Umbralized Umbral Operators
In this paper we shall present a theory which arises from the "umbralization" of umbra I operators. The English translation of the Latin word "umbra" is "shadow". Let {an}n;;.O denote a sequence of numbers or polynomials. Umbral operations on the given sequence {an}n;;.O were originally understood as formal manipulations in which, first, one "pretends" that the sequence in question is the power sequence {a n}n;;.O, then one performs the desired operations, and finally one lowers all the exponents to their "shadows" (or subscripts), thereby obtaining valid identities for the sequence {an}n;;.O. One such umbral operation is the umbral composition of polynomial sequences. Let {an (x )}n;;.O and {bn (x )}n;;.O denote sequences of polynomials, and suppose