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Kerstin Jordaan

Publications and source records attributed to Kerstin Jordaan.

16 recordsLinked to original sources
arXiv2026

Two-point completion of zero interlacing and Wronskian-type bounds, with applications to Jacobi and other orthogonal polynomials

We investigate completed interlacing of zeros for pairs of polynomial sequences that fail to interlace by exactly two points. Using suitable mixed recurrence relations, we identify two additional points, called the completion points, required to obtain completed interlacing and establish general results for polynomials with real zeros. We also show that completed interlacing yields Wronskian-type bounds for the original polynomial pairs. The completed-interlacing results are applied to Jacobi, Meixner-Pollaczek, and Pseudo-Jacobi polynomials. In the Jacobi case, we improve earlier results by determining two explicit additional points that complete the interlacing of $P_n^{(α,β)}$ and $P_{n+1}^{(α+1,β+1)}$. We also analyze the locations of these points and show that the common-zero cases are non-generic in the Jacobi parameter domain. For this doubly-shifted Jacobi pair, the Wronskian-type bounds yield cross-family Turán-type inequalities and lead to an associated Stieltjes representation and complete-monotonicity consequences. For Meixner-Pollaczek polynomials, we address an open question concerning consecutive degrees with the parameter increased by one, and we obtain analogous completed interlacing results for Pseudo-Jacobi polynomials.

math.CA↗
arXiv2026

Characterization of classical orthogonal polynomials in two continuous variables

For a family of polynomials in two continuous variables, orthogonal with respect to a weight function, we prove, under suitable conditions, the equivalence of the following properties: the matrix Pearson equation of the weight, the second order linear partial differential equation, the orthogonality of the gradients, the matrix Rodrigues formula involving tensor products of matrices, and the so-called first structure relation. We then introduce a notion of classical orthogonal polynomials in two variables and relate the corresponding theory for weight functions and moment functionals. Finally, we present a nontrivial example that illustrates and delineates our contribution to the field.

math.CA↗
arXiv2026

Separating zeros of polynomials using an added interlacing point

Following a systematic analysis of existing results, we investigate when complete interlacing between the zeros of distinct polynomial sequences, $\{\mathcal{P}_n\}$ and $\{\mathcal{G}_n\}$ can be achieved by using a naturally arising extra point. Specifically, we analyse several general mixed recurrence relations that ensure the $n+1$ zeros of the polynomial $(x-E)\mathcal{P}_n(x)$ interlace with the $k$ zeros of $\mathcal{G}_k$, where $k=n$ or $n+1$. In addition, we show that imposing specific conditions on the extra point $E$ yields full interlacing between the zeros of $\mathcal{P}_n$ and $\mathcal{G}_k$ for a suitable choice of $n$. The approach provides a consolidated framework broadly applicable to both orthogonal and non-orthogonal polynomials and we illustrate this with new interlacing results for zeros of Krawtchouk, Meixner, and Narayana polynomials. We also illustrate that this general approach can be used to recover and refine existing results regarding the complete interlacing of zeros for classical Jacobi and Laguerre polynomials.

math.CA↗
arXiv2025

Symmetric Sextic Freud Weight

This paper investigates properties of the sequence of coefficients $(β_n)_{n\geq0}$ in the recurrence relation satisfied by the sequence of monic symmetric polynomials, orthogonal with respect to the symmetric sextic Freud weight $$ω(x;τ, t) = \exp(-x^6 +τx^4 + t x^2), \qquad x \in \mathbb{R},$$ with real parameters $τ$ and $t$. It is known that the recurrence coefficients $β_n$ satisfy a fourth-order nonlinear discrete equation, which is a special case of the second member of the discrete Painlevé I hierarchy, often known as the ''string equation''. The recurrence coefficients have been studied in the context of Hermitian one-matrix models and random symmetric matrix ensembles with researchers in the 1990s observing ''chaotic, pseudo-oscillatory'' behaviour. More recently, this ''chaotic phase'' was described as a dispersive shockwave in a hydrodynamic chain. Our emphasis is a comprehensive study of the behaviour of the recurrence coefficients as the parameters $τ$ and $t$ vary. Extensive computational analysis is carried out, using Maple, for critical parameter ranges, and graphical plots are presented to illustrate the behaviour of the recurrence coefficients as well as the complexity of the associated Volterra lattice hierarchy. The corresponding symmetric sextic Freud polynomials are shown to satisfy a second-order differential equation with rational coefficients. The moments of the weight are examined in detail, including their integral representations, differential equations, and recursive structure. Closed-form expressions for moments are obtained in several special cases in terms of generalised hypergeometric functions and modified Bessel functions. The results highlight the rich algebraic and analytic structures underlying the Freud weight and its connections to integrable systems.

nlin.SI↗
arXiv2024

Interlacing of zeros from different sequences of Meixner-Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials

In this paper we consider interlacing of the zeros of polynomials from different sequences $\{p_n\}$ and $\{g_n\}$. In our main result we consider a mixed recurrence equation necessary for existence of a linear term $(x-A)$ so that the $(n+1)$ zeros of $(x-A)g_n(x)$ interlace with the $n$ zeros of $p_n$. We apply our result to Meixner-Pollaczek, Pseudo-Jacobi and Continuous Hahn polynomials to obtain new interlacing results for the zeros of polynomials of the same degree from different polynomial sequences.

