Matrix Gegenbauer Polynomials: the $2\times 2$ Fundamental Cases

Abstract

In this paper, we exhibit explicitly a sequence of $2\times 2$ matrix valued orthogonal polynomials with respect to a weight $W_{p,n}$, for any pair of real numbers $p$ and $n$ such that $0<p<n$. The entries of these polynomiales are expressed in terms of the Gegenbauer polynomials $C_k^{\lambda }$. Also the corresponding three-term recursion relations are given and we make some studies of the algebra of differential operators associated with the weight $W_{p,n}$.

References in corpus (2)

• Time and Band Limiting for Matrix Valued Functions, an Example
• The Algebra of Differential Operators for a Gegenbauer Weight Matrix

Cited by in corpus (12)

• Bispectrality and Time-Band-Limiting: Matrix valued polynomials
• Matrix-valued orthogonal polynomials related to the quantum analogue of $(SU(2)\times SU(2),\text{diag})$
• Time and band limiting for matrix valued functions: an integral and a commuting differential operator
• The Algebra of Differential Operators for a Gegenbauer Weight Matrix
• $2\times 2$ Hypergeometric operators with diagonal eigenvalues
• Singular examples of the Matrix Bochner Problem
• Darboux transformations and the algebra $\mathcal{D}(W)$
• Asymptotic Linear Programming Lower Bounds for the Energy of Minimizing Riesz and Gauss Configurations
• The algebra $\mathcal{D}(W)$ via strong Darboux transformations
• A $3\times 3$ singular solution to the Matrix Bochner Problem with $\mathcal{D}(W)$ not of the form $\mathbb{C}[D]$
• Structural formulas for matrix-valued orthogonal polynomials related to $2\times 2$ hypergeometric operators
• An evolution of matrix-valued orthogonal polynomials