A Note on Hermite-Bernoulli Polynomials

Using the concepts and formalism of different families of Hermite polynomials, we discuss here some generalizations of polynomials belonging to the Bernoulli class, and we also show how to represent the action of the operators involving fractional derivatives. In particular, by using the method of generating function, we introduce generalized Bernoulli polynomials by operating in their generating function with the formalism of the two-variable Hermite polynomials. In addition, we extend some operational techniques in order to derive different forms of Bernoulli numbers and polynomials. Finally, we explore some general properties of generalized Bernoulli polynomials, focusing on their extension to the 2D case, and we introduce a family of polynomials strictly related to the Hermite polynomials in order to compute the effect of fractional operators on a given function.