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Asymptotic zero distribution of biorthogonal polynomials

2015
*
Journal of Approximation Theory
*

We study the

doi:10.1016/j.jat.2014.01.001
fatcat:a6klpo5ycbgjlnsduwrluo4c74
*distribution**of**zeros**of*P n as n → ∞, and related potential theory. ... Let P n be a*polynomial**of*degree n determined by the*biorthogonality*conditions 1 0 P n (x) ψ (x) j dx = 0, j = 0, 1, . . . , n − 1. ... Using steepest descent, the strong*asymptotics**of*P n , and their*zero**distribution*, were established in [14] .*Asymptotics*for more general*polynomials**of*this type were analyzed by Elbert [7] . ...##
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Page 8499 of Mathematical Reviews Vol. , Issue 99m
[page]

1999
*
Mathematical Reviews
*

The

*asymptotic*expressions are so good that they allow the authors to rigorously prove universality conjectures concerning*distributions**of*energy levels*of*quantum systems. ... Akhiezer, who announced*asymptotic*formulas for such*polynomials*on and off the support*of*the orthogonality measure in a short note [Soviet Math. Dokl. 1 (1960), 31-34; MR 22 #8263]. ...##
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Page 3514 of Mathematical Reviews Vol. , Issue 2003e
[page]

2003
*
Mathematical Reviews
*

The properties

*of*these*zeros*akin to the well-known properties*of*the*zeros**of*orthogonal*polynomials*on the real line, such as separation and*asymptotic**distribution*, are considered. ... In particular, an important theorem for the*asymptotic*behavior*of*the*zeros**of*the associated orthogonal*polynomials*is obtained. ...##
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Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

1986
*
Constructive approximation
*

An

doi:10.1007/bf01893438
fatcat:wgo2cexvxbbwxedz5odtkyh4fe
*asymptotic*expansion including error bounds is given for*polynomials*{P., Q"} that are*biorthogonal*on the unit circle with respect to the weight function (1-ei•)a+ll(1-e-' 8 )a-13 • The*asymptotic*... The point z = 1 is an interesting point, where the*asymptotic*behavior*of*the*polynomials*strongly changes. The approximants in the expansions are confluent hypergeometric functions. ... The author thanks the referees and Tom Koomwinder (editor) for their valuable comments on an earlier version*of*the paper. ...##
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Page 1014 of Mathematical Reviews Vol. 37, Issue 5
[page]

1969
*
Mathematical Reviews
*

The notion

*of**biorthogonal*functions is utilized for calcu- lating infinite medium flux*distributions*in radiation penetration theory. ... Similarly the location*of*the*zeros**of*the nth Laguerre*polynomial*determines the density*of*eigenvalues for an ensemble*of*n-dimensional positive matrices, and the*zeros**of*the jnth Cebyéev poly- nomial ...##
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Page 2703 of Mathematical Reviews Vol. , Issue 2000d
[page]

2000
*
Mathematical Reviews
*

For the MDF an orthog-
onal

*polynomial*sequence (OPS) can be constructed. The*zeros**of*these OPs are knots*of*Gaussian quadrature formulas. ... Summary: “*Biorthogonal**polynomials*P,”’ include as particular cases vector orthogonal*polynomials**of*dimension d and —d (d € N). We pay special attention to the cases*of*dimension | and —1. ...##
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Asymptotic properties of biorthogonal polynomials systems related to Hermite and Laguerre polynomials
[article]

2015
*
arXiv
*
pre-print

In this paper, the structures to a family

arXiv:1503.05387v1
fatcat:go3o2bu2mnf75cb2hrrr25iyi4
*of**biorthogonal**polynomials*that approximate to the Hermite and Generalized Laguerre*polynomials*are discussed respectively. ... Therefore, the*asymptotic*relation between several orthogonal*polynomials*and combinatorial*polynomials*are derived from the systems, which in turn verify the Askey scheme*of*hypergeometric orthogonal ... These limits give insight in the location*of*the*zeros*for large values*of*the limit parameter, and the*asymptotic*relation with the Hermite*polynomials*if the parameter N become large and x is properly ...##
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Page 4088 of Mathematical Reviews Vol. , Issue 83j
[page]

