Bispectral Jacobi type polynomials

We study the bispectrality of Jacobi type polynomials, which are eigenfunctions of higher-order differential operators and can be defined by taking suitable linear combinations of a fixed number of consecutive Jacobi polynomials. Jacobi type polynomials include, as particular cases, the Krall-Jacobi polynomials. As the main results we prove that the Jacobi type polynomials always satisfy higher-order recurrence relations (i.e., they are bispectral). We also prove that the Krall-Jacobi families are the only Jacobi type polynomials which are orthogonal with respect to a measure on the real line. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper studies the bispectrality of Jacobi type polynomials and proves that they satisfy higher-order recurrence relations. It also discovers that the Krall-Jacobi families are the only Jacobi type polynomials that are orthogonal with respect to a certain measure.