A Characterization of L-orthogonal Polynomials from Three Term Recurrence Relations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Polynomials generated by a three term recurrence relation: bounds for complex zeros

This paper deals with the zeros of polynomials generated by a certain three term recurrence relation. The main objective is to find bounds, in terms of the coefficients of the recurrence relation, for the regions where the zeros are located. In most part, the zeros are explored through an Eigenvalue representation associated with a corresponding Hessenberg rnatrix. Applications to Szego polynomials, para-orthogonal polynomials and polynomials with non-zero complex coefficients are considered. (C) 2004 Elsevier B.V. All rights reserved.

A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Szego polynomials: some relations to L-orthogonal and orthogonal polynomials

We consider the real Szego polynomials and obtain some relations to certain self inversive orthogonal L-polynomials defined on the unit circle and corresponding symmetric orthogonal polynomials on real intervals. We also consider the polynomials obtained when the coefficients in the recurrence relations satisfied by the self inversive orthogonal L-polynomials are rotated. (C) 2002 Elsevier B.V. B.V. All rights reserved.

Rational approximations of the Arrhenius integral using Jacobi fractions and gaussian quadrature

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Monotonicity of the zeros of orthogonal polynomials through related measures

Relation between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), is well known. We use this relation to study the monotonicity properties of the zeros of generalized orthogonal polynomials. As examples, the Jacobi, Laguerre and Charlier polynomials are considered. (c) 2005 Elsevier B.V. All rights reserved.

On linear combinations of L-orthogonal polynomials associated with distributions belonging to symmetric classes

This paper deals with the classes S-3(omega, beta, b) of strong distribution functions defined on the interval [beta(2)/b, b], 0 < beta < b <= infinity, where 2 omega epsilon Z. The classification is such that the distribution function psi epsilon S-3(omega, beta, b) has a (reciprocal) symmetry, depending on omega, about the point beta. We consider properties of the L-orthogonal polynomials associated with psi epsilon S-3(omega, beta, b). Through linear combination of these polynomials we relate them to the L-orthogonal polynomials associated with some omega epsilon S-3(1/2, beta, b). (c) 2004 Elsevier B.V. All rights reserved.

Modified Chebyshev algorithm: some applications

Two applications of the modified Chebyshev algorithm are considered. The first application deals with the generation of orthogonal polynomials associated with a weight function having singularities on or near the end points of the interval of orthogonality. The other application involves the generation of real Szego polynomials.

Companion orthogonal polynomials

We give some properties relating the recurrence relations of orthogonal polynomials associated with any two symmetric distributions d phi(1)(x) and d phi(2)(x) such that d phi(2)(x) = (I + kx(2))d phi(1)(x). AS applications of these properties, recurrence relations for many interesting systems of orthogonal polynomials are obtained.

Zeros of a family of hypergeometric para-orthogonal polynomials on the unit circle

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. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Long-Term Recurrence of Nonmelanoma Skin Cancer after Topical Methylaminolevulinate Photodynamic Therapy in a Dermato-Oncology Department

BACKGROUND: Most available studies on the efficacy of topical photodynamic therapy focus on short-to medium-term results. Long-term data are scarce. OBJECTIVE: To evaluate the long-term efficacy of photodynamic therapy with topical methylaminolevulinate to treat Bowen's disease and basal cell carcinoma in the clinical practice setting of a dermato-oncology department. METHODS: The study included patients diagnosed with Bowen's disease or basal cell carcinoma, and who received photodynamic therapy from 2004 to 2008. Treatment protocol and clinical follow-up were standardized. The primary endpoint was clinically observed recurrence in a previous photodynamic therapy-treated area. Descriptive and survival analyses were performed. RESULTS: A total of 31 Bowen's disease lesions and 44 superficial basal cell carcinoma were treated, with a median follow-up of 43.5 months. Recurrence was observed in 14 Bowen's disease lesions (53.8%) and in 11 superficial basal cell carcinoma (33.3%). Significantly higher estimates for recurrence rates were found in patients with Bowen's disease (p=0.0036) or those aged under 58 years (p=0.039). The risk of recurrence was higher in patients with Bowen's disease than in those with superficial basal cell carcinoma and younger patients. CONCLUSIONS: Recurrence should be considered when choosing to treat non-melanoma skin cancer with photodynamic therapy. Younger age and Bowen's disease were independent predictors for long-term recurrence, suggesting the need to establish an extended period of follow-up for this subset of patients.

Orthogonal polynomials and recurrence equations, operator equations and factorization

This article surveys the classical orthogonal polynomial systems of the Hahn class, which are solutions of second-order differential, difference or q-difference equations. Orthogonal families satisfy three-term recurrence equations. Example applications of an algorithm to determine whether a three-term recurrence equation has solutions in the Hahn class - implemented in the computer algebra system Maple - are given. Modifications of these families, in particular associated orthogonal systems, satisfy fourth-order operator equations. A factorization of these equations leads to a solution basis.

On perturbed Szego recurrences

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Does Full Wound Rupture following Median Pilonidal Closure Alter Long-Term Recurrence Rate?

OBJECTIVE The purpose of this study was to examine the recurrence rate of wound rupture in primary pilonidal sinus disease (PSD) after median closure. SUBJECTS AND METHODS A total of 583 patients from the German military cohort were interviewed. We compared the choice of surgical therapy, wound dehiscence (if present) and long-term recurrence-free survival for patients with primary open treatment, marsupialization and primary median treatment (closed vs. secondary open, respectively). Actuarial recurrence rate was determined using the Kaplan-Meier calculation with a follow-up of up to 20 years after primary PSD surgery. RESULTS Patients with excision followed by primary open wound treatment showed a significantly lower 5- than 10-year recurrence rate (8.3 vs. 11.2%) compared to the patients with primary midline closure (17.4 vs. 20.5%, p = 0.03). The 20-year recurrence rate was 28% in primary open wound treatment versus 44% in primary midline closure without wound rupture. In contrast to these findings, long-term recurrence rates following secondary open wound treatment (12.2% at 5 years vs. 17.1% at 10 years) tended to be higher (although not significantly, p = 0.57) compared to primary open treatment (8.3% at 5 years vs. 11.2% at 10 years). There was no statistical difference in long-term recurrence rates between secondary open and primary midline closure (p = 0.7). Hence, despite only a short wound closure time experienced before wound rupture, the patient does not fully benefit from an open wound treatment in terms of recurrence rate. CONCLUSION The postoperative pilonidal sinus wound rupture of primary midline closures did not significantly increase the 5- and 10-year long-term recurrence rates compared to uneventfully healing primary midline closures.