The higher-order differential operator for the generalized Jacobi polynomials - new representation and symmetry

For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by Koornwinder’s generalized Jacobi polynomials with four parameters $\alpha ,\beta \in {\mathbb{N}}_0$ and $M,N\ge 0$ determining the orthogonality measure on the interval $-1\le x\le 1$. The corresponding differential equation of order $2\alpha +2\beta +6$ is presented here as a linear combination of four elementary components which make the corresponding differential operator widely accessible for applications. In particular, we show that this operator is symmetric with respect to the underlying scalar product and thus verify the orthogonality of the eigenfunctions.