Comportement asymptotique des polynômes polaires primitives des polynômes orthogonaux sur le contour

Abstract

Let μ be a finite positive measure defined on the Borelian σ− algebra of C, μ is absolutely continuous with respect to the Lebesgue measure |dξ| on contour E. i.e : dμ = ρ(ξ)|dξ|,

∀ β ∈ B(C) : β(B) = Z B∩E ρ(ξ)|dξ|

Let us consider {Ln(z)}n∈N

, the system of monic orthogonal polynomial with respect to μ.

Ln(z) = z

n + ......., Z E Ln(ξ)ξ p dμ = 0 ; p = 0, 1, 2, ...., n − 1. We introduce a new class of polynomials {Pn,α} , that we call polar polynomials associated to {Ln(z)}n∈N . For a fixed complex number α, Pn,α (z) is solution of the following differential equation (n + 1) Ln (z) = Pn,α (z) + (z − α) P 0 n,α (z). We study algebraic and asymptotic properties of the polar polynomials.