Operational techniques for Laguerre and Legendre polynomials

By starting from the concepts and the related formalism of the Monomiality Principle, we exploit methods of operational nature to describe different families of Laguerre polynomials, ordinary and generalized, and to introduce the Legendre polynomials through a special class of Laguerre polynomials themselves. Many of the identities presented, involving families of different polynomials were derived by using the structure of the operators who satisfy the rules of a Weyl group. In this paper, we first present the Laguerre and Legendre polynomials, and their generalizations, from an operational point of view, we discuss some operational identities and further we derive some interesting relations involving an exotic class of orthogonal polynomials in the description of Legendre polynomials.