Factorized A₂-Leonard pair

2023/12/13 by Nicolas Crampé, Crampe, Nicolas, Meri Zaimi +1 · 3 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2312.08312

openalex publication_date 2023/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The notion of factorized A₂-Leonard pair is introduced. It is defined as a rank 2 Leonard pair, with actions in certain bases corresponding to the root system of the Weyl group A₂, and with some additional properties. The functions arising as entries of transition matrices are bivariate orthogonal polynomials (of Tratnik type) with bispectral properties. Examples of factorized A₂-Leonard pairs are constructed using classical Leonard pairs associated to families of orthogonal polynomials of the (q-)Askey scheme. The most general examples are associated to an intricate product of univariate (q-)Hahn and dual (q-)Hahn polynomials.

Citations

• Hidden Symmetry of the Racah and Clebsch-Gordan Problems for the Quantum Algebra sl_q(2)
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials
• Multivariate P- and/or Q-polynomial association schemes
• λ-Griffiths polynomials: Bispectrality and biorthogonality

Cited by

• Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way
• The rank two Jacobi algebra
• Contiguity relations for finite families of orthogonal polynomials in the Askey scheme

Related

• Leonard pairs and the q-Racah polynomials
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials
• A way to treat dual Hahn polynomials as Racah polynomials via the theory of Leonard pairs
• Dual -1 Hahn polynomials: "classical" polynomials beyond the Leonard duality
• Leonard pairs and the Askey-Wilson relations