Multiple orthogonal polynomials associated with the exponential integral

Multiple orthogonal polynomials

We investigate the asymptotic behaviour of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree d + 1. The polynomials that we investigate are multiple orthogonal with respect to a system of d analytic weights defined on a symmetric (d + 1)-star centred at the origin. In the first part we analyse in detail a vector equilibrium problem involving a system of d interacting measures (μ1, …, μd) supported on star-like sets in the plane. We show that in the subcritical regime, the first component of the solution to this problem is the asymptotic zero distribution of the multiple orthogonal polynomials. It also characterizes the domain where the eigenvalues in the normal matrix model accumulate, in the sense that the Schwarz function associated with the boundary of this domain can be expressed explicitly in terms of . The second part of the paper is devoted to the asymptotic analysis of the multiple orthogonal polynomials. The asymptotic results are obtained again in the subcritical regime, and they follow from the Deift/Zhou steepest descent analysis of a Riemann–Hilbert problem of size (d + 1) × (d + 1). The vector equilibrium problem and the Riemann–Hilbert problem that we investigate are generalizations of those studied recently by Bleher–Kuijlaars in the case d = 2.

In this paper, we obtain the ladder operators and associated compatibility conditions for types I and II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials. Our approach is based on Riemann–Hilbert problems and the Christoffel–Darboux formula for multiple orthogonal polynomials, and the nearest-neighbor recurrence relations. As an illustration, we give several explicit examples involving multiple Hermite and Laguerre polynomials, and multiple orthogonal polynomials with exponential weights and cubic potentials.

Multiple orthogonal polynomials are a generalization of orthogonal polynomials in which the orthogonality is distributed among a number of orthogonality weights. They appear in random matrix theory in the form of special determinantal point processes that are called multiple orthogonal polynomial (MOP) ensembles. The correlation kernel in such an ensemble is expressed in terms of the solution of a Riemann-Hilbert problem, that is of size (r + 1) × (r + 1) in the case of r weights. A number of models give rise to a MOP ensemble, and we discuss recent results on models of non-intersecting Brownian motions, Hermitian random matrices with external source, and the two matrix model. A novel feature in the asymptotic analysis of the latter two models is a vector equilibrium problem for two or more measures, that describes the limiting mean eigenvalue density. The vector equilibrium problems involve both an external field and an upper constraint.

We extend the Mehler-Heine type formula of Jacobi polynomials to a class of multiple orthogonal polynomials of type II. The Mehler-Heine type formulas show standard orthogonal polynomials or multiple orthogonal polynomials near the endpoints of the interval of orthogonality.

Generalizing previous work of Hora (1998) on the asymptotic spectral analysis for the Hamming graph $H(n,q)$ which is the $n^{\mathrm{th}}$ Cartesian power $K_q^{\square n}$ of the complete graph $K_q$ on $q$ vertices, we describe the possible limits of the joint spectral distribution of the pair $(G^{\square n},\overline{G}\vphantom{G}^{\square n})$ of the $n^{\mathrm{th}}$ Cartesian powers of a strongly regular graph $G$ and its complement $\overline{G}$, where we let $n\rightarrow\infty$, and $G$ may vary with $n$. This result is an analogue of the bivariate central limit theorem, and we obtain in this way the bivariate Poisson distributions and the standard bivariate Gaussian distribution, together with the product measures of univariate Poisson and Gaussian distributions. We also report a family of bivariate hypergeometric orthogonal polynomials with respect to the last distributions, which we call the bivariate Charlier-Hermite polynomials, and prove basic formulas for them. This family of orthogonal polynomials seems previously unnoticed, possibly because of its peculiarity.

A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour

Multiple orthogonal polynomials are traditionally studied because of their connections to number theory and approximation theory. In recent years they were found to be connected to certain models in random matrix theory. In this paper we introduce the notion of a multiple orthogonal polynomial ensemble (MOP ensemble) and derive some of their basic properties. It is shown that Angelesco and Nikishin systems give rise to MOP ensembles and that the equilibrium problems that are associated with these systems have a natural interpretation in the context of MOP ensembles.

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms of τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions, which are double Wronskians of generalized hypergeometric series. The shifts in the spectral parameter for type II and type I multiple orthogonal polynomials are described using banded matrices. It is demonstrated that these polynomials offer solutions to multicomponent integrable extensions of the nonlinear Toda equations. Additionally, the paper characterizes extensions of the Nijhoff–Capel totally discrete Toda equations. The hypergeometric τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions are shown to provide solutions to these integrable nonlinear equations. Furthermore, the authors explore Laguerre–Freud equations, nonlinear equations for the recursion coefficients, with a particular focus on the multiple Charlier, generalized multiple Charlier, multiple Meixner II, and generalized multiple Meixner II cases.

The paper examines two examples of multiple orthogonal polynomials generalizing orthogonal polynomials of a discrete variable, meaning thereby the Meixner polynomials. One example is bound up with a discrete Nikishin system, and the other leads to essentially new effects. The limit distribution of the zeros of polynomials is obtained in terms of logarithmic equilibrium potentials and in terms of algebraic curves.Bibliography: 9 titles.

Difference equations for discrete classical multiple orthogonal polynomials

Multiple orthogonal polynomials, irrationality and transcendence

Rational approximation of Euler’s constant using multiple orthogonal polynomials

In recent years, one of the most interesting developments in quantum mechanics has been the construction of new exactly solvable potentials connected with the appearance of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials, which start with a constant, the EOP families begin with some polynomial of degree greater than or equal to one, but still form complete, orthogonal sets with respect to some positive-definite measure. We show how they may appear in the bound-state wavefunctions of some rational extensions of well-known exactly solvable quantum potentials. Such rational extensions are most easily constructed in the framework of supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class of translationally shape invariant potentials. We review the most recent results in this field, which use higher-order SUSYQM. We also comment on some recent re-examinations of the shape invariance condition, which are independent of the EOP construction problem.

A family of multivariate orthogonal polynomials generalizing the standard (univariate) Charlier polynomials is shown to arise in the matrix elements of the unitary representation of the Euclidean group E(d) on oscillator states. These polynomials in d discrete variables are orthogonal on the product of d Poisson distributions. The accent is put on the d = 2 case and the group theoretical setting is used to obtain the main properties of the polynomials: orthogonality and recurrence relations, difference equation, raising/lowering relations, generating function, hypergeometric and integral representations and explicit expression in terms of standard Charlier and Krawtchouk polynomials. The approach is seen to extend straightforwardly to an arbitrary number of variables. The contraction of SO(3) to E(2) is used to show that the bivariate Charlier polynomials correspond to a limit of the bivariate Krawtchouk polynomials.