Umbral Calculus and New Trigonometries

Abstract

The term Umbra 1 has been diffused by S. Roman and G.C. Rota [1] in the second half of the 20th century to underline, in the emerging field of operational calculus, the practice of replacing the representation in series of certain function 𝑓𝑓𝑓𝑓(𝑥𝑥𝑥𝑥) with the formal exponential series 2 . Earlier, other mathematicians like J. Blissard in Theory of Generic Equations or Edouard Lucas in Théorie nouvelle des nombres de Bernoulli et d'Euler, established the rules which allowed for viewing different functions through the same abstract entity and showed its usefulness for a wide range of applications. In the current research, the same starting point has been used but with methods closer to the Heaviside point of view [2], by proposing strategies combining different technicalities and operational methods which yield the formulation of a powerful "different mathematical language" [3]. The umbral image, the key element to establish the rules to replace higher transcendental functions in terms of elementary functions, and rewrite complex problems into simplified exercises, will be the starting point to fix the criteria to take advantage from such a replacement, based on the Laplace and Borel transform Theory and the Principle of Permanence of the Formal Properties.To introduce the umbral operator, we have to provide some preliminary definitions. The operator 𝑐𝑐𝑐𝑐, called umbral, is a shift operator 𝑐𝑐𝑐𝑐̂= 𝑒𝑒𝑒𝑒 𝜕𝜕𝜕𝜕 𝑧𝑧𝑧𝑧 acting on an appropriate chosen vacuum 3 , which is a function that will change according to the problem to solve [3]. If the initial vacuum is, for example, the function:1 It is assonant to the term "Ombra" in Latin which means "shadow" in English.