Asymptotics for polynomials orthogonal in an indefinite metric

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 ...

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...

Asymptotics for Orthogonal Polynomials off the Circle

We study the strong asymptotics of orthogonal polynomials with respect to a measure of the type dμ/2π + ∑∞ j=1Ajδ(z− zk), where μ is a positive measure on the unit circle Γ satisfying the Szegö condition and {zj}j=1 are fixed points outside Γ. The masses {Aj}j=1 are positive numbers such that ∑∞ j=1Aj < +∞. Our main result is the explicit strong asymptotic formulas for the corresponding orthogo...

Asymptotics of Orthogonal Polynomials via Recurrence Relations