A description of the Laguerre-Hahn orthogonal polynomials of class zero revisited with two new families analogous to Bessel and new results for structure relations and differential equations

Abstract

This paper has a threefold aim. On the one hand, we provide a complete description of Laguerre-Hahn forms of class zero. This fills a gap in the literature: more precisely, up to an affine change of variables, there are ten families, including two new ones analogous to the Bessel classical family. On the other hand, we establish the structure relations for all these families, correcting those that have previously been reported in the literature. At last, as an application, using an algorithm previously obtained, with the aid of symbolic computations, we derive four new structure relations and a new fourth-order differential equation for one of the new families analogous to Bessel.