Asymptotic properties of polar polynomials

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 dθ on [−π, +π]. Let us consider {L n (z)} n∈AE , the system of monic orthogonal polynomial with respect to μ. We introduce a new class of polynomials {P n } , that we call polar polynomials associated to {L n (z)} n∈AE. For a fixed complex number α, P n (z) is solution of the following differential equation (n + 1) L n (z) = P n (z) + (z − α) P n (z). we study algebraic and asymptotic properties of the polar polynomials {P n } n∈AE .

Asymptotic of Second Order Polar Polynomials

“…A similar study has been done by A. Fundora, H. Pijeira and W. Urbina [1], in the case of [−1, +1]. More information on the history and applications of this concept of polar may be found in [1,10,12].…”