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.

     

Structure relations for monic orthogonal polynomials in two discrete variables

In this paper, extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given, by using an appropriate vector–matrix notation. Three-term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type. Structure relations, difference representations as well as lowering and raising operators are obtained. Finally, expressions for all matrix coefficients appearing in these finite-type relations are explicitly presented for a finite set of Hahn and Kravchuk orthogonal polynomials.     

Combinatorial formula for Macdonald polynomials and generic Macdonald polynomials

We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, and which depend on d additional parameters and specialize to all Macdonald polynomials of degree d. The form of these generic polynomials is that of a Bethe eigenfunction and they imitate, on a more elementary level, the R-matrix construction of quantum immanants.     

Multiple orthogonal polynomials for classical weights

A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to 1$"> weights satisfying Pearson's equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order . We also obtain explicit formulas and recurrence relations for these polynomials.

     

A characterization of classical orthogonal Laurent polynomials

In [3] certain Laurent polynomials of ₂F₁ genus were called “Jacobi Laurent polynomials”. These Laurent polynomials belong to systems which are orthogonal with respect to a moment sequence ((a)_n/(c)_n)_nεℤ where a, c are certain real numbers. Together with their confluent forms, belonging to systems which are orthogonal with respect to 1/(c)_n)_nεℤ respectively ((a)_n)_nεℤ, these Laurent polynomials will be called “classical”. The main purpose of this paper is to determine all the simple (see section 1) orthogonal systems of Laurent polynomials of which the members satisfy certain second order differential equations with polynomial coefficients, analogously to the well known characterization of S. Bochner [1] for ordinary polynomials.     

Menke points on the real line and their connection to classical orthogonal polynomials

We investigate the properties of extremal point systems on the real line consisting of two interlaced sets of points solving a modified minimum energy problem. We show that these extremal points for the intervals [−1,1], [0,∞) and (−∞,∞), which are analogues of Menke points for a closed curve, are related to the zeros and extrema of classical orthogonal polynomials. Use of external fields in the form of suitable weight functions instead of constraints motivates the study of “weighted Menke points” on [0,∞) and (−∞,∞). We also discuss the asymptotic behavior of the Lebesgue constant for the Menke points on [−1,1].     

A new property of reproducing kernels for classical orthogonal polynomials

We exhibit a second-order differential operator commuting with the reproducing kernel $\text{∑}{}_{\mathrm{n\; ?\; 0}}{}^{\mathrm{T}}\text{φ}{}_{\mathrm{n}}\text{(λ)}\text{φ}{}_{\mathrm{n}}\text{(μ)}\text{h}{}_{\mathrm{n}}$ each time that {φ_n(λ)} is one of the classical orthogonal polynomials: Jacobi, Laguerre, Hermite and Bessel. This is the analog of a known property in the study of time and band-limited signals.     

Diophantine equations for classical continuous orthogonal polynomials

Let A, B, C denote rational numbers with AB ≠ 0 and m > n ≥ 3 arbitrary rational integers. We study the Diophantine equation AP_m(x) + Bp_n(y) = C, in x, y ? , where {Pk(x)}I is one of the three classical continuous orthogonal polynomial families, i.e. Laguerre polynomials, Jacobi polynomials (including Gegenbauer, Legendre or Chebyshev polynomials) and Hermite polynomials. We prove that with exception of the Chebyshev polynomials for all such polynomial families there are at most finitely many solutions (x, y) ? ² provided n > 4. The tools are besides the criterion [3], a theorem of Szeg— [14] on monotonicity of stationary points of polynomials which satisfy a second order Sturm-Liouville differential equation,

     

A Christoffel–Darboux formula for multiple orthogonal polynomials

Bleher and Kuijlaars recently showed that the eigenvalue correlations from matrix ensembles with external source can be expressed by means of a kernel built out of special multiple orthogonal polynomials. We derive a Christoffel–Darboux formula for this kernel for general multiple orthogonal polynomials. In addition, we show that the formula can be written in terms of the solution of the Riemann–Hilbert problem for multiple orthogonal polynomials, which will be useful for asymptotic analysis.     

Coefficients of multiplication formulas for classical orthogonal polynomials

In this paper using both analytic and algorithmic approaches, we derive the coefficients \(D_m(n,a)\) of the multiplication formula

$$\begin{aligned} p_n(ax)=\sum _{m=0}^nD_m(n,a)p_m(x) \end{aligned}$$

or the translation formula

$$\begin{aligned} p_n(x+a)=\sum _{m=0}^nD_m(n,a)p_m(x), \end{aligned}$$

where \(\{p_n\}_{n\ge 0}\) is an orthogonal polynomial set, including the classical continuous orthogonal polynomials, the classical discrete orthogonal polynomials, the \(q\)-classical orthogonal polynomials, as well as the classical orthogonal polynomials on a quadratic lattice and a \(q\)-quadratic lattice. We give a representation of the coefficients \(D_m(n,a)\) as a single, double or triple sum whereas in many cases we get simple representations.

     

A class of bivariate orthogonal polynomials and cubature formula

Summary. The existence of Gaussian cubature for a given measure depends on whether the corresponding multivariate orthogonal polynomials have enough common zeros. We examine a class of orthogonal polynomials of two variables generated from that of one variable. Received February 9, 1993 / Revised version received January 18, 1994     

Difference equations for discrete classical multiple orthogonal polynomials

For discrete multiple orthogonal polynomials such as the multiple Charlier polynomials, the multiple Meixner polynomials, and the multiple Hahn polynomials, we first find a lowering operator and then give a (r+1)th order difference equation by combining the lowering operator with the raising operator. As a corollary, explicit third order difference equations for discrete multiple orthogonal polynomials are given, which was already proved by Van Assche for the multiple Charlier polynomials and the multiple Meixner polynomials.     

Sharp bounds for the extreme zeros of classical orthogonal polynomials

Bounds for the extreme zeros of the classical orthogonal polynomials are obtained by a surprisingly simple method. Nevertheless, it turns out that, in most cases, the estimates obtained in this note are better than the best limits known in the literature.     

An interpolation-theoretical characterization of the classical orthogonal polynomials

Acta Mathematica Hungarica -     

A product formula for orthogonal polynomials associated with infinite distance-transitive graphs

The infinite, locally finite distance-transitive graphs form an extension of homogeneous trees and are described by two discrete parameters. The associated orthogonal polynomials may be regarded as spherical functions of certain Gelfand pairs or as characters of some polynomial hypergroups; they are certain Bernstein polynomials and admit a discrete nonnegative product formula. In this paper we use the graph-theoretic origin of these polynomials to derive the existence of positive dual continuous product and transfer formulas. The dual product formulas will be computed explicitly.     

Bounds for extreme zeros of some classical orthogonal polynomials

We use mixed three term recurrence relations typically satisfied by classical orthogonal polynomials from sequences corresponding to different parameters to derive upper (lower) bounds for the smallest (largest) zeros of Jacobi, Laguerre and Gegenbauer polynomials.