Making Multiplicative Calculus Operational: A Tabular Framework for Infinite Product Identities

In 1985, the author discovered and developed a new differential operator, here named the Volonterio Operator, defined as Vg=exp(g′/g). This operator measures multiplicative growth and exhibits a rich multiplicative algebraic structure. The present work, based on original manuscripts dated 7 April 1985, provides the first complete formalization of this operator, including its definition via a limit, fundamental properties, an explicit inverse operator, systematic tables of transformations, and applications to functions defined by infinite products. To the best of our knowledge, this constitutes the first systematic treatment of such an operator as an autonomous calculus.