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Asymptotic zero distribution of biorthogonal polynomials

D.S. Lubinsky, A. Sidi, H. Stahl
2015 Journal of Approximation Theory
We study the 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] .  ...

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].  ...

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.  ...

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

N. M. Temme
1986 Constructive approximation
An 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.  ...

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  ...

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.  ...

Asymptotic properties of biorthogonal polynomials systems related to Hermite and Laguerre polynomials [article]

Yan Xu
2015 arXiv   pre-print
In this paper, the structures to a family 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  ...

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.  ...

A note on the limiting mean distribution of singular values for products of two Wishart random matrices

Lun Zhang
2013 Journal of Mathematical Physics
With known results on 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  ...

Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

D. S. Lubinsky, A. Sidi
2008 Constructive approximation
We analyze 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.  ...

Average characteristic polynomials of determinantal point processes

2015 Annales de l'I.H.P. Probabilites et statistiques
For a subclass 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.  ...

Orthogonal polynomials and some q-beta integrals of Ramanujan

P.I. Pastro
1985 Journal of Mathematical Analysis and Applications
Actually as a consequence 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.  ...

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.  ...

On the computation of density and two-point correlation functions of a class of random matrix ensembles [article]

Kazi Alam, Swapnil Yadav, K. A. Muttalib
2020 arXiv   pre-print
The method is suitable for solving log-gas models with 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.  ...

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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