Bispectral Jacobi type polynomials

TL;DR

This paper investigates the bispectral properties of Jacobi type polynomials, demonstrating their higher-order recurrence relations and characterizing the orthogonal cases as Krall-Jacobi families.

Contribution

It proves that Jacobi type polynomials are always bispectral with higher-order recurrence relations and uniquely orthogonal as Krall-Jacobi polynomials among them.

Findings

Jacobi type polynomials satisfy higher-order recurrence relations.

Krall-Jacobi polynomials are the only orthogonal Jacobi type polynomials.

All Jacobi type polynomials are bispectral.

Abstract

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.