Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials

Uniform Asymptotics for Orthogonal Polynomials

We consider asymptotics of orthogonal polynomials with respect to a weight e ?Q(x) dx on R, where either Q(x) is a polynomial of even order with positive leading coeecient, or Q(x) = NV (x), where V (x) is real analytic on R and grows suuciently rapidly as jxj ! 1. We formulate the orthogonal polynomial problem as a Riemann-Hilbert problem following the work of Fokas, Its and Kitaev. We employ ...

Asymptotics of Orthogonal Polynomials via Recurrence Relations

Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials

The asymptotic contracted measure of zeros of a large class of orthogonal polynomials is explicitly given in the form of a Lauricella function. The polynomials are defined by means of a three-term recurrence relation whose coefficients may be unbounded but vary regularly and have a different behaviour for even and odd indices. Subclasses of systems of orthogonal polynomials having their contrac...

Comparative Asymptotics for Perturbed Orthogonal Polynomials

Let {Φn}n∈N0 and {Φ̃n}n∈N0 be such systems of orthonormal polynomials on the unit circle that the recurrence coefficients of the perturbed polynomials Φ̃n behave asymptotically like those of Φn. We give, under weak assumptions on the system {Φn}n∈N0 and the perturbations, comparative asymptotics as for Φ̃n(z)/Φ ∗ n(z) etc., Φ ∗ n(z) := z Φ̄n( 1 z ), on the open unit disk and on the circumference ma...

Strong asymptotics for Gegenbauer-Sobolev orthogonal polynomials

We study the asymptotic behaviour of the monic orthogonal polynomials with respect to the Gegenbauer-Sobolev inner product (f, g)S = 〈f, g〉 + λ〈f ′, g′〉 where 〈f, g〉 = ∫ 1 −1 f(x)g(x)(1 − x 2)α−1/2dx with α > −1/2 and λ > 0. The asymptotics of the zeros and norms of these polynomials is also established. The study of the orthogonal polynomials with respect to the inner products that involve der...