Orthogonal polynomials and Gaussian quadrature rules related to oscillatory weight functions

In this paper we consider polynomials orthogonal with respect to an oscillatory weight function w ( x ) = x e i m π x on [ - 1 , 1 ] , where m is an integer. The existence of such polynomials as well as several of their properties (three-term recurrence relation, differential equation, etc.) are proved. We also consider related quadrature rules and give applications of such quadrature rules to some classes of integrals involving highly oscillatory integrands.