Continuous Sheffer families I

A convolution locally convex algebra U of holomorphic functions is introduced as a natural setting where to place special functions, which are continuously indexed counterparts to sequences of the classical orthogonal polynomials arising in the umbral calculus. In this way, such functions become semigroups, or Sheffer-type classes associated with semigroups, in the algebra U. A central role in this approach is to be played by the Gamma function. We also discuss the Hermite and Lerch functions to illustrate the theory.