Orthogonal polynomials related to the oscillatory-chebyshev weight function

G. V. Milovanović, A. S. Cvetković

In this paper we discuss the existence question for polynomials orthogonal with respect to the moment functional \[ L(p)=\int_{-1}^1 p(x) x (1-x^2)^{-1/2}e^{\ij \zeta x} d x,\quad \zeta\in \RR.\] Since the weight function alternates in sign in the interval of orthogonality, the existence of orthogonal polynomials is not assured. A nonconstructive proof of the existence is given. The three-term recurrence relation for such polynomials is investigated and the asymptotic formulae for recursion coefficients are derived.