An evolution of matrix-valued orthogonal polynomials

2024/11/27 by Erik Koelink, Koelink, Erik, Pablo Román +3 · 1 citation
Computer Science · #33C45 #33C47 #33E30 #33F10 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2411.18362

openalex publication_date 2024/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating functions, distribution of zeros for individual entries of the matrices and new type of differential-difference structure. We further speculate about other potentials of the connection formulas found. Part of our proofs makes use of creative telescoping in a matrix setting-the strategy which is not yet developed algorithmically.

Cited by

• Algebraic interpretation of discrete families of matrix valued orthogonal polynomials

Related

• Matrix-valued Gegenbauer polynomials
• Explicit construction of matrix-valued orthogonal polynomials of arbitrary size
• Matrix valued polynomials generated by the scalar-type Rodrigues' formulas
• Change of Basis between Classical Orthogonal Polynomials
• Recovering orthogonality from quasi-nature of Spectral transformations