Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1108.3531

Jordan algebras and orthogonal polynomials

Satoshi Tsujimoto·Luc Vinet·Alexei Zhedanov
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 that has this operator (up to constants) as one of its three generators and whose defining relations are given in terms of anticommutators. It is a special case of the Askey-Wilson algebra AW(3). We show how the structure and recurrence relations of the big -1 Jacobi polynomials are obtained from the representations of this algebra. We also present ladder operators for these polynomials and point out that the big -1 Jacobi polynomials satisfy the Hahn property with respect to a generalized Dunkl operator.

Explore related subjects

math.CAmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov. 2011-08-17. Jordan algebras and orthogonal polynomials. https://doi.org/10.1063/1.3653482

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Regular specular differentiation in Euclidean spaces

We study the regular specular derivative, a generalized derivative defined at every point where both one-sided derivatives exist and are finite. Geometrically, it is the slope of the mirror that reflects the left tangent ray into the right one. In one variable we derive computational formulas, prove inverse function and rotation rules, establish Quasi-Rolle's Theorem and the Quasi-Mean Value Theorem, and obtain a derivative-limit theorem, which shows that twice regularly specularly differentiable functions are continuously differentiable. We also prove both parts of the Fundamental Theorem of Calculus. In several variables we introduce specular gradients, directional derivatives, tangent hyperplanes, and normal vectors, show that a continuous specular gradient forces classical differentiability, and characterize when the specular tangent hyperplane is unique.

math.CA↗
arXiv2026

Prevalent smoothness in inhomogeneous Besov spaces

In this article, we prove that, under some assumptions on the so-called environment, prevalent functions in inhomogeneous Besov spaces recently introduced by Barral-Seuret in 2023 are multifractal, with a singularity spectrum that we determine. This completes the previous Baire generic results already obtained.

math.CA↗
arXiv2026

Lebesgue Covering Theorem and level sets of continuous functions

We formulate and prove a dimension-theoretic generalization of a version of the Lebesgue Covering Theorem. A generalized $n$-dimensional version of the Steinhaus Chessboard Theorem, recently proved algorithmically by Turzański and Ziajor, is a particular case of this result. Moreover, we study two types of sets associated with a continuous function $g \colon [0,1]^n \to \mathbb{R}$. Namely, the set of all points $p \in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ connects $i$th opposite faces of $[0,1]^n$, and the set of all points $p \in \mathbb{R}$ such that the fiber $g^{-1}[\left\{p\right\}]$ separates $i$th opposite faces of $[0, 1]^n$.

math.CA↗