The objective of this manuscript is to study directly the Favard type theorem associated with the three term recurrence formulais a real sequence and {d n } ∞ n=1 is a positive chain sequence. We establish that there exists an unique nontrivial probability measure µ on the unit circle for which {R n (z) − 2(1 − m n )R n−1 (z)} gives the sequence of orthogonal polynomials. Here, {m n } ∞ n=0 is the minimal parameter sequence of the positive chain sequence {d n } ∞ n=1 . The element d 1 of the chain sequence, which does not effect the polynomials R n , has an influence in the derived probability measure µ and hence, in the associated orthogonal polynomials on the unit circle. To be precise, if {M n } ∞ n=0 is the maximal parameter sequence of the chain sequence, then the measure µ is such that M 0 is the size of its mass at z = 1. An example is also provided to completely illustrates the results obtained.

a b s t r a c t Keywords:Orthogonal polynomials on the real line Co-polynomials on the real line Zeros Transfer matricesIn this paper, we study new algebraic and analytic aspects of orthogonal polynomials on the real line when finite modifications of the recurrence coefficients, the socalled co-polynomials on the real line, are considered. We investigate the behavior of their zeros, mainly interlacing and monotonicity properties. Furthermore, using a transfer matrix approach we obtain new structural relations, combining theoretical and computational advantages. Finally, a connection with the theory of orthogonal polynomials on the unit circle is pointed out.

Motivated by the works of Genin (1988, 1991), who studied paraorthogonal polynomials associated with positive definite Hermitian linear functionals and their corresponding recurrence relations, we provide paraorthogonality theory, in the context of quasidefinite Hermitian linear functionals, with a recurrence relation and the analogous result to the classical Favard's theorem or spectral theorem. As an application of our results, we prove that for any two monic polynomials whose zeros are simple and strictly interlacing on the unit circle, with the possible exception of one of them which could be common, there exists a sequence of paraorthogonal polynomials such that these polynomials belong to it. Furthermore, an application to the computation of Szegő quadrature formulas is also discussed.

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