Orthogonality and recurrence for ordered Laurent polynomial sequences

We consider orderings of nested subspaces of the space of Laurent polynomials on the real line, more general than the balanced orderings associated with the ordered bases { 1 , z − 1 , z , z − 2 , z 2 , … } and { 1 , z , z − 1 , z 2 , z − 2 , … } . We show that with such orderings the sequence of orthonormal Laurent polynomials determined by a positive linear functional satisfies a three-term recurrence relation. Reciprocally, we show that with such orderings a sequence of Laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional.