  We develop a new point of view to introduce families of functions, which can
be identified as generalization of the ordinary trigonometric or hyperbolic
functions. They are defined using a procedure based on umbral methods, inspired
to the Bessel Calculus of Bochner, Cholewinsky and Haimo. We propose further
extensions of the method and of the relevant concepts as well and obtain new
families of integral transform allowing the framing of the previous concepts
within the context of generalized Borel tranforms.
