No abstract

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.

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.

Abstract. Szegő polynomials with respect to the weight function ω(θ) = e ηθ [sin(θ/2)] 2λ , where η, λ ∈ R and λ > −1/2 are considered. Many of the basic relations associated with these polynomials are given explicitly. Two sequences of para-orthogonal polynomials with explicit relations are also given.

When a nontrivial measure µ on the unit circle satisfies the symmetry dµ(e i(2π−θ) ) = −dµ(e iθ ) then the associated OPUC, say S n , are all real. In this case, in [12], Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials {zS n (z) + S * n (z)} and {zS n (z) − S * n (z)} satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [−1, 1]. The same authors, in [13], have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently in [10] and then [8] the extension associated with the para-orthogonal polynomials zS n (z)− S * n (z) was thoroughly explored, especially from the point of view of the three term recurrence, and chain sequences play an important part in this exploration. The main objective of the present manuscript is to provide the theory surrounding the extension associated with the para-orthogonal polynomials zS n (z) + S * n (z) for any nontrivial measure on the unit circle. Like in [10] and [8] chain sequences also play an important role in this theory. Examples and applications are also provided to justify the results obtained.

We consider certain quadrature rules of highest algebraic degree of precision that involve strong Stieltjes distributions (i.e., strong distributions on the positive real axis). The behavior of the parameters of these quadrature rules, when the distributions are strong c-inversive Stieltjes distributions, is given. A quadrature rule whose parameters have explicit expressions for their determination is presented. An application of this quadrature rule for the evaluation of a certain type of integrals is also given. (1991): 33A70, 65D32 Mathematics Subject Classification

No abstract

No abstract