Orthogonal polynomials with complex-valued weight function, II

Abstract

In this paper we continue our study of the asymptotic behavior of polynomialsQ _mn(z), m, n ∈N, of degree≤n satisfying the orthogonal relation

$$( * )\oint_c {\zeta ^l Q_{mn} (\zeta )} \frac{{f(\zeta )d\zeta }}{{\omega _{m + n} (\zeta )}} = 0,l = 0,...,n - 1,$$

((*))

and all its singularities are supposed to be contained in a set$E \subseteq \hat C$ of capacity zero, ω _m+n (z) is a polynomial of degreem+n+1 with all its zeros contained inV, and the path of integrationC separatesV from the setE. We state and prove results concerning the asymptotic magnitude of the integral in (*) forl=n,n+1,⋯.