1983
*
Mathematical Reviews
*

Madrid, to appear] involving

*distributional*weight functions. It is shown that the Gegenbauer*polynomials*G> are*distributionally*orthogonal even for A<-1. A connection*of*some*polynomials**of*Ya. L. ... The authors introduce and study a pair*of**biorthogonal**polynomials*suggested by the classical Jacobi*polynomials*. ...##
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A note on the limiting mean distribution of singular values for products of two Wishart random matrices

2013
*
Journal of Mathematical Physics
*

With known results on

doi:10.1063/1.4818978
fatcat:bxn7l4hy2bexjoanlz46pxduo4
*asymptotic**zero**distribution**of*these*polynomials*and general theory on multiple orthogonal*polynomial*ensembles, it is then easy to obtain an explicit expression for the*distribution*... certain*biorthogonal**polynomials*that can be explicitly constructed. ... For each P (γ,κ) k,n , we can associate the normalized counting*zero*measure defined by ν(P (γ,κ) k,n ) = 1 k P (γ,κ) k,n (x)=0 δ x . (2.10) A measure ν ξ is called the*asymptotic**zero**distribution**of*...##
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Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

2008
*
Constructive approximation
*

We analyze

doi:10.1007/s00365-008-9014-2
fatcat:o6lmvsmtjjgzniq5iwfy2fo7u4
*polynomials*Pn that are*biorthogonal*to exponentials e n;j x n j=1 , in the sense that Z 1 0 ... 1 d d exp 1 1 d : This is independent*of*; , and since we know that for = = 0, the*asymptotic**zero**distribution*is given by (5.5-5.6), the same*zero**distribution*holds for all ; . ... In Section 3, we state an extension*of*a result*of*Van Assche, Fano and Ortolani, and a consequence for*zero**distribution**of*composite*polynomials*. In Section 4, we prove the results*of*Section 3. ...##
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Average characteristic polynomials of determinantal point processes

2015
*
Annales de l'I.H.P. Probabilites et statistiques
*

For a subclass

doi:10.1214/13-aihp572
fatcat:vkijoektfbeg3kns42n6b3n6du
*of*point processes, which contains Orthogonal*Polynomial*Ensembles and Multiple Orthogonal*Polynomial*Ensembles, we provide a sufficient condition for its limiting*zero**distribution*to match ... As another 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 ... in an earlier version*of*this work, and the anonymous referee for improving the readability*of*the paper. ...##
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Orthogonal polynomials and some q-beta integrals of Ramanujan

1985
*
Journal of Mathematical Analysis and Applications
*

Actually as a consequence

doi:10.1016/0022-247x(85)90261-6
fatcat:ukeaoyas7fdzves2yqyvnda5ya
*of*the real*biorthogonality*the*polynomials*in (4.9) have*zeros*on the real line and on the contrary many*of*the*polynomials*defined in (3.1) have complex*zeros*. ... I, p. 3041 there are two families*of**biorthogonal**polynomials*with respect to the beta*distribution*(2.6). ... So there exists a set*of*orthogonal*polynomials*related to S(x). These*polynomials*are J,(t)= i (4 -7 q)r r:O (4; 4MW2; 4L (q (r,2)+ It)r. ...##
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Page 1013 of Mathematical Reviews Vol. , Issue 83c
[page]

1983
*
Mathematical Reviews
*

positive

*polynomials*have*biorthogonal*companions*of*all orders, in a formal sense. ... Muldoon (Downsview, Ont.) van Rossum, H. 83c:33011 Formally*biorthogonal**polynomials*. ...##
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On the computation of density and two-point correlation functions of a class of random matrix ensembles
[article]

2020
*
arXiv
*
pre-print

The method is suitable for solving log-gas models with

arXiv:2010.10330v1
fatcat:f3cpe5hqinfbfmocp5eyigkkdq
*biorthogonal*type two-body interactions and arbitrary potentials. ... We demonstrate a method to solve a general class*of*random matrix ensembles numerically. ... Inset shows details*of*the convergence near*zero*. ...##
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Page 42 of Mathematical Reviews Vol. , Issue 89C
[page]

1989
*
Mathematical Reviews
*

We also consider loci

*of**zeros*, existence*of*Rodrigues-type formulae and reducibility to standard orthogonality. The paper is accompanied by several examples*of**biorthogonal*systems.” L. ... (N-NTH-N) On the theory*of**biorthogonal**polynomials*. Trans. Amer. Math. Soc. 306 (1988), no. 2, 455-474. This paper is a substantial and valuable work. ...
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