Umbral Calculus A Different Mathematical Language

Abstract

The thesis is aimed at a thorough exposition of the Umbral Method, relevant in the theory of special functions, for the solution of ordinary and partial differential equations, including those of fractional nature. It will provide an account of the theory and applications of Operational Methods allowing the "translation" of the theory of special functions and polynomials into a "different" mathematical language. The language we are referring to is that of symbolic methods, largely based on a formalism of umbral type which provides a tremendous simplification of the derivation of the associated properties, with significant advantages from the computational point of view, either analytical or to derive efficient numerical methods to handle integrals, ordinary and partial differential equations, special functions and physical problems solutions. The strategy we will follow is that of establishing the rules to replace higher trascendental functions in terms of elementary functions, taking advantage from such a recasting. Albeit the point of view discussed here is not equivalent to that developed by Rota and coworkers, we emphasize that it deepens its root into the Heaviside operational calculus and into the methods introduced by the operationalists (Sylvester, Boole, Glaisher, Crofton and Blizard) of the second half of the XIX century. The method has opened new avenues to deal with rational, trascendental and higher order trascendental functions, by the use of the same operational forms. The technique had been formulated in general enough terms to be readily extended to the fractional calculus. The starting point of our theory is the use of the Borel transform methods to put the relevant mathematical foundation on rigorous grounds. Our target is the search for a common thread between special functions, the relevant integral representation, the differential equations they satisfy and their group theoretical interpretation, by embedding all the previously quoted features within the same umbral formalism. The procedure we envisage allows the straightforward derivation of (not previously known) integrals involving e.g. the combination of special functions or the Cauchy type partial differential equations (PDE) by means of new forms of solution of evolution operator, which are extended to fractional PDE. It is worth noting that our methods allow a new definition of fractional forms of Poisson distributions different from those given in processes involving fractional kinetics. A noticeable amount of work has been devoted to the rigorous definition of the evolution operator and in particular the problem of its hermiticity properties and more in general of its invertibility. Much effort is devoted to the fractional ordering problem, namely the use of non-commuting operators in fractional evolution equations and to time ordering. We underscore the versatility and the usefulness of the proposed procedure by presenting a large number of application of the method in different fields of Mathematics and Physics .