Monomiality Principle and Related Operational Techniques for Orthogonal Polynomials and Special Functions

The concepts and the related aspects of the monomiality principle are presented in this paper to explore different approaches for some classes of orthogonal polynomials. The associated operational calculus introduced by the monomiality principle allows us to reformulate the theory of Hermite, Laguerre and Legendre polynomials from a unified point of view. They are indeed shown to be particular cases of more general polynomials, whose usefulness in purely mathematical and applied context is discussed. The powerful tool represented by the Hermite and Laguerre polynomials allows us to derive classes of isospectral problems in applied mathematics and economics.