In the last years, the study of orthogonal polynomials and related topics like quadrature rules and Padé approximants with respect to a complex weight function, has become an interesting research topic where the first author of the paper under review, and his co-workers, have greatly contributed. In this paper, the case of the complex weight function \(w(x)= x\exp(imx)\) on \([-1,1]\) where \(m\) is an integer, is analyzed by proving the existence of a sequence of orthogonal polynomials along with algebraic and analytic properties. Gaussian quadrature rules are also proposed and numerically applied in the computation of integrals involving highly oscillatory integrands in connection with the calculation of the Fourier coefficients of a given function. The paper also contains a large number of numerical experiments which enables us to conjecture about properties satisfied by the coefficients in the three-term recurrence relation and the localization and multiplicity of the zeros of the orthogonal polynomials.