Vol. 2, No. 3, 2020

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The Markov sequence problem for the Jacobi polynomials and on the simplex

Dominique Bakry and Lamine Mbarki

Vol. 2 (2020), No. 3, 535–566

The Markov sequence problem aims at the description of possible eigenvalues of symmetric Markov operators with some given orthonormal basis as eigenvector decomposition. A fundamental tool for their description is the hypergroup property. We first present the general Markov sequence problem and provide the classical examples, most of them associated with the classical families of orthogonal polynomials. We then concentrate on the hypergroup property, and provide a general method to obtain it, inspired by a fundamental work of Carlen, Geronimo and Loss. Using this technique and a few properties of diffusion operators having polynomial eigenvectors, we then provide a simplified proof of the hypergroup property for the Jacobi polynomials (Gasper’s theorem) on the unit interval. We finally investigate various generalizations of this property for the family of Dirichlet laws on the simplex.

Markov sequences, hypergroups, orthogonal polynomials, Dirichlet measures
Mathematical Subject Classification 2010
Primary: 33C45, 43A62
Secondary: 43A90, 46H99, 60J99
Received: 12 December 2018
Revised: 1 May 2019
Accepted: 18 May 2019
Published: 9 October 2019
Dominique Bakry
Institut de Mathématiques de Toulouse
Université Paul Sabatier
Lamine Mbarki
Département de mathématiques
Université des sciences El Manar