On generalized Stieltjes-Wigert and related orthogonal polynomials

The Charlier, Wall, and generalized Stieltjes-Wigert polynomials are characterized by a property involving the concept of kernel polynomials. This characterization leads to consideration of a certain functional equation satisfied by solutions of the associated Stieltjes moment problem. All distribution functions which satisfy this functional equation are found up to singular functions. This yields new distribution functions, both discrete and absolutely continuous, with respect to which generalized Stieltjes-Wigert polynomials are orthogonal.     

On the linear functionals associated to linearly related sequences of orthogonal polynomials

An inverse problem is solved, by stating that the regular linear functionals u and v associated to linearly related sequences of monic orthogonal polynomials n(Pn) and n(Qn), respectively, in the sense

     

Special linear combinations of orthogonal polynomials

The paper lists a number of problems that motivate consideration of special linear combinations of polynomials, orthogonal with the weight p(x) on the interval (a,b). We study properties of the polynomials, as well as the necessary and sufficient conditions for their orthogonality. The special linear combinations of Chebyshev orthogonal polynomials of four kinds with absolutely constant coefficients hold a distinguished place in the class of such linear combinations.     

Asymptotic properties of generalized Laguerre orthogonal polynomials

In the present paper we deal with the polynomials L_n^(α,M,N) (x) orthogonal with respect to the Sobolev inner product

     

N-symmetric Chebyshev polynomials in a composite model of a generalized oscillator

We continue to study a composite model of a generalized oscillator generated by an N-periodic Jacobi matrix. The foundation of the model is a system of orthogonal polynomials connected to this matrix for N = 3, 4, 5. We show that such polynomials do not exist for N ?? 6.     

Chain sequences and symmetric generalized orthogonal polynomials

In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K_n^(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials K_n^(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K_n^(λ,M,k) in relation to M and k are also given.     

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACT

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.     

Multivariate generalized Bernstein polynomials: identities for orthogonal polynomials of two variables

We introduce polynomials $B^n_{k}(\boldmath{x};\omega|q)$ of total degree n, where $\boldmath{k} = (k_1,\ldots,k_d)\in\mathbb N_0^d, \; 0\le k_1+\ldots+k_d\le n$ , and $\boldmath{x}=(x_1,x_2,\ldots,x_d)\in\mathbb R^d$ , depending on two parameters q and ω, which generalize the multivariate classical and discrete Bernstein polynomials. For ω=0, we obtain an extension of univariate q-Bernstein polynomials, introduced by Phillips (Ann Numer Math 4:511–518, 1997). Basic properties of the new polynomials are given, including recurrence relations, q-differentiation rules and de Casteljau algorithm. For the case d=2, connections between $B^n_{k}(\boldmath{x};\omega|q)$ and bivariate orthogonal big q-Jacobi polynomials—introduced recently by the first two authors—are given, with the connection coefficients being expressed in terms of bivariate q-Hahn polynomials. As limiting forms of these relations, we give connections between bivariate q-Bernstein and Dunkl’s (little) q-Jacobi polynomials (SIAM J Algebr Discrete Methods 1:137–151, 1980), as well as between bivariate discrete Bernstein and Hahn polynomials.     

Orthogonality properties of linear combinations of orthogonal polynomials

Let {P _n } be a sequence of orthogonal polynomials with respect to the measured on the unit circle and letP _n =P _n + _j ^=1l _nj P _n–j fornl, where _n,j . It is shown that the sequence of linear combinations {P _n },n2l, is orthogonal with respect to a positive measured if and only ifd is a Bernstein-Szegö measure andd is the product of a unique trigonometric polynomial and the Bernstein-Szegö measured. Furthermore for a given sequence ofP _n 's an algorithm for the calculation of the _n,j 's is provided.Supported by Dirección General de Investigación Cientifica y Técnica (DGICYT) of Spain and Österreichischer Akademischer Austauschdienst of Austria with grant 4B/1995.Also supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, project-number P9267-PHY.     

Extremal measures for a system of orthogonal polynomials

We characterize the extremal measures of an indeterminate moment problem associated with a system of orthogonal polynomials defined by a three-term recurrence relation.     

The zeros of linear combinations of orthogonal polynomials

Let {p_n} be a sequence of monic polynomials with p_n of degree n, that are orthogonal with respect to a suitable Borel measure on the real line. Stieltjes showed that if m<n and x₁,…,x_n are the zeros of p_n with x₁<<x_n then there are m distinct intervals f the form (x_j,x_j+1) each containing one zero of p_m. Our main theorem proves a similar result with p_m replaced by some linear combinations of p₁,…,p_m. The interlacing of the zeros of linear combinations of two and three adjacent orthogonal polynomials is also discussed.     

Repeated modifications of orthogonal polynomials by linear divisors

Algorithms are developed for computing the coefficients in the three-term recurrence relation of repeatedly modified orthogonal polynomials, the modifications involving division of the orthogonality measure by a linear function with real or complex coefficient. The respective Gaussian quadrature rules can be used to account for simple or multiple poles that may be present in the integrand. Several examples are given to illustrate this.     

The generalized Bochner condition about classical orthogonal polynomials revisited

We bring a new proof for showing that an orthogonal polynomial sequence is classical if and only if any of its polynomial fulfils a certain differential equation of order 2k, for some k?1. So, we build those differential equations explicitly. If k=1, we get the Bochner's characterization of classical polynomials. With help of the formal computations made in Mathematica, we explicitly give those differential equations for k=1,2 and 3 for each family of the classical polynomials. Higher order differential equations can be obtained similarly.     

A note on the Geronimus transformation and Sobolev orthogonal polynomials

In this note we recast the Geronimus transformation in the framework of polynomials orthogonal with respect to symmetric bilinear forms. We also show that the double Geronimus transformations lead to non-diagonal Sobolev type inner products.     

Convex linear combinations of sequences of monic orthogonal polynomials

For a sequence of monic orthogonal polynomials (SMOP), with respect to a positive measure supported on the unit circle, we obtain necessary and sufficient conditions on a SMOP in order that a convex linear combination with be a SMOP with respect to a positive measure supported on the unit circle.

     

Orthogonality properties of linear combinations of orthogonal polynomials II

Let and be polynomials orthogonal on the unit circle with respect to the measures dσ and dμ, respectively. In this paper we consider the question how the orthogonality measures dσ and dμ are related to each other if the orthogonal polynomials are connected by a relation of the form , for , where . It turns out that the two measures are related by if , where and are known trigonometric polynomials of fixed degree and where the 's are the zeros of on . If the 's and 's are uniformly bounded then (under some additional conditions) much more can be said. Indeed, in this case the measures dσ and dμ have to be of the form and , respectively, where are nonnegative trigonometric polynomials. Finally, the question is considered to which weight functions polynomials of the form where denotes the reciprocal polynomial of , can be orthogonal. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

A norm estimation for the generalized quasitranslation operator by orthogonal polynomials

Moscow Institute of Construction Engineering. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 1, pp. 61–63, January–March, 1992.