Plancherel–Rotach Asymptotics for q-Orthogonal Polynomials

We establish the Plancherel–Rotach-type asymptotics around the largest zero (the soft edge asymptotics) for some classes of polynomials satisfying three-term recurrence relations with exponentially increasing coefficients. As special cases, our results include this type of asymptotics for q ^?1-Hermite polynomials of Askey, Ismail, and Masson; q-Laguerre polynomials; and the Stieltjes–Wigert polynomials. We also introduce a one-parameter family of solutions to the q-difference equation of the Ramanujan function.     

Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics

We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Weyl solution. We also prove L² convergence of Szegő asymptotics on the spectrum.     

Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory

We consider polynomials orthogonal on [0,∞) with respect to Laguerre-type weights w(x) = x^α e^-Q(x), where α > -1 and where Q denotes a polynomial with positive leading coefficient. The main purpose of this paper is to determine Plancherel-Rotach-type asymptotics in the entire complex plane for the orthonormal polynomials with respect to w, as well as asymptotics of the corresponding recurrence coefficients and of the leading coefficients of the orthonormal polynomials. As an application we will use these asymptotics to prove universality results in random matrix theory. We will prove our results by using the characterization of orthogonal polynomials via a 2 × 2 matrix valued Riemann--Hilbert problem, due to Fokas, Its, and Kitaev, together with an application of the Deift-Zhou steepest descent method to analyze the Riemann-Hilbert problem asymptotically.     

Asymptotics of derivatives of orthogonal polynomials on the real line

We show that uniform asymptotics of orthogonal polynomials on the real line imply uniform asymptotics for all their derivatives. This is more technically challenging than the corresponding problem on the unit circle. We also examine asymptotics in the L ₂ norm. Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353.     

Orthogonal polynomials and a generalized Szegő condition

Asymptotical properties of orthogonal polynomials from the so-called Szeg? class are very well-studied. We obtain asymptotics of orthogonal polynomials from a considerably larger class and we apply this information to the study of their spectral behavior. To cite this article: S. Denisov, S. Kupin, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Asymptotics of the Best Polynomial Approximation of?|x|p and?of?the Best Laurent Polynomial Approximation of sgn(x) on Two Symmetric Intervals

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of |x| ^p on two symmetric intervals. Note that, in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of sgn(x) on [−1,−a]∪[a,1] by Laurent polynomials.      

SCALED ASYMPTOTICS FOR SOME q-SERIES

This work, investigates the asymptotics for Euler’s q-exponentialE_q(z), Ramanujan’s function A_q(z), Jackson’s q-Besselfunction J_v⁽²⁾ (z; q), the Stieltjes–Wigert orthogonalpolynomials S_n(x; q) and q-Laguerre polynomials L_n⁽⁾ (x; q)as q approaches 1.     

Strong asymptotics for Krawtchouk polynomials

We determine the strong asymptotics for the class of Krawtchouk polynomials on the real line. We show how our strong asymptotics describes the limiting distribution of the zeros of the Krawtchouk polynomials.     

Flat Cyclotomic Polynomials of Order Three

A cyclotomic polynomial _n is said to be of order 3 if n = pqrfor three distinct odd primes p, q, and r. We establish theexistence of an infinite family of such polynomials whose coefficientsdo not exceed 1 in modulus. 2000 Mathematics Subject Classification11B83, 11C08.     

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights w_n(x)dx = e^−nV(x) dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [19, 20]. The Riemann‐Hilbert problem is analyzed in turn using the steepest‐descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμ_V for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.     

Christoffel functions and universality on the boundary of the ball

We establish asymptotics for Christoffel functions, and universality limits, associated with multivariate orthogonal polynomials, on the boundary of the unit ball in ? ^d .     

Simultaneous prime specializations of polynomials over finite fields

Recently the author proposed a uniform analogue of the Bateman–Hornconjectures for polynomials with coefficients from a finitefield (that is, for polynomials in F_q[T] rather than Z[T]).Here we use an explicit form of the Chebotarev density theoremover function fields to prove this conjecture in particularranges of the parameters. We give some applications includingthe solution of a problem posed by Hall.     

Counting Horoballs and Rational Geodesics

Let M be a geometrically finite pinched negatively curved Riemannianmanifold with at least one cusp. The asymptotics of the numberof geodesics in M starting from and returning to a given cusp,and of the number of horoballs at parabolic fixed points inthe universal cover of M, are studied in this paper. The caseof SL(2, Z), and of Bianchi groups, is developed. 2000 MathematicsSubject Classification 53C22, 11J06, 30F40, 11J70.     

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [−1,1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann–Hilbert (RH) problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the RH problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order 1/n terms in the expansions. A critical step in the analysis of the RH problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order. In addition, the RH approach gives us also strong asymptotics of the orthogonal polynomials near the algebraic singularities in terms of Bessel functions.     

Plancherel-Rotach asymptotics for certain basic hypergeometric series

In this work we study the chaotic and periodic asymptotics for the confluent basic hypergeometric series. For a fixed q∈(0,1), the asymptotics for Euler's q-exponential, q-Gamma function Γq(x), q-Airy function of K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Ramanujan function (q-Airy function), Jackson's q-Bessel function of second kind, Ismail-Masson orthogonal polynomials (q⁻¹-Hermite polynomials), Stieltjes-Wigert polynomials, q-Laguerre polynomials could be derived as special cases.     

Funk-Hecke Formula for Orthogonal Polynomials on Spheres and on Balls

Analogues of the Funk–Hecke formula for spherical harmonicsare proved for Dunkl's h-harmonics associated to the reflectiongroups, and for orthogonal polynomials related to h-harmonicson the unit ball. In particular, an analogue and its applicationare discussed for the weight function (1–|x|²)^µ–1/2on the unit ball in R^d. 2000 Mathematics Subject Classification33C50, 33C55, 42C10.     

On ratio asymptotics for general polynomials

In this paper some characterizations of the ratio asymptotics for general polynomials are given. These results are extensions and improvements of the ratio asymptotics for orthogonal polynomials and are applicable to the ratio asymptotics for polynomials with disturbed nodes.     

Interval-polynomial stability theory and its applications in testing the strict positive realness of interval transfer functions

Based on value-set geometry and vector operations in the complexplane, this paper improves some early results on the robustD-stability of an interval polynomial. Almost strong Kharitonov-typeresults for some typical stability regions D are presented.Some connections between the critical vertex polynomials withrespect to these stability regions are established. Explicitupper bounds for the number of critical vertex polynomials associatedwith each stability region are derived. We also present a simpledirect procedure for construction of the critical vertex polynomialswith respect to the left-sector stability region. Illustrativeexamples are given. Using the stability theory of interval polynomials,some strong Kharitonov-type results are obtained for strictpositive realness of interval rational functions.     

Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities

We briefly review some asymptotics of orthonormal polynomials. Then we derive the Bernstein–Szeg, the Riemann–Hilbert (or Fokas–Its–Kitaev), and Rakhmanov projection identities for orthogonal polynomials and attempt a comparison of their applications in asymptotics.     

Asymptotics of partition functions in a fermionic matrix model and of related q‐polynomials

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix‐like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes‐Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general q‐polynomials.