Orthogonal polynomials for modified Gegenbauer weight and corresponding quadratures

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