Bruna Germano

Touchard like polynomials and generalized Stirling numbers

Abstract The theory of Touchard polynomials is generalized using a method based on the definition of exponential operators, which extend the notion of the shift operator. The proposed technique, along with the use of the relevant operational formalism, allows the straightforward derivation of properties of this family of polynomials and their relationship to different forms of Stirling numbers.

Hermite polynomials with more than two variables and associated bi-orthogonal functions

We show that under appropriate conditions the Hermite polynomials, with more than two variables, belong to bi-orthogonal sets. We extend the result to the case of Bell-type polynomials.

A novel theory of Legendre polynomials

We reformulate the theory of Legendre polynomials using the method of integral transforms, which allow us to express them in terms of Hermite polynomials. We show that this allows a self consistent point of view to their relevant properties and the possibility of framing generalized forms like the Humbert polynomials within the same framework. The multi-index multi-variable case is touched on.

The negative derivative operator

Abstract We discuss the use of the negative derivative operator formalism to derive new series expansion for special functions and combinatorial identities.

Integrals of Bessel functions

Abstract We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique that we propose is based on the formal reduction of functions in this family to Gaussians.

Bell polynomials and generalized Blissard problems

We introduce two possible generalizations of the classical Blissard problem and we show how to solve them by using the second order and multi-dimensional Bell polynomials, whose most important properties are recalled.

Laguerre orthogonal functions from an operational point of view

We discuss the properties of Laguerre orthogonal functions by means of an operational point of view, which is shown to be a fairly powerful tool to investigate the relevant properties and further generalizations.

Negative derivatives and special functions

Abstract The formalism underlying the concept of negative derivatives is proved to be a powerful tool to study new series expansions of special functions defined by integral representations. Its importance is also discussed within the context of multiple integrations and of integral equations.

Legendre polynomials: Lie methods and monomiality

We combine the Lie algebraic methods and the technicalities associated with the monomialty principle to obtain new results concerning Legendre polynomial expansions.

Umbral Methods and Harmonic Numbers

The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.