A transformation of orthogonal polynomial sequences into orthogonal Laurent polynomial sequences

The uses of orthogonal Laurent polynomials in the mathematical and natural sciences and engineering are only beginning to be discovered. Indeed, this thesis is the first study dedicated to a comprehensive treatment of special classes of orthogonal Laurent polynomials on the real line.; Laurent polynomial analogues of the Poisson, Jacobi, Hermite, and Laguerre orthogonal polynomials are investigated, principally via the introduction and exploitation of a simple, yet general, transformation of systems of orthogonal polynomials. For fixed {dollar}lambda, gamma > 0{dollar}, {dollar}{dollar}upsilon(x) := {lcub}1overlambda{rcub} (x - {lcub}gammaover x{rcub}){dollar}{dollar}is called "the doubling transformation". Employing {dollar}v(x){dollar} and utilizing other previously known techniques, a variety of explicit expressions, formulas, and relations pertaining to each of the classical Laurent polynomial analogues are obtained. Included are results on moments, zeros, recurrence relations, L-polynomial coefficients. Rodrigues' type formulas, generating functions, and differential and difference equations.