# An elementary representation of the higher-order Jacobi-type differential equation

## TL;DR

This paper provides an elementary representation and new factorization of higher-order Jacobi-type differential equations, clarifying their orthogonality and enabling recurrence relations for the order of the equations.

## Contribution

It introduces a completely elementary form of the Jacobi-type differential operator of any even order and establishes a new factorization linking to recurrence relations.

## Key findings

- Elementary representation of the higher-order Jacobi-type differential operator.
- New factorization leading to recurrence relations.
- Connection between orthogonality and differential equations clarified.

## Abstract

We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval $[-1,1]$ with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order equations was introduced by J. and R. Koekoek in 1999 essentially by using special function methods. In this paper, a completely elementary representation of the Jacobi-type differential operator of any even order is given. This enables us to trace the orthogonality relation of the Jacobi-type polynomials back to their differential equation. Moreover, we establish a new factorization of the Jacobi-type operator which gives rise to a recurrence relation with respect to the order of the equation.