An application in stochastics of the Laguerre-type polynomials

We explain how an inner product derived from a perturbation of a weight function by the addition of a delta distribution is used in the orthogonalization procedure of a sequence of martingales related to a Lévy process. The orthogonalization is done by isometry. The resulting set of pairwise strongly orthogonal martingales involved are used as integrators in the so-called (extended) chaotic representation property. As example, we analyse a Lévy process which is a combination of Brownian motion and the Gamma process and encounter the Laguerre-type polynomials introduced by Littlejohn.