  In the enduring, fruitful research on spectral differential equations with
polynomial eigenfunctions, Koornwinder's generalized Laguerre polynomials are
playing a prominent role. Being orthogonal on the positive half-line with
respect to the Laguerre weight and an additional point mass $N \ge 0$ at the
origin, these polynomials satisfy, for any $\alpha\in\mathbb{N}_{0}$, a linear
differential equation of order $2\alpha+4$. In the present paper we establish a
new elementary representation of the corresponding 'Laguerre-type' differential
operator and show its symmetry with respect to the underlying weighted scalar
product. Furthermore, we discuss various other representations of the operator,
mainly given in factorized form, and show their equivalence. Finally, by
applying a limiting process to the Laguerre-type equation, we deduce new
elementary representations for the higher-order differential equation satisfied
by the Bessel-type functions on the positive half-line.

Classical Analysis and ODEs