Antonio J. Durán

Orthogonal matrix polynomials and higher-order recurrence relations

Abstract It is well known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel measure on the real line. We extend this result and show that every system of polynomials satisfying some (2 N +1)-term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are N × N matrices. We apply this result to polynomials orthogonal with respect to a discrete Sobolev inner product and other inner products in the linear space of polynomials. As an application we give a short proof of Kreins characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points.

Orthogonal matrix polynomials satisfying second-order differential equations

We develop a general method that allows us to introduce families of orthogonal matrix polynomials of size N × N satisfying second-order differential equations. The presence of this extra property should make these orthogonal polynomials into useful tools in several areas of mathematics and its applications. Historically, this has certainly been the case for their scalar-valued versions. The subtlety of the noncommutative algebra of matrices can be exploited to yield many different such families, almost dwarfing the scalar situation by comparison. All these families form a nice and rich hierarchy starting from the classical Jacobi, Hermite, and Laguerre families, N=1, and increasing in number and variety as the size N increases. We illustrate the use of our method by giving large classes of generic examples of arbitrary size N.

On orthogonal polynomials with respect to a positive definite matrix of measures

In this paper, we prove that any sequence of polynomials (pn ) n for which dgr(pn ) = n which satisfies a (2N + l)-term recurrence relation is orthogonal with respect to a positive definite N × N matrix of measures. We use that result to prove asymptotic properties of the kernel polynomials associated to a positive measure or a positive definite matrix of measures. Finally, some examples are given.

The index of determinacy for measures and the ²-norm of orthonormal polynomials

For determinate measures # having moments of every order we define and study an index of determinacy which checks the stability of determinacy under multiplication by even powers of It z for z a complex number. Using this index of determinacy, we solve the problem of determining for which z E C: the sequence (P(m)(Z))n ( m E N ) belongs to t2, where (Pn)n is the sequence of orthonormal polynomials associated with the measure ,u.

A transformation from Hausdorff to Stieltjes moment sequences

We introduce a non-linear injective transformation τ from the set of non-vanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formulaT[(an)n=1∞]n = 1/a1 ...an. Special cases of this transformation have appeared in various papers on exponential functionals of Lévy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related toq-series.

Full length article: Exceptional Meixner and Laguerre orthogonal polynomials

Using Casorati determinants of Meixner polynomials (m n )n , we construct for each pair F = (F1, F2) of finite sets of positive integers a sequence of polynomials ma,c;F n , n ∈ σF , which are eigenfunctions of a second order difference operator, where σF is certain infinite set of nonnegative integers, σF N. When c and F satisfy a suitable admissibility condition, we prove that the polynomials ma,c;F n , n ∈ σF , are actually exceptional Meixner polynomials; that is, in addition, they are orthogonal and complete with respect to a positive measure. By passing to the limit, we transform the Casorati determinant of Meixner polynomials into a Wronskian type determinant of Laguerre polynomials (Ln )n . Under the admissibility conditions for F and α, these Wronskian type determinants turn out to be exceptional Laguerre polynomials. c ⃝ 2014 Elsevier Inc. All rights reserved.

Full length article: Using D-operators to construct orthogonal polynomials satisfying higher order difference or differential equations

We introduce the concept of D-operators associated to a sequence of polynomials (pn)n and an algebra A of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate families of orthogonal polynomials which are eigenfunctions of a higher order difference or differential operator. Indeed, given a classical discrete family (pn)n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we form a new sequence of polynomials (qn)n by considering a linear combination of two consecutive pn: qn = pn + βnpn−1, βn ∈ R. Using the concept of D-operator, we determine the structure of the sequence (βn)n in order that the polynomials (qn)n are common eigenfunctions of a higher order difference operator. In addition, we generate sequences (βn)n for which the polynomials (qn)n are also orthogonal with respect to a measure. The same approach is applied to the classical families of Laguerre and Jacobi polynomials.

Laguerre expansions of tempered distributions and generalized functions

Abstract Let (Lnα(t))n be for α > −1, the sequence of generalized Laguerre polynomials. Then we consider the orthonormal system (( n! Γ(n + α + 1) ) 1 2 L n α (t) t α 2 e −t 2 ) n in L2([0, + ∞)). We study the expansions of certain spaces of generalized functions with respect to this orthonormal system (for α = 0, this space of generalized functions is the space of tempered distributions with positive support). We characterize the sequences of Fourier-Laguerre coefficients which appear in these expansions. Finally we give some applications.

Density questions for the truncated matrix moment problem

For a truncated matrix moment problem, we describe in detail the set of positive definite matrices of measures n in V2n (this is the set of solutions of the problem of degree 2n) for which the polynomials up to degree n are dense in the corresponding space L2(n). These matrices of measures are exactly the extremal measures of the set Vn. This work has been partially supported by DGICYT ref. PB93-0926. Received by the editors October 2, 1995; revised July 9, 1996. AMS subject classification: 42C05, 44A60. c Canadian Mathematical Society 1997.

Exceptional Charlier and Hermite orthogonal polynomials

Using Casorati determinants of Charlier polynomials (ca n )n , we construct for each finite set F of positive integers a sequence of polynomials cF n , n ∈ σF , which are eigenfunctions of a second order difference operator, where σF is certain infinite set of nonnegative integers, σF ( N. For suitable finite sets F (we call them admissible sets), we prove that the polynomials cF n , n ∈ σF , are actually exceptional Charlier polynomials; that is, in addition, they are orthogonal and complete with respect to a positive measure. By passing to the limit, we transform the Casorati determinant of Charlier polynomials into a Wronskian determinant of Hermite polynomials. For admissible sets, these Wronskian determinants turn out to be exceptional Hermite polynomials. c ⃝ 2014 Elsevier Inc. All rights reserved.