Upward Extension of the Jacobi Matrix for Orthogonal Polynomials

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrixrnew rows and columns, so that the original Jacobi matrix is shifted downward. Thernew rows and columns contain 2rnew parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials, and the 2rnew parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the 1?orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials.     

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.     

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 Stieltjes type theorems and location of common zeros of multivariate orthogonal polynomials

Invariant factors of bivariate orthogonal polynomials inherit most of the properties of univariate orthogonal polynomials and play an important role in the research of Stieltjes type theorems and location of common zeros of bivariate orthogonal polynomials. The aim of this paper is to extend our study of invariant factors from two variables to several variables. We obtain a multivariate Stieltjes type theorem, and the relationships among invariant factors, multivariate orthogonal polynomials and the corresponding Jacobi matrix. We also study the location of common zeros of multivariate orthogonal polynomials and provide some examples of tri-variate.     

Elliptic polynomials orthogonal on the unit circle with a dense point spectrum

We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of the Jacobi elliptic functions. We find explicit expression for these polynomials in terms of elliptic hypergeometric functions. We show that the obtained polynomials are orthogonal on the unit circle with respect to a dense point measure. We also construct corresponding explicit systems of polynomials orthogonal on the interval of the real axis with respect to a dense point measure. They can be considered as an elliptic generalization of the Askey-Wilson polynomials of a special type.      

Characterizations of Classical Orthogonal Polynomials

We give a simple unified proof and an extension of some of the characterization theorems of classical orthogonal polynomials of Jacobi, Bessel, Laguerre, and Hermite. In particular, we prove that the only orthogonal polynomials whose derivatives form a weak orthogonal polynomial set are the classical orthogonal polynomials.     

Computing orthogonal polynomials on a triangle by degree raising

We give an algorithm for computing orthogonal polynomials over triangular domains in Bernstein–Bézier form which uses only the operator of degree raising and its adjoint. This completely avoids the need to choose an orthogonal basis (or tight frame) for the orthogonal polynomials of a given degree, and hence the difficulties inherent in that approach. The results are valid for Jacobi polynomials on a simplex, and show the close relationship between the Bernstein form of Jacobi polynomials, Hahn polynomials and degree raising.     

Matrix Pearson Equations Satisfied by Koornwinder Weights in Two Variables

We consider Koornwinder’s method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder’s construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with these polynomials, and consequently, we deduce the second order linear partial differential operators for classical Koornwinder polynomials.     

On the orthogonality of residual polynomials\newline of minimax polynomial preconditioning

Summary. This paper studies polynomials used in polynomial preconditioning for solving linear systems of equations. Optimum preconditioning polynomials are obtained by solving some constrained minimax approximation problems. The resulting residual polynomials are referred to as the de Boor-Rice and Grcar polynomials. It will be shown in this paper that the de Boor-Rice and Grcar polynomials are orthogonal polynomials over several intervals. More specifically, each de Boor-Rice or Grcar polynomial belongs to an orthogonal family, but the orthogonal family varies with the polynomial. This orthogonality property is important, because it enables one to generate the minimax preconditioning polynomials by three-term recursive relations. Some results on the convergence properties of certain preconditioning polynomials are also presented. Received February 1, 1992/Revised version received July 7, 1993     

Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.     

Exceptional Laurent biorthogonal polynomials through spectral transformations of generalized eigenvalue problems

A formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second-order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional-type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen–van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar to the cases of exceptional extensions of classical orthogonal polynomials, both state-deletion and state-addition occur.     

A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

In this paper we deal with a family of nonstandard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so-called Δ-Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the Δ-Meixner–Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre–Sobolev orthogonal polynomials.     

