Distribution approximation and modelling via orthogonal polynomial sequences

A given weight function \(p(x)\) on an interval \([a, b]\) defines uniquely, subject to normalization, a sequence of orthogonal polynomials \(P_n(x)\) and their corresponding sequence of functions of the second kind $$\begin{aligned} Q_n(x)=\int _a^b\frac{P_n(t)p(t)dt}{x-t},\quad \ x\in \mathbf{C } \backslash [a,b]. \end{aligned}$$ corresponding to them. This paper focuses on the reverse problem of characterizing orthogonal polynomials by means of functions of the second kind, together with the properties of such functions of the second kind. Functions of the second kind for orthogonal polynomials are also of particular interest in that they differ only slightly from the second solution of the differential equation satisfied by the orthogonal polynomials.

Let (P ν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ωυ) the monic associated polynomials of (P v), and letA andB be self-reciprocal polynomials. We show that the sequence of polynomials (APυλ+BΩυλ)/Aλ, λ stuitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as theP ν's from a certain index value onward, and determine the orthogonality measure explicity. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order ofP ν and of the corresponding orthogonality measure and Szego function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally necessary and suficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.

Orthogonality of linear combinations of two orthogonal polynomial sequences

Companion orthogonal polynomials: some applications

The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential polynomial systems (X-Jacobi DPSs) of series J1, J2, J3, and W. The first three X-DPSs formed by pseudo-Wronskians of two Jacobi polynomials contain both exceptional orthogonal polynomial systems (X-Jacobi OPSs) on the interval (−1, +1) and the finite EOP sequences on the positive interval (1, ∞). On the contrary, the X-DPS of series W formed by Wronskians of two Jacobi polynomials contains only (infinitely many) finite EOP sequences on the interval (1, ∞). In addition, the paper rigorously examines the three isospectral families of the associated Liouville potentials (rationally extended hyperbolic Pöschl-Teller potentials of types a, b, and a′) exactly quantized by the EOPs in question.

Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities

This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials.

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

We study properties of the form (linear functional) u = λ(x – a)–1 ν + δb , where ν is a regular form. We give a necessary and sufficient condition for the regularity of the form u. The coefficients of a three-term recurrence relation, satisfied by the corresponding sequence of orthogonal polynomials, are given explicitly. The semi-classical character of the founded families is studied. We conclude by giving some examples.

ABSTRACTWe investigate the sequence of random polynomials generated by the three-term recurrence relation , , with initial conditions , , assuming that is a sequence of positive i.i.d. random variables. is a sequence of orthogonal polynomials on the real line, and is the characteristic polynomial of a Jacobi matrix . We investigate the relation between the common distribution of the recurrence coefficients and two other distributions obtained as weak limits of the averaged empirical and spectral measures of . Our main result is a description of combinatorial relations between the moments of the aforementioned distributions in terms of certain classes of coloured planar trees. Our approach is combinatorial, and the starting point of the analysis is a formula of P. Flajolet for weight polynomials associated with labelled Dyck paths.

Numerical computation of BCOPs in two variables for solving the vibration problem of a CF-elliptical plate

In this paper, we determine all pairs of orthogonal polynomial sequences $\{ p_n (x)\} $ and $\{ q_n (x)\} $, such that their convolution, \[ Q_n (x,y) = \sum_{k = 0}^n {p_k (x)q_{n - k} (y)} ,\quad n \geqq 0.\] defines $\{ Q_n (x,y)\} $ as an orthogonal polynomial sequence in x for all y. All such triples are determined explicitly in terms of their three-term recurrence formulas. Generating functions and “explicit” representation formulas are obtained. The resulting sequences are found to consist of a class of orthogonal polynomials characterized by J. Meixner (which class includes the Laguerre and Hermite polynomials) together with a new class of orthogonal polynomials which includes the orthogonal q-polynomials of Al-Salam and Carlitz. Explicit orthogonality relations are found for one new special case of this latter class.

Let α and β be any two real numbers and let be the Jacobi polynomial sequences. For any non-zero real number a, is an orthogonal polynomial sequence with respect to convolution if and only if either (i) b = 1, α = 0 and β + 1 is not equal to a negative integer or (ii) b = — 1, β = 0 and α + 1 is not equal to a negative integer.

On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials

In polynomial interpolation, the choice of the polynomial basis and the location of the interpolation points play an important role numerically, even more so in the multivariate case. We explore the concept of spherical orthogonality for multivariate polynomials in more detail on the disk. We focus on two items: on the one hand the construction of a fully orthogonal cartesian basis for the space of multivariate polynomials starting from this sequence of spherical orthogonal polynomials, and on the other hand the connection between these orthogonal polynomials and the Lebesgue constant in multivariate polynomial interpolation on the disk. We point out the many links of the two topics under discussion with the existing literature. The new results are illustrated with an example of polynomial interpolation and approximation on the unit disk. The numerical example is also compared with the popular radial basis function interpolation.