The symmetric form of the Koekoeks' Laguerre type differential equation (Q1893573)

The classical Laguerre polynomials \(L^n_\alpha\) are orthogonal on \([0,+ \infty)\) with respect to \(x^\alpha e^{-x}\) and are eigenfunctions of the second-order differential operator \(L_\alpha [y]= -xy''+ (x-1- \alpha)y'\). On multiplying by \(x^\alpha e^{-x}\) this operator becomes formally symmetric. \textit{J. Koekoek} and \textit{R. Koekoek} [Proc. Am. Math. Soc. 112, No. 4, 1045-1054 (1991; Zbl 0737.33003)] proved that the Laguerre type polynomials \(L_n^{\alpha, N}\), which are orthogonal on \([0,+ \infty)\) with respect to \((x^\alpha e^{-x}/ \alpha!+ N\delta (x)) dx\) are the eigenfunctions of a differential operator \(L^{2 \alpha+4, N}\) of order \(2\alpha +4\), where \(\alpha\) is a nonnegative integer. The authors announce that this operator also becomes formally symmetric when multiplied by \(x^\alpha e^{-x}\). They give explicitly the formally symmetric differential expression of \(x^\alpha e^{-x} L_{2 \alpha+4, N}\) in the form \(\sum (-1)^n (b^n (x) y^{(n)} (x))^{(n)}\). A criterion for a general differential expression to have a symmetry factor is also stated. Proofs will appear in a forthcoming paper by the second and the third author.