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The Umbral Calculus (Pure and Applied Mathematics): Steven Roman

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Have you come across the umbral calculus? The umbra (Latin: "shadow") is the darkest part of a shadow. But in mathematics? Perhaps a generating function is the shadow of its distribution? Or the term umbral calculus refers to surprising similarities between otherwise unrelated polynomial equations, and certain shadowy techniques used to prove them.

It is used in the study of orthogonal polynomials, in differential equations, in computations generally, and in the study of probability distributions. In fact it is an old (classical!) theory, but it undergoes periodic revivals.

An especially notable revival was pioneered by Gian-Carlo Rota in the 1970ties. He put the subject on a firm (rigorous!) foundation; with axioms, operators, diagrams, and duality arrows.

Previously it had been more like a bag of tricks involving formal power series and ad hoc computations with generating functions.

Motivated by Rota, Steven Roman came out with a different book: offering a wider view, a striking elegance, and including numerous new and intriguing applications. Roman's book was now reprinted in 2005 by Dover in this attractive little volume. Review by Palle Jorgensen, September 2007.

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