Orthogonal polynomials for modified Gegenbauer weight and corresponding quadratures

In this paper we consider polynomials orthogonal with respect to the linear functional L:P→CL:P→C, defined by L[p]=∫−11p(x)(1−x2)λ−1/2exp(iζx)dx, where PP is a linear space of all algebraic polynomials, λ>−1/2λ>−1/2 and ζ∈Rζ∈R. We prove the existence of such polynomials for some pairs of λλ and ζζ, give some their properties, and finally give an application to numerical integration of highly oscillatory functions.