R II type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle

Abstract

We consider a sequence of polynomials {P n } n≥0 satisfying a special R II type recurrence relation where the zeros of P n are simple and lie on the real line. It turns out that the polynomial P n , for any n≥2, is the characteristic polynomial of a simple n×n generalized eigenvalue problem. It is shown that with this R II type recurrence relation one can always associate a positive measure on the unit circle. The orthogonality property satisfied by P n with respect to this measure is also obtained. Finally, examples are given to justify the results.