Differential operators and the Laguerre type polynomials

In 1940, all fourth-order differential equations which have a sequence of orthogonal polynomial eigenfunctions were classified by H. L. Krall, up to a linear change of variable. One of these equations was subsequently named the Laguerre type equation and various properties of the orthogonal polynomial solutions and the right-definite boundary value problem were studied by A. M. Krall in 1981. In this paper, the Laguerre type expression is further studied in the right-definite setting and the appropriate left-definite problem associated with the fourth-order Laguerre type differential expression is discussed in detail.