Orthogonal Laurent Polynomials of Jacobi, Hermite and Laguerre Types

We will outline in this paper a general procedure for constructing whole systems of orthogonal Laurent polynomials on the real line from systems of orthogonal polynomials. To further explain our intentions, we proceed with some basic definitions and results, all of which appear, or are modifications of those that appear, in the literature. In particular, [1,2,3,4,5,6,7,8,9,10,11,12] were consulted in our preparation. If f :D → R, where D is a subset of the set of real numbers R, then the set σ (f) := [x ∈ D : There is an 2 > 0 such that (x− 2, x + 2) ⊆ D and f(x + δ)− f(x− δ) > 0 for all δ > 0 such that δ < 2] is called the spectrum of f . If ψ:R → R is a bounded, non-decreasing function with an infinite spectrum σ(ψ) such that the moments μn(ψ) defined by the RiemannStieltjes integrals