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Large n limit of Gaussian random matrices with external source, part II
, 2004
"... We continue the study of the Hermitian random matrix ensemble with external source ..."
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Cited by 95 (26 self)
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We continue the study of the Hermitian random matrix ensemble with external source
A ChristoffelDarboux formula for multiple orthogonal polynomials
 J. Approx. Theory
"... Bleher and Kuijlaars recently showed that the eigenvalue correlations from matrix ensembles with external source can be expressed by means of a kernel built out of special multiple orthogonal polynomials. We derive a ChristoffelDarboux formula for this kernel for general multiple orthogonal polynom ..."
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Cited by 35 (10 self)
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Bleher and Kuijlaars recently showed that the eigenvalue correlations from matrix ensembles with external source can be expressed by means of a kernel built out of special multiple orthogonal polynomials. We derive a ChristoffelDarboux formula for this kernel for general multiple orthogonal polynomials. In addition, we show that the formula can be written in terms of the solution of the RiemannHilbert problem for multiple orthogonal polynomials, which will be useful for asymptotic analysis. 1
Multiple orthogonal polynomial ensembles
, 2009
"... 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 ens ..."
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Cited by 27 (7 self)
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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.
Some Discrete Multiple Orthogonal Polynomials
 J. Comput. Appl. Math
, 2001
"... In this paper we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Ha ..."
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Cited by 26 (2 self)
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In this paper we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn [4] [6] [7]. These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order di#erence equation, and an explicit expression from which the coe#cients of the threeterm recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied in [10] [3]. The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally there also exists a recurrence relation of order r + 1 for these multiple orthogonal polynomials of type II. We compute the coe#cients of the recurrence relation explicitly when r = 2. 1 Classical discrete orthogonal polynomials Orthogonal polynomials {p n (x) : n = 0, 1, 2, . . .} corresponding to a positive measure on the real line are such that p n has degree n and satisfies the conditions Z p n (x)x j d(x) = 0, j = 0, 1, . . . , n  1. This defines the polynomial up to a multiplicative factor. In the case of discrete orthogonal polynomials, we have a discrete measure (with finite moments) = N X k=0 # k # x k , # k > 0, x k # R and N # N # {+#}, which is a linear combination of Dirac measures on the N + 1 points x 0 , . . . , xN . The orthogonality conditions of a discrete orthogonal polynomial p n on the set {x k = k : k ...
Multiple orthogonal polynomials associated with the modified Bessel functions of the first kind
 CONSTR. APPROX
, 2001
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