Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type

In this paper we consider polynomials orthogonal with respect to the linear functional L : P -> C, defined on the space of all algebraic polynomials P by L[p] = integral(1)(-1)p(x)(1 - x)(alpha-1/2)(1 + x)(beta-1/2)exp(i zeta x)dx, where alpha, beta > 1/2 are real numbers such that l = vertical bar beta - alpha vertical bar is a positive integer, and zeta is an element of R\{0}. We prove the existence of such orthogonal polynomials for some pairs of alpha and zeta and for all nonnegative integers l. For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations. For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered. Also, some numerical examples are included.