Matrix valued orthogonal polynomials of Jacobi type: the role of group representation theory

Grünbaum, F. Alberto; Pacharoni, Inés; Alfredo Tirao, Juan

Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2051-2068.

Le résultat principal du présent article est la construction d’une nouvelle famille de polynômes orthogonaux matriciels, du type de Jacobi. Ces polynômes proviennent du groupe sous-jacent $SU\left(n\right)$ et de ses représentations : ce sont des fonctions propres d’un opérateur différentiel symétrique du second ordre, hypergéométrique et à valeurs matricielles. Le résultat final est valable pour des valeurs arbitraires des paramètres $\alpha ,\beta >-1$, mais est dérivé uniquement pour les valeurs provenant de la théorie des groupes.

The main purpose of this paper is to present new families of Jacobi type matrix valued orthogonal polynomials obtained from the underlying group $SU\left(n\right)$ and its representations. These polynomials are eigenfunctions of some symmetric second order hypergeometric differential operator with matrix coefficients. The final result holds for arbitrary values of the parameters $\alpha ,\beta >-1$, but it is derived only for those values that come from the group theoretical setup.

Classification : 33C45, 22E45
Keywords: Matrix valued orthogonal polynomials, Jacobi polynomials
Mot clés : polynômes orthogonaux matriciels, polynômes de Jacobi

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