On Fourth-order Difference Equations for Orthogonal Polynomials of a Discrete Variable: Derivation,Factorization and Solutions

We derive and factorize the fourth-order difference equations satisfied by orthogonal polynomials obtained from some modifications of the recurrence coefficients of classical discrete orthogonal polynomials such as: the associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified classical orthogonal polynomials. Moreover, we find four linearly independent solutions of these fourth-order difference equations, and show how the results obtained for modified classical discrete orthogonal polynomials can be extended to modified semi-classical discrete orthogonal polynomials. Finally, we extend the validity of the results obtained for the associated classical discrete orthogonal polynomials with integer order of association from integers to reals.     

On the Christoffel–Darboux formula for generalized matrix orthogonal polynomials

We obtain an extension of the Christoffel–Darboux formula for matrix orthogonal polynomials with a generalized Hankel symmetry, including the Adler–van Moerbeke generalized orthogonal polynomials.     

On a class of polynomials orthogonal with respect to a discrete Sobolev inner product

This paper analyzes polynomials orthogonal with respect to the Sobolev inner product with and (x)is a weight function.We study this family of orthogonal polynomials, as linked to the polynomials orthogonal with respect to (x) and we find the recurrence relation verified by such a family. If the weight is semiclassical we obtain a second order differential equation for these polynomials. Finally, an illustrative example is shown.     

An analogue of the big <Emphasis Type="Italic">q</Emphasis>-Jacobi polynomials in the algebra of symmetric functions

It is well known how to construct a system of symmetric orthogonal polynomials in an arbitrary finite number of variables from an arbitrary system of orthogonal polynomials on the real line. In the special case of the big q-Jacobi polynomials, the number of variables can be made infinite. As a result, in the algebra of symmetric functions, there arises an inhomogeneous basis whose elements are orthogonal with respect to some probability measure. This measure is defined on a certain space of infinite point configurations and hence determines a random point process.     

On Koornwinder classical orthogonal polynomials in two variables

In 1975, Tom Koornwinder studied examples of two variable analogues of the Jacobi polynomials in two variables. Those orthogonal polynomials are eigenfunctions of two commuting and algebraically independent partial differential operators. Some of these examples are well known classical orthogonal polynomials in two variables, such as orthogonal polynomials on the unit ball, on the simplex or the tensor product of Jacobi polynomials in one variable, but the remaining cases are not considered classical by other authors. The definition of classical orthogonal polynomials considered in this work provides a different perspective on the subject. We analyze in detail Koornwinder polynomials and using the Koornwinder tools, new examples of orthogonal polynomials in two variables are given.     

Orthogonal Structure on a Wedge and on the Boundary of a Square

Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier orthogonal expansions is studied. These are used to establish analogous results for orthogonal polynomials on the boundary of the square. As an application, we study the statistics of the associated determinantal point process and use the basis to calculate Stieltjes transforms.

     

Computing orthogonal polynomials in Sobolev spaces

Summary. Numerical methods are considered for generating polynomials orthogonal with respect to an inner product of Sobolev type, i.e., one that involves derivatives up to some given order, each having its own (positive) measure associated with it. The principal objective is to compute the coefficients in the increasing-order recurrence relation that these polynomials satisfy by virtue of them forming a sequence of monic polynomials with degrees increasing by 1 from one member to the next. As a by-product of this computation, one gains access to the zeros of these polynomials via eigenvalues of an upper Hessenberg matrix formed by the coefficients generated. Two methods are developed: One is based on the modified moments of the constitutive measures and generalizes what for ordinary orthogonal polynomials is known as "modified Chebyshev algorithm". The other - a generalization of "Stieltjes's procedure" - expresses the desired coefficients in terms of a Sobolev inner product involving the orthogonal polynomials in question, whereby the inner product is evaluated by numerical quadrature and the polynomials involved are computed by means of the recurrence relation already generated up to that point. The numerical characteristics of these methods are illustrated in the case of Sobolev orthogonal polynomials of old as well as new types. Based on extensive numerical experimentation, a number of conjectures are formulated with regard to the location and interlacing properties of the respective zeros. Received July 13, 1994 / Revised version received September 26, 1994     

Krall-type orthogonal polynomials in several variables

For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional defined in the linear space of polynomials in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials.