Related papers: Matrix-valued Gegenbauer polynomials
In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2)\times SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The…
Matrix valued Laguerre polynomials are introduced via a matrix weight function involving several degrees of freedom using the matrix nature. Under suitable conditions on the parameters the matrix weight function satisfies matrix Pearson…
In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the…
We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $2\times 2$ introduced by C. Calder\'on et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric…
The matrix-valued spherical functions for the pair (K x K, K), K=SU(2), are studied. By restriction to the subgroup A the matrix-valued spherical functions are diagonal. For suitable set of representations we take these diagonals into a…
The properties of matrix valued polynomials generated by the scalar-type Rodrigues' formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence…
Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of $(SU(2) \times SU(2), \text{diag})$ are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised…
In this paper, we exhibit explicitly a sequence of $2\times2$ 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…
For a family of polynomials in two continuous variables, orthogonal with respect to a weight function, we prove, under suitable conditions, the equivalence of the following properties: the matrix Pearson equation of the weight, the second…
The space of polynomials in two real variables with values in a 2-dimensional irreducible module of a dihedral group is studied as a standard module for Dunkl operators. The one-parameter case is considered (omitting the two-parameter case…
We consider Koornwinder's method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder's construction…
In this paper, we explicitly provide expressions for a sequence of orthogonal polynomials associated with a weight matrix of size $N$ constructed from a collection of scalar weights $w_{1}, \ldots, w_{N}$: $$W(x) =…
The main purpose of this paper is to obtain an explicit expression of a family of matrix valued orthogonal polynomials {P_n}_n, with respect to a weight W, that are eigenfunctions of a second order differential operator D. The weight W and…
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…
In this paper we study matrix valued orthogonal polynomials of one variable associated with a compact connected Gelfand pair (G,K) of rank one, as a generalization of earlier work by Koornwinder and subsequently by Koelink, van Pruijssen…
In this work we classify all the order-two Hypergeometric operators $D$, symmetric with respect to some $2\times 2$ irreducible matrix-weight $W$ such that $DP_n=P_n\left(\begin{smallmatrix} \lambda_n&0\\0&\mu_n \end{smallmatrix} \right)$…
It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order…
The problem of finding weight matrices $W(x)$ of size $N \times N$ such that the associated sequence of matrix-valued orthogonal polynomials are eigenfunctions of a second-order matrix differential operator is known as the Matrix Bochner…
In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially…
We develop a unified construction of matrix-valued orthogonal polynomials associated with discrete weights, yielding bispectral sequences as eigenfunctions of second-order difference operators. This general framework extends the discrete…