# Bispectral Laguerre type polynomials

## TL;DR

This paper investigates Laguerre type polynomials, demonstrating their bispectrality and higher-order recurrence relations, and establishing the uniqueness of Krall-Laguerre polynomials as the only orthogonal families on the real line.

## Contribution

It proves that Laguerre type polynomials are bispectral with higher-order recurrence relations and characterizes Krall-Laguerre polynomials as the unique orthogonal families on the real line.

## Key findings

- Laguerre type polynomials satisfy higher-order recurrence relations.
- They are eigenfunctions of higher-order differential operators.
- Krall-Laguerre polynomials are uniquely orthogonal on the real line.

## Abstract

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