This book covers different aspects of umbral calculus and of its more recent developments. It discusses the technical details in depth, including its relevant applications. The book has therefore manyfold scopes to introduce a mathematical tool, not widespread known as it should be; to present a complete account of the relevant capabilities through the use of different examples of applications; to provide a formal bridge between different fields of research in pure and applied.

This is a book about Mathematics but not a book of Mathematics. It is an attempt, between the serious and facetious, of conveying the idea that a mathematical thought is the result of different experiences, geographical and social factors. Even though it is not clear when Mathematics had started, it is evident that it had been used at an early stage of human history and by ancient Babylonians and Egyptians who have already developed a sophisticated corpus of mathematical items, which were the workhorse tools in engineering, navigation, trades and astronomy. The book sweeps across the mathematical minds of the Greek and Arab traditions, concepts by Assyro-Babylonians, and ancient Indian Vedic culture. The mathematical mind has modeled the evolution of societies and has been modeled by it. It is now in the midst of a great revolution and it is not clear where it will bring us. The current new epoch needs new mathematical tools and, above this, a new way of looking at Mathematics. This book tells the tale of what went on and what might go on.

The book covers different aspects of mathematical methods for Physics. It is designed for graduate courses but a part of it can also be used by undergraduate students. The leitmotiv of the book is the search for a common mathematical framework for a wide class of apparently disparate physical phenomena. An important role, within this respect, is provided by a nonconventional formulation of special functions and polynomials. The proposed methods simplify the understanding of the relevant technicalities and yield a unifying view to their applications in Physics as well as other branches of science.The chapters are not organized through the mathematical study of specific problems in Physics, rather they are suggested by the formalism itself. For example, it is shown how the matrix formalism is useful to treat ray Optics, atomic systems evolution, QED, QCD and Feynman diagrams. The methods presented here are simple but rigorous. They allow a fairly substantive tool of analysis for a variety of topics and are useful for beginners as well as the more experienced researchers.

The book covers different aspects of mathematical methods for Physics. It is designed for graduate courses but a part of it can also be used by undergraduate students. The leitmotiv of the book is the search for a common mathematical framework for a wide class of apparently disparate physical phenomena. An important role, within this respect, is provided by a nonconventional formulation of special functions and polynomials. The proposed methods simplify the understanding of the relevant technicalities and yield a unifying view to their applications in Physics as well as other branches of science.The chapters are not organized through the mathematical study of specific problems in Physics, rather they are suggested by the formalism itself. For example, it is shown how the matrix formalism is useful to treat ray Optics, atomic systems evolution, QED, QCD and Feynman diagrams. The methods presented here are simple but rigorous. They allow a fairly substantive tool of analysis for a variety of topics and are useful for beginners as well as the more experienced researchers.

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