math.CA↗
arXiv2023

Generalised higher-order Freud weights

We discuss polynomials orthogonal with respect to a semi-classical generalised higher order Freud weight \[ω(x;t,λ)=|x|^{2λ+1}\exp\left(tx^2-x^{2m}\right),\qquad x\in\mathbb{R},\] with parameters $λ> -1$, $t\in\mathbb{R}$ and $m=2,3,\dots$\ . The sequence of generalised higher order Freud weights for $m=2,3,\dots$, forms a hierarchy of weights, with associated hierarchies for the first moment and the recurrence coefficient. We prove that the first moment can be written as a finite partition sum of generalised hypergeometric $_1F_m$ functions and show that the recurrence coefficients satisfy difference equations which are members of the first discrete Painlevé hierarchy. We analyse the asymptotic behaviour of the recurrence coefficients and the limiting distribution of the zeros as $n \to \infty$. We also investigate structure and other mixed recurrence relations satisfied by the polynomials and related properties.

math.CA↗
arXiv2022

Structure Relations of Classical Orthogonal Polynomials in the Quadratic and $q$-Quadratic Variable

We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$, the orthogonality of the second derivatives $\{\mathbb{D}_{x}^2P_n\}_{n= 2}^{\infty}$ and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tend to $\infty$. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the first structure relation.

math.CA↗
arXiv2021

Generalised Airy Polynomials

We consider properties of semi-classical orthogonal polynomials with respect to the generalised Airy weight \[ω(x;t,λ)=x^λ\exp\left(-\tfrac13x^3+tx\right),\qquad x\in \mathbb{R}^+,\] with parameters $λ>-1$ and $t\in \mathbb{R}$. We also investigate the zeros and recurrence coefficients of the polynomials. The generalised sextic Freud weight \[ω(x;t,λ)=|x|^{2λ+1}\exp\left(-x^6+tx^2\right), \qquad x\in \mathbb{R},\] arises from a symmetrisation of the generalised Airy weight and we study analogous properties of the polynomials orthogonal with respect to this weight.

math.CA↗
arXiv2020

A Generalised Sextic Freud Weight

We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalised sextic Freud weight \[ω(x;t,λ)=|x|^{2λ+1}\exp\left(-x^6+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $λ>-1$ and $t\in\mathbb{R}$. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalised hypergeometric functions ${}_1F_2(a_1;b_1,b_2;z)$. We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlighting a fascinating connection between generalised quartic, sextic, octic and decic Freud weights when expressing their first moments in terms of generalised hypergeometric functions.

nlin.SI↗
arXiv2018

A Characterization of Askey-Wilson polynomials

We show that the only monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$ that satisfy $$π(x)\mathcal{D}_{q}^2P_{n}(x)=\sum_{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\; x=\cosθ,\;~ a_{n,n-2}\neq 0,~ n=2,3,\dots,$$ where $π(x)$ is a polynomial of degree at most $4$ and $\mathcal{D}_{q}$ is the Askey-Wilson operator, are Askey-Wilson polynomials and their special or limiting cases. This completes and proves a conjecture by Ismail concerning a structure relation satisfied by Askey-Wilson polynomials. We use the structure relation to derive upper bounds for the smallest zero and lower bounds for the largest zero of Askey-Wilson polynomials and their special cases.

math.CA↗
arXiv2017

Properties of Generalized Freud Polynomials

We consider the semi-classical generalized Freud weight function \[w_λ(x;t) = |x|^{2λ+1}\exp(-x^4 +tx^2),\qquad x\in\mathbb{R},\] with $ λ>-1$ and $t\in\mathbb{R}$ parameters. We analyze the asymptotic behavior of the sequences of monic polynomials that are orthogonal with respect to $w_λ(x;t)$, as well as the asymptotic behavior of the recurrence coefficient, when the degree, or alternatively, the parameter $t$, tend to infinity. We also investigate existence and uniqueness of positive solutions of the nonlinear difference equation satisfied by the recurrence coefficients and prove properties of the zeros of the generalized Freud polynomials.

nlin.SI↗
arXiv2016

Zeros of Quasi-Orthogonal Jacobi Polynomials

We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by $α>-1$, $-2<β<-1$. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials $P_n^{(α, β)}$ and $P_{n}^{(α,β+2)}$ are interlacing, holds when the parameters $α$ and $β$ are in the range $α>-1$ and $-2<β<-1$. We prove that the zeros of $P_n^{(α, β)}$ and $P_{n+1}^{(α,β)}$ do not interlace for any $n\in\mathbb{N}$, $n\geq2$ and any fixed $α$, $β$ with $α>-1$, $-2<β<-1$. The interlacing of zeros of $P_n^{(α,β)}$ and $P_m^{(α,β+t)}$ for $m,n\in\mathbb{N}$ is discussed for $α$ and $β$ in this range, $t\geq 1$, and new upper and lower bounds are derived for the zero of $P_n^{(α,β)}$ that is less than $-1$.

math.CA↗
arXiv2015

A Generalized Freud Weight

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight \[w(x;t)=|x|^{2λ+1}\exp\left(-x^4+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $λ>-1$ and $t\in\mathbb{R}$, and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlevé equation. Further we derive a second-order linear ordinary differential equation and a differential-difference equation satisfied by the generalized Freud polynomials.

math.CA↗
arXiv2013

The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions which arise in the description of special function solutions of the fourth Painlevé equation.

nlin.SI↗
arXiv2009

Polynomial solutions of differential-difference equations

We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1 respectively. We address the question of when the zeros are real and simple and whether the zeros of polynomials of adjacent degree are interlacing. Our result holds for general classes of polynomials but includes sequences of classical orthogonal polynomials as well as Euler-Frobenius, Bell and other polynomials.

math.CA↗