difference equation, and we obtain recurrence relations.

Here, we investigate type II multiple big q-Jacobi orthogonal polynomials. We provide their explicit formulae in terms of basic hypergeometric series, raising and lowering operators, Rodrigues formulae, third-order q-difference equation, and we obtain recurrence relations. [ABSTRACT FROM AUTHOR]

In this study, we introduce a new family of discrete multiple orthogonal polynomials, namely ω-multiple Meixner polynomials of the first kind, where ω is a positive real number. Some structural properties of this family, such as the raising operator, Rodrigue's type formula and an explicit representation are derived. The generating function for ω-multiple Meixner polynomials of the first kind is obtained and by use of this generating function we find several consequences for these polynomials. One of them is a lowering operator which will be helpful for obtaining a difference equation. We give the proof of the lowering operator by use of new technique which is a more elementary proof than the proof of Lee in (J. Approx. Theory 150:132–152, 2008). By combining the lowering operator with the raising operator we obtain the difference equation which has the ω-multiple Meixner polynomials of the first kind as a solution. As a corollary we give a third order difference equation for the ω-multiple Meixner polynomials of the first kind. Also it is shown that, for the special case ω = 1 , the obtained results coincide with the existing results for multiple Meixner polynomials of the first kind. In the last section as an illustrative example we consider the special case when ω = 1 / 2 and, for the 1 / 2 -multiple Meixner polynomials of the first kind, we state the corresponding result for the main theorems. [ABSTRACT FROM AUTHOR]

We investigate type I multiple orthogonal polynomials on r intervals that have a common point at the origin and endpoints at the r roots of unity ω j , j = 0 , 1 , ... , r - 1 , with ω = exp (2 π i / r) . We use the weight function | x | β (1 - x r) α , with α , β > - 1 , for the multiple orthogonality relations. We give explicit formulas for the type I multiple orthogonal polynomials, the coefficients in the recurrence relation, and the differential equation, and we obtain the asymptotic distribution of the zeros. [ABSTRACT FROM AUTHOR]

We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as 2 -orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a 3 -star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them 2 -orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni. [ABSTRACT FROM AUTHOR]

Multiple orthogonal polynomials generalize standard orthogonal polynomials by requiring orthogonality with respect to several inner products. This paper discusses an application to the approximation of matrix functions and presents quadrature rules that generalize the anti-Gauss rules proposed by Laurie. [ABSTRACT FROM AUTHOR]

We show that solution to the Hermite–Padé type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev–Petviashvili) system and of its adjoint linear problem. Our result explains the appearance of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorithms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite–Padé approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite–Padé problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations. [ABSTRACT FROM AUTHOR]

We consider two families of type II multiple orthogonal polynomials. Each family has orthogonality conditions with respect to a discrete vector measure. The r components of each vector measure are q-analogues of Meixner measures of the first and second kind, respectively. These polynomials have lowering and raising operators, which lead to the Rodrigues formula, difference equation of order r+1, and explicit expressions for the coefficients of recurrence relation of order r+1. Some limit relations are obtained.

In this paper, we investigate a numerical method for the construction of an optimal set of quadrature rules in the sense of Borges (Numer. Math. 67, 271-288, 1994) for two or three definite integrals with the same integrand and interval of integration, but with different weight functions, related to an arbitrary multi-index. The presented method is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]