Related papers: Differential properties of Jacobi-Sobolev polynomi…
We prove that similarly to the standard case, the equilibrium measure of Julia sets of exceptional Jacobi polynomials tends to the equilibrium measure of the interval of orthogonality in weak-star sense.
We characterize the atomic probability measure on $\mathbb{R}^d$ which having a finite number of atoms. We further prove that the Jacobi sequences associated to the multiple Hermite (resp. Laguerre, resp. Jacobi) orthogonal polynomials are…
We study non-symmetric Jacobi polynomials of type $BC_1$ by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials allows us to introduce shift operators for the…
We study the relation between certain non-degenerate lower Hessenberg infinite matrices $\mathcal{G}$ and the existence of sequences of orthogonal polynomials with respect to Sobolev inner products. In other words, we extend the well-known…
We illustrate how Jordan algebras can provide a framework for the interpretation of certain classes of orthogonal polynomials. The big -1 Jacobi polynomials are eigenfunctions of a first order operator of Dunkl type. We consider an algebra…
Given a measure $\mu$ on the unit sphere $\partial\mathbb{B}^d$ in $\mathbb{C}^d$ with Lebesgue decomposition ${\rm d} \mu = w \, {\rm d} \sigma + {\rm d} \mu_s$, with respect to the rotation-invariant Lebesgue measure $\sigma$ on $\partial…
We study the bispectrality of Jacobi type polynomials, which are eigenfunctions of higher-order differential operators and can be defined by taking suitable linear combinations of a fixed number of consecutive Jacobi polynomials. Jacobi…
In this paper we derive generalizations of different properties of monic polynomial families of binomial type, i.e. families of monic polynomials, for which the binomial theorem holds $$ p_n(\alpha+\beta)=\sum_{k=0}^n…
The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's…
We study the analogue of polynomials (solutions to $\Delta^{n+1} u =0$ for some $n$) on the Sierpinski gasket ($SG$) with respect to a family of symmetric, self-similar Laplacians constructed by Fang, King, Lee, and Strichartz, extending…
We study the regularity of the roots of complex univariate polynomials whose coefficients depend smoothly on parameters. We show that any continuous choice of the roots of a $C^{n-1,1}$-curve of monic polynomials of degree $n$ is locally…
A two-variable generalization of the Big $-1$ Jacobi polynomials is introduced and characterized. These bivariate polynomials are constructed as a coupled product of two univariate Big $-1$ Jacobi polynomials. Their orthogonality measure is…
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…
Below the normalized weighted reciprocal of the Christoffel function with respect to exceptional Jacobi polynomials is investigated. It is proved that it tends to the equilibrium measure of the interval of orthogonality in weak-star sense.…
The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in $n+m$ variables, which reduce to the Jack polynomials when $n=0$ or $m=0$ and provide joint eigenfunctions of the quantum integrals of the…
Let $\mathcal{T}_{\mu}$ be the Dunkl operator. A pair of symmetric measures $(u, v)$ supported on a symmetric subset of the real line is said to be a symmetric Dunkl-coherent pair if the corresponding sequences of monic orthogonal…
We consider two sequences of orthogonal polynomials $(P_n)_{n\geq 0}$ and $(Q_n)_{n\geq 0}$ with respect regular functionals ${\bf u}$ and ${\bf v}$, respectively. We assume that $$\sum_{j=1} ^{M} a_{j,n}\mathrm{D}_x ^k P_{k+n-j}…
We identify a class of remarkable algebraic relations satisfied by the zeros of the Krall orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two. Given an orthogonal polynomial family…
We study the monic polynomials orthogonal with respect to a symmetric perturbed Gaussian weight $$ w(x;t):=\mathrm{e}^{-x^2}\left(1+t\: x^2\right)^\lambda,\qquad x\in \mathbb{R}, $$ where $t> 0,\;\lambda\in \mathbb{R}$. This weight is…
We study a family of symmetric polynomials that we refer to as the Boolean product polynomials. The motivation for studying these polynomials stems from the computation of the characteristic polynomial of the real matroid spanned by the…