The symmetric form of the Koekoeksʹ Laguerre type differential equation

Koekoek and Koekoek (1991) proved that the Laguerre type polynomials {Lnα,N(x)}n=0∞, which are orthogonal on [0,∞) with respect to the measure dσα,N defined on [0,∞) by dσα,N(x) = (xαexΓ(α + 1) + Nδ(x))dx, satisfy a differential equation L2α+4,N[y][x] = λny(x) of order 2α + 4 when α is a nonnegative integer. In this paper, we announce that the differential expression L2α+4,N[·] is symmetrizable on (0, ∞) with symmetry factor xαe−x and explicitly give this formally symmetric expression xαe−xL2α+4,N[y].