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We consider the indeterminate Stieltjes moment problem associated with the Stieltjes-Wigert polynomials. After a presentation of the well-known solutions, we study a transformation T of the set of solutions. All the classical solutions turn out to be fixed under this transformation but this is not the case for the so-called canonical solutions. Based on generating func
We show that for a positive linear operator acting in ℓ2 and defined from anxn+1 + bn xn + an-1xn-1 its so-called Friedrichs and Krein extensions may be explicitly characterized by boundary conditions as n → ∞.
We consider indeterminate symmetric moment problems related to birth and death processes with quartic rates. The Nevanlinna matrices are presented and the entire functions B and D are expressed in terms of the trigonometric functions of order 4. Several examples of symmetric and nonsymmetric solutions are given.
We study the Al-Salam-Chihara polynomials when q > 1. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with re
The symmetric Al-Salam-Chihara polynomials for q > 1 are associated with an indeterminate moment problem. There is a self-adjoint second-order difference operator on 2(Z) to which these polynomials are eigenfunctions. We determine the spectral decomposition of this self-adjoint operator. This leads to a class of discrete orthogonality measures, which have been obtained previously by Christiansen a
We consider Jacobi matrices whose essential spectrum is a finite union of closed intervals. We focus on Szego{double acute}'s theorem, Jost solutions, and Szego{double acute} asymptotics for this situation. This announcement describes talks the authors gave at OPSFA 2007.
The generalized Stieltjes-Wigert polynomials depending on parameters 0≤;p<1 and 0
Let e ⊂ ℝ be a finite union of disjoint closed intervals. We study measures whose essential support is e and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szego{double acute} condition is equivalent to (this includes prior results of Widom and Peherstorfer-Yuditskii). Using Remling's extension of the Denisov-Rakhmanov theorem and an analysis of Jost functions, we provide a
Let e ⊂ ℝ be a finite union of disjoint closed intervals. In the study of orthogonal polynomials on the real line with measures whose essential support is e, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a
We study the spectral properties of a class of Sturm-Liouville type operators on the real line where the derivatives are replaced by a q-difference operator which has been introduced in the context of orthogonal polynomials. Using the relation of this operator to a direct integral of doubly-infinite Jacobi matrices, we construct examples for isolated pure point, dense pure point, purely absolutely
The Stieltjes-Wigert polynomials, which correspond to an indeterminate moment problem on the positive half-line, are eigenfunctions of a second order q-difference operator. We consider the orthogonality measures for which the difference operator is symmetric in the corresponding weighted L2-spaces. Under some additional assumptions these measures are exactly the solutions to the q-Pearson equation
Contact allergy (CA) is not uncommon in the population, including to various fragrance allergens. If not diagnosed correctly, allergic contact dermatitis (ACD) may ensue, because targeted allergen avoidance is not possible. The primary objective of the study is to estimate the prevalence of CA to seven fragrance materials in patients with suspected ACD across Europe. Based on the outcome, a conclu
Background: Eating disorders significantly impact physical health and have negative psychosocial consequences. The global burden of anorexia nervosa and bulimia nervosa among young females is estimated to account for 1.9 million disability-adjusted life years. Early identification of disordered eating attitudes is critical for preventing the progression to more severe forms of eating disorders; ho
We survey results on Chebyshev polynomials centered around the work of H. Widom. In particular, we discuss asymptotics of the polynomials and their norms and general upper and lower bounds for the norms. Several open problems are also presented.
This is the first of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 1 contains most of the material on orthogonal polynomials, from the classical orthogonal polynomials of Hermite, Laguerre and Jacobi to the Askey–Wilson polynomials, which are the most general basic hypergeometric orthogonal polynomials. Separate chapter
De kristnas historia i Mellanöstern under tvåtusen år.
