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On inherited properties and Julia sets of exceptional Jacobi polynomials. (English) Zbl 1490.42028

The notion of exceptional orthogonal polynomials, motivated by the study of certain problems in quantum mechanics, was introduced in 2009 by several authors. Similar to the classical case, these polynomials are eigenfunctions of Sturm-Liouville-type differential operators but unlike the classical ones, the coefficients of these operators are rational functions.
Exceptional orthogonal polynomials constitute a complete system of polynomials, orthogonal with respect to a positive measure. As opposed to the classical case, these sequences start with a polynomial of degree one. Exceptional orthogonal polynomials also possess a Bochner-type characterization as each family can be derived from one of the classical families applying finitely many Darboux transformations.
Taking into account this characterization, along with the paper the author investigates some properties which are inherited to exceptional Jacobi polynomials from the classical ones. Besides new results, he gives some simple proofs to known ones in the exceptional orthogonal Jacobi case.
As an application of these properties, the author provides the weak-star limit of the equilibrium measure of Julia sets of exceptional Jacobi polynomials. In the standard case, the sequence of equilibrium measures of Julia sets of polynomials orthonormal to a (probability) measure supported on a compact set (of positive capacity) on the complex plane belongs to the family of measure-sequences which tends to the equilibrium measure of the set of orthogonality. The other members of this family are for instance, the normalized counting measure based on the zeros of the orthogonal polynomials in question, the weighted reciprocal of the Christoffel functions as the sequence of densities, the normalized counting measure based on the eigenvalues of the truncated multiplication operator, and the normalized counting measure based on the zeros of the average characteristic polynomials.
Similar theorems can be derived from exceptional Jacobi polynomials. The last results of the author fit into this family.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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