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In the framework of the ‘monomiality principle’, we introduce a class of Bessel-type functions which can be derived by applying the properties of an isomorphism, related to the so called Laguerre-type exponentials. Further extensions and possible generalizations to the multivariable case are mentioned.

We discuss the theory of Lagrange polynomials and of associated generalized forms, using two different approaches based on the integral transform method and the Umbral Calculus. We will show that the proposed methods offer a powerful tool to investigate the properties of this family of polynomials.

We show that most of the properties of the monomial polynomials can be extended quite straightforwardly to the monumbral case. We show that the formalism underlying the Monumbrality principle allows the derivation of differential equations satisfied by large classes of polynomials ranging from Bernoulli to Laguerre type. We will finally prove the existence of generalized forms of Rodrigues formulae yielding the operational definition of differential types of conventional and generalized polynomials.

Second level exponential operators are exponential operators involving exponentials of exponential operators. We discuss the relevant operational rules and show that they can be exploited to derive new families of generating functions and to study the properties of a large family of Appell type polynomials.

We use the Borel theorem to find new families of generating functions, involving Hermite, Laguerre and Tchebycheff polynomials of one or more variables. We also use operational methods and integral representations to discuss the properties of the relativistic Laguerre and Bessel polynomials and to derive generating functions involving sums of the type