Tunisian Journal of Mathematics

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

Dominique Bakry and Lamine Mbarki

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Abstract

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.

Article information

Source
Tunisian J. Math., Volume 2, Number 3 (2020), 535-566.

Dates
Received: 12 December 2018
Revised: 1 May 2019
Accepted: 18 May 2019
First available in Project Euclid: 13 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.tunis/1576206289

Digital Object Identifier
doi:10.2140/tunis.2020.2.535

Mathematical Reviews number (MathSciNet)
MR4041283

Subjects
Primary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 43A62: Hypergroups
Secondary: 43A90: Spherical functions [See also 22E45, 22E46, 33C55] 46H99: None of the above, but in this section 60J99: None of the above, but in this section

Keywords
Markov sequences hypergroups orthogonal polynomials Dirichlet measures

Citation

Bakry, Dominique; Mbarki, Lamine. The Markov sequence problem for the Jacobi polynomials and on the simplex. Tunisian J. Math. 2 (2020), no. 3, 535--566. doi:10.2140/tunis.2020.2.535. https://projecteuclid.org/euclid.tunis/1576206289


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