Multiple orthogonal polynomials associated with macdonald functions

Assche, Walter Van; Yakubovich, Semyon

doi:10.1080/10652460008819257

Cited by 64 publications

(72 citation statements)

References 5 publications

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“…Multiple orthogonal polynomials associated with the two weights (2.6)-(2.7) were considered by Van Assche and Yakubovich [38] for which they obtained four term recurrence relations; see also [17] and [40] for asymptotic results for these polynomials. In the random matrix context (i.e., the case where…”

Section: Special Case M =mentioning

confidence: 99%

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Kuijlaars

Zhang

2014

Commun. Math. Phys.

126

261

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Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M = 2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy-Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.

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Section: Special Case M =mentioning

confidence: 99%

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Kuijlaars

Zhang

2014

Commun. Math. Phys.

126

261

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“…Recently they also appeared in random matrix theory for matrix ensembles with external source [3,7,8] and Wishard ensembles [6]. The particular limiting behavior which we are considering appears in the examples Jacobi-Piñeiro, Laguerre I [29] and the example associated with modified Bessel functions [30].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic zero distribution for a class of multiple orthogonal polynomials

Coussement

Assche

2008

Trans. Amer. Math. Soc.

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Abstract. We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples of multiple orthogonal polynomials, in particular Jacobi-Piñeiro, Laguerre I and the example associated with modified Bessel functions. We also discuss an application to Toeplitz matrices.

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“…This problem has been recently studied by Van Assche and Yakubovich [21]. Hence the two papers bring, independently, an answer to that open problem.…”

Section: Jomentioning

confidence: 91%

“…The authors wish to thank the referees for their useful comments, in particular, for the information about [21].…”

mentioning

confidence: 99%

On two-orthogonal polynomials related to the Bateman’s $J^{u,v}_{n}$-function

Douak¹,

Cheikh²

2000

Methods and Applications of Analysis

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Abstract. The purpose of the present paper is to study the sequence of hypergeometric polynomials {Bn^}n^o defined by B%'P(x) = liM-n; 1 + a, 1 + /Jjx), where a, /3 g -N* , which are (with the change of notations u = a, v + u/2 = /3) related to the Bateman's function J% ,v . We show that it constitutes a two-parameter family of 2-orthogonal classical polynomials of Laguerre type. Among other things, we give their recurrence relation and the third order differential equation satisfied by each polynomial. When a ^ /3 > -1, we obtain that these polynomials are 2-orthogonal on the interval (0,-|-oo) with respect to a pair of weight functions W a>^ and (xWa^Y, where W a> /3 is a positive function involving the MacDonald function Ku (v = a-ft). For a suitable choice of the parameters a and /?, we obtain that Wa,p is the ultra-exponential function introduced by Voronoi and then encounter an open problem posed by Prudnikov for which we give an answer. In addition, we obtain by means of the technique of cubic decomposition that the polynomials in question are connected to another kind of 2-orthogonal polynomials of Hermite type.

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Multiple orthogonal polynomials associated with macdonald functions

Cited by 64 publications

References 5 publications

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Asymptotic zero distribution for a class of multiple orthogonal polynomials

On two-orthogonal polynomials related to the Bateman’s $J^{u,v}_{n}$-